Physics 21st Century A Compilation of Contemporary and Emerging Technologies
Physics 21st Century A Compilation of Contemporary and Emerging Technologies
Physics 21st Century A Compilation of Contemporary and Emerging Technologies
Batterson
Caton
Davidson
Fetsko
Jackson
Kar
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
CK-12 Foundation is a non-prot organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. Copyright 2011 CK-12 Foundation, www.ck12.org The names CK-12 and CK12 and associated logos and the terms FlexBook, and FlexBook Platform, (collectively CK-12 Marks) are trademarks and service marks of CK-12 Foundation and are protected by federal, state and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non-Commercial/Share Alike 3.0 Unported (CC-by-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the CC License), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: August 9, 2011
Authors
James Batterson, Randall Caton, Bruce Davidson, Michael Fetsko, Andrew Jackson, Tapas Kar, John Ochab, David Slykhuis, David P. Stern, Tony Wayne
Contributor
James Batterson
Editors
Christopher E. Giersch, John H. Stadler
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Contents
Foreword Foreword Foreword Foreword Foreword 1 VA Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Background and Overview of Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pilot FlexBook Outcomes Expanded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Organization of this FlexBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FlexBook Chapter Synopses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi vii viii ix x 1 1 2 3 3 4 4 5
2 Toward Understanding Gravitation. 2.1 2.2 2.3 PrefaceA Note to the Teacher and Student Regarding Background Information and Pedagogy Toward an Understanding of Gravitation (With a Few Interesting Side Trips) . . . . . . . . Virginia Physics Standards of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 8 10 28 30 30 30 30 34 34
3 Nuclear Energy 3.1 3.2 3.3 3.4 3.5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Nuclear Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Foundations: Atoms and Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Nuclear Binding Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fission of Heavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 38 39 41 44 45 46 48 48 49 50 50
3.10 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Review Answers 3.12 Controlling the Nuclear Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Final Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 References / Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Virginia Physics Standards of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Standard Model of Particle Physics 4.1 4.2 4.3 4.4 Visual Overview for The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . In the Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References / Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virginia Physics Standards of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51 51 51 76 76 78 78 78 79 80 82 85 86 89 93 93 95
5 The Standard Model and Beyond 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminology and Some Background Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . What is a Collider? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Hadron Collider, LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LHC Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is Mass? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Super Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dark Matter and Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 References / Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Virginia Physics Standards of Learning 6 A Brief Synopsis of Modern Physics 6.1 6.2 6.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 1: What is Modern Physics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97 97 98
Section 2: What Parts of Modern Physics are Still Being Researched? . . . . . . . . . . . . 109
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Section 3: What are the Implications of Some of Modern Physics (Including String Theory, Nanoscience, Dark Matter, Black Holes, Parallel Universes, and The Graviton)? . . . . . . . 110 References / Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Virginia Physics Standards of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 115
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 References / Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Virginia Physics Standards of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9 Kinematics: Motion, Work, and Energy 9.1 9.2 9.3 9.4 9.5
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Linear Motion and How to Describe It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Energy and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Virginia Physics Standards of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10 Laboratory Activities 10.2 Virginia Physics Standards of Learning 11 Statistical Physics and Random Walks
191
11.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 11.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 12 Modeling and Simulation in the Physics Classroom 12.1 Introduction 12.2 Squeak 12.3 STELLA
206
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
12.4 Virginia Physics Standards of Learning 13 Modeling and Simulating NASAs Launch Abort System
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13.2 Describing One-Dimensional Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 13.3 Force: The Cause of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 13.4 Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 13.5 Contact Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 13.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
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Foreword
The 21st Century Physics FlexBook is a collaborative eort of the Secretaries of Education and Technology and the Department of Education that seeks to elevate the quality of physics instruction across the Commonwealth of Virginia.
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Foreword
The 21st Century Physics FlexBook is a collaborative eort of the Secretaries of Education and Technology and the Department of Education that seeks to elevate the quality of physics instruction across the Commonwealth of Virginia.
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Foreword
The 21st Century Physics FlexBook is a collaborative eort of the Secretaries of Education and Technology and the Department of Education that seeks to elevate the quality of physics instruction across the Commonwealth of Virginia.
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Foreword
The 21st Century Physics FlexBook is a collaborative eort of the Secretaries of Education and Technology and the Department of Education that seeks to elevate the quality of physics instruction across the Commonwealth of Virginia.
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Foreword
The 21st Century Physics FlexBook is a collaborative eort of the Secretaries of Education and Technology and the Department of Education that seeks to elevate the quality of physics instruction across the Commonwealth of Virginia.
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Chapter 1 VA Introduction
1.1 Background and Overview of Goals
Welcome to Virginias 21st Century Physics FlexBook: A Compilation of Contemporary and Emerging Technologies, a result of Virginias FlexBook Pilot Project. This project was motivated by the conuence of two independent desires and capabilities: The recommendations of a 2007 Standards of Learning (SOL) review panel of practicing scientists and engineers that VA SOL should include contemporary and emerging science content as well as laboratory activities that incorporate industry state-of-the-practice equipment; and that Virginia should support an open-source software platform, such as a Wiki, for the timely publication of teacher-developed curriculum. The mission of the CK-12 Foundation to provide a collaborative online authoring environment that enables the production of free and open content aligned to curriculum standards and customizable for each student. This particular pilot FlexBook aims at several outcomes: Supplementing currently used Virginia physics textbooks by making valuable contemporary and emerging physics ideas available to all teachers at a single URL. Making laboratory activities that employ industry state-of-the-practice equipment available to all teachers. Providing a path for continuous improvement from teachers themselves through comments and new ideas after using a chapter with their physics classes. This pilot FlexBook project seeks many other outcomes: Can working teachers provide useful contemporary, emerging, and laboratory curriculum content in addition to their normal teaching duties? What intellectual property (IP) issues may be barriers to or facilitators of open-source content?
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Is the CK-12 FlexBook a good open-content platform for Virginias purposes? What additional features would make the CK-12 FlexBook even more useful to Virginia? What quality assurance process is required to make appropriate content available to all teachers and students? Is a book of many chapters by many authors in many voices readable and comprehensible by most students? Does this FlexBook provide valuable contemporary and emerging physics content that supplements current physics SOL? Is the content readily available to ALL of Virginias physics teachers at a single Web-based source? Can we provide timely and valuable feedback to CK-12 that will help them continually improve their FlexBook system for teachers use? Can we provide suggestions from Virginias teachers and students to CK-12 regarding Web 2.0 needs? Can we supply Virginias education policy-makers with concrete examples of the 2007 physics panels recommendations to help inform their 2010 review of Virginia physics SOL? Does this project give us a sense of the qualitative value of e-formats replacing some textbook purchases? Can we determine whether to extend this type of project to the instruction side of the DOE and to other disciplines?
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The FlexBook laboratory chapters are addressed to three audiences: Teachers who have little or no experience with labs Teachers who teach labs but may be using obsolete equipment and technology Teachers who would like to use the FlexBook labs as a jumping-o point in developing their own labs For the rst group of teachers, some of whom have limited experience and prociency in lab science in general or physics labs in particular, the FlexBook write-ups should provide equipment lists and cookbook instruction. This will at least provide for some hands-on work with state-of-the-practice technology. The second group of teachers will be introduced to new equipment manufacturers and taught how to incorporate state-of-the-practice technology into engaging physics laboratories. The third group of teachers may nd some of the equipment and its capabilities to be new and can use this information to develop their own labs with more advanced technology.
A technical review by a university research physicist Peer review by three other authors Review by several students including three 10th grade high school students and a college freshman (non-science major)
Version 1.0: All chapters in Release 1.0 underwent one additional level of review via the public feedback we received from our open mailing list. All content is conguration controlled. While it can be copied and edited by users on the CK-12 FlexBook Platform http://flexbooks.ck12.org/flexr/book/vaflexbook the original FlexBook content cannot be changed by readers. The chapters will be updated from time to time based on the authors experiences and comments from readers and users. These updates will be noted by their release numbers.
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Biophysics (Medical Imaging) by David Slykhuis, James Madison University; Mark Mattson, James Madison University; and Tom ONeill, Shenandoah Valley Governors School. Today we have access to incredibly advanced non-invasive imaging technology for the analysis of our health. However, to most students, methods such as xrays, MRI, and ultrasound are just black boxes that give the doctor a magic result. This chapter addresses these three major medical imaging technologies and their foundations in physics. Ultrasound is available in the rst FlexBook release (v1.0), followed by sections on MRI and xray in later releases. Kinematics by John Ochab, J. Sargeant Reynolds Community College. Understanding how things move is fundamental to our understanding of the physical universe. Critical to this understanding is the ability to portray motion in a manner that is clear, accurate, precise, eicient, and reproducible. In the rst part of the chapter, Motion and How to Describe It, we identify the terms used to characterize motion and illustrate the graphical methods used to represent motion visually. In the second part of the chapter, we study the work done by one or more forces on one or more bodies, determine the types of energy involved, and draw connections between the work done on the bodies and the energy changes in the bodies. Information is presented in tutorial format and includes an introduction to using motion sensors with a computer. Laboratory Activities by Bruce Davidson, Newport News City Schools. This chapter presents 15 physics experiments that utilize 21st century technology to conduct investigations that can be used in the high school classroom. The PASCO Xplorer GLX handheld interface is highlighted with downloadable labs on linear motion, Newtons laws of motion, friction, momentum, conservation of energy, kinetic energy, energy transfer, and sound waves. Modeling and Simulation by Mark Clemente, Virginia Beach City Schools/National Institute of Aerospace. Modeling and simulation have been used for design, test, evaluation, and training in the industry for several decades. With the advances in technology and computer capabilities in recent years, modeling and simulation are now tools for instruction that are accessible to most classroom teachers. This chapter presents several examples of how physics content can be taught using modeling and simulation. Modeling and Simulating NASAs Launch Abort System by Randall Caton, Bigfork, Minnesota. Complex systems abound in our world and it is valuable to model and simulate them to better understand how they work and improve their design. Student learners will modify a model based on Newtons Laws and simplifying assumptions that can be applied in a computer environment (Etoys) to simulate the motion of NASAs Launch Abort System. The concepts of position, velocity, acceleration, force, and mass are introduced in the context of Newtons Laws. Students will learn by doing by starting with a simple model using constant acceleration and modify the model to simulate air drag, the varying force of gravity, the real rocket, the 2 dimensional case and a two-stage rocket.
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modeling and simulation demonstration school project. The purpose of this project is to use modeling and simulation as an instructional strategy within mathematics and science instruction and to demonstrate ways to integrate mathematics and science instruction through the use of models and simulations. Bruce Davidson: Author, Newport News, Virginia Bruce Davidson has an MS in physical science education from Old Dominion University. A retired physics and biology teacher, he is currently working part-time for Newport News Public Schools in Newport News, Virginia. He currently works with new as well as experienced science teachers integrating technology and the hands-on experience into classroom instruction. He also provides professional development to science teachers using handheld data collectors to enhance students experimental experience. Outside of the classroom you will nd him kayaking, biking and hiking. He currently lives with his wife and son (17 years) in Newport News, VA. Michael Fetsko: Author, Henrico County Schools, Virginia Mike Fetsko is currently a physics teacher at Godwin High School in Richmond, Virginia. He received his BS in multiple science from LeMoyne College and an MST in physics from the State University of New York at Plattsburgh. He has been teaching all levels of high school physics since 1993 and he is always looking at ways to incorporate innovative ideas and content into his curriculum. Andrew Jackson: Author, Harrisonburg City Schools, Virginia Andy Jackson teaches physics and astronomy at Harrisonburg High School in Harrisonburg, Virginia. He teaches half-time, is the K-12 science coordinator for Harrisonburg City Public Schools, and part-time physics lab instructor at James Madison University. Andy received his BS in physics from JMU in 1987 and has been teaching various levels of physics since. Andy has been an active member of the Virginia Instructors of Physics since its inception and served as president from 19982006. He is a life member of the Virginia Association of Science Teachers (VAST) and has served VAST as Physics Chair, PDI Chair, and was President of VAST in 2008. Tapas Kar, PhD: Author, Utah State University, Utah Tapas Kar is an Assistant Professor with the Department of Chemistry and Biochemistry at Utah State University (USU). Prior to working at USU he taught and did research at Southern Illinois University Carbondale (SIUC). Tapas focuses his research and teaching in the area of nanoscience and nanotechnolgy. He introduced nanotechnology courses at USU and currently teaches nanochemistry courses. John S. Ochab, Jr., PhD: Author, J. Sargeant Reynolds Community College, Virginia John Ochab was born in a suburb of Boston, MA. He attended the University of Massachusetts (at Boston) and obtained a BA in Biology. He worked as a biochemiocal laboratory technician for 3 years (with journal aknowledgements) and as a toxicologist for one year. He then decided to go into physics. After taking courses in advanced mathematics and physics (at M.I.T. and at Boston University), he enetered graduate school at Clark university, (Worcester, MA) where he obtained an MA in physics (nuclear solid state). He then entered the University of Maine (at Orono) were he obtained a PhD in experimental surface physics. Upon graduation, he worked in the industry for such companies as Spectra Physics, GTE Sylvania, as well as smaller companies. He also did research in high temperature superconducting thin lms at Brookhaven National Laboratory in Long Island, NY. Due to the nancial crises of the late 1980s, he moved to California, where he trained process engineers in semiconductor metrology and taught physics part-time at local community colleges. John then moved to West Virginia and taught physics, physical and engineering physics, and after getting married, moved with his wife to Virginia. He has been teaching algebra and calculus-based physics at J. Sargeant Reynolds Community college ever since. He has rst-author publications in Journal of Surface Science, and coauthored publications in the Physical Review Letters, Journal of Applied Physics, and Physicsa C. He is a member of the American Association of Physics Teachers, the Virginia Academy of Science, and was a www.ck12.org
long-standing member of the American Institute of Physics. Dr. David A. Slykhuis: Author, James Madison University, Virginia Dr. David Slykhuis is Chair of the Physics/Physical Science Academy. Dr. Slykhuis has been at James Madison University since the fall of 2004. His primary responsibilities lie in the preparation of science teachers in the middle and secondary education program. His research interest involves the use of technology in K-16 science classrooms to increase student achievement. Dr. Slykhuis received his PhD in science education from North Carolina State University in May of 2004. He has ve years of high school classroom experience, teaching primarily chemistry and physics. David P. Stern: Author, Greenbelt, Maryland Dr. Stern received his MS in physics from the Hebrew University in Jerusalem, his doctorate from the Israel Institute of Technology, and retired after 40 years of research with NASA Goddard SFC on the Earths magnetosphere. He has produced extensive education resources on the Web, including From Stargazers to Starships. He has also written space-related history, poems and a middle-school mathematics enrichment text, Math Squared. Randall Caton: Author, Bigfork, Minnesota Randy Caton was born in Minnesota and went to the University of Minnesota, the University of Pennsylvania, and the City University of New York, where he received his doctorate in Physics. He has worked in experimental solid-state physics in the areas of electrical properties of solids, heat capacity, low temperature physics, dilute magnetic alloys, superconductive materials, rare-earth alloys, and metallic glasses. He has taught introductory and advanced physics courses and laboratories to classes ranging from 5 to 700 students for 30 years and has incorporated Peer Instruction and Just-In-Time-Teaching and other learning tools. He has directed several science education programs for teachers and students from 1986 to 2008. He is currently retired and lives in northern Minnesota. He has used Etoys (a free, open-source multimedia authoring environment) to develop web-based activities for NASA programs, physics courses and the chapter in this online book. Jim Batterson: Project Manager, Newport News, Virginia Jim Batterson taught high school physics and mathematics, worked as a scientic programmer for LTV Corporation, and, from 1980 until his retirement in 2008, was a research engineer at NASA Langley Research Center. At NASA he was responsible for ight research on the dynamics and control of aerospace vehicles, served as Head of the Dynamics and Control Branch, and later as Deputy Director for Strategic Development. He has also served on a number of community boards including the Newport News (Virginia) School Board and New Horizons Regional Education Center Board. While at NASA, he served on assignments to the Oice of Science and Technology Policy, the National Nanotechnology Coordination Oice, NASA Headquarters, and, most recently, to the Oice of Virginias Secretary of Education.
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2.1 PrefaceA Note to the Teacher and Student Regarding Background Information and Pedagogy
Nearly every physics textbook has an adequate section regarding Newtons universal gravitation, Cavendishs work and an introduction of Keplers laws of planetary motion. Therefore, in this work I will not attempt to teach those topics, but will assume that students have a basic understanding of the physics involved as it pertains to an understanding of gravity. Many textbooks do not contain a treatment on current understanding and development of the ideas regarding gravitation. Those that do often place this material as footnotes to a chapter or as chapters late in the text that a typical class may never cover. This chapter of the 21 st Century Physics FlexBook will attempt to address our changing understanding of gravitation and in doing so also introduce the student to a few interesting areas of astronomy and cosmology. This chapter should be an appropriate extension to a study of Newtons universal law of gravitation. The presentation deals with gravitation from a purely conceptual approach. The appropriate high school level mathematical treatment would pertain to Newtons universal law of gravitation and it is assumed that the students will study from traditional text or with their teachers. The chapter is set up in dialogue style. This technique has a wonderful heritage in physics going back to Galileos Dialogue Concerning the Two Chief World Systems published in 1632. Bold Print statements represent questions asked by a student with the appropriate answers following. It is my practice and suggestion that a treatment of universal gravitation in a high school physics class be approached in a historical manner starting with Aristotle and extending to as near the present understanding as possible.
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Figure 2.1: Aristotle Contemplating the Bust of Homer - painted by Rembrandt in 1653 at a time when scientists understood the planets to orbit the sun, but had no concept that a force called caused their motion and were only beginning to abandon Aristotelian beliefs of motion.
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1. All planets move in elliptical orbits with the Sun at one focus. 2. A line connecting a planet to the Sun sweeps out equal areas of space in equal amounts of time. 3. The period of a planets orbit squared is directly proportional to the cube of its orbital radius. It is so easy to state and learn these laws that it may lead you to think they were easy to gure out. If you would like to gain a little understanding of Keplers accomplishments you should look over the details of how he came to these three conclusions at http://www-groups.dcs.st-and.ac.uk/~history/Extras/ Keplers_laws.html
Figure 2.2: Johannes Kepler as painted in 1610 by an unknown artist. He would soon hear of a revolutionary new tool for astronomythe telescope. It is also very useful for you to have a good understanding of these laws and the nature of ellipses, so here is a little project for you to do.
Elliptical Homework
Get a scrap piece of cardboard, two push-pins, a loop of string, and a pencil. Push the two pins into the cardboard. Place the loop of string on the cardboard with the two pins in the loop. Use a pencil to pull the loop away from the pens to make the loop tight against the pencil and the two pins. Move the pencil around in a circle (its an ellipse) keeping the loop tight as you draw. The shape you have is an ellipse. The two pin holes are the two foci. Mark one as Sun. The drawn curve is the path of a planet around the Sun. See if you can sketch in the idea of Keplers Second Law of Planetary Motion. The eccentricity of an elliptical orbit is found by measuring the two distances shown and dividing the dierence between them by the larger. Therefore, the eccentricity of a circle is equal to zero.
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Figure 2.4: This ellipse has an eccentricity of . The sun would be at one focus.
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The eccentricity for Mars is about 0.09, which is much larger than Earths. What does that tell you about how elliptical the orbits of the planets are? Can you use a string, pin, and pencil to create an ellipse with e = 0.09? If you make the loop of string a bit shorter and draw another ellipse it will represent the path of another planet. See if you can apply an understanding of Keplers third law of planetary motion to the two ellipses. This would be an excellent thing to talk through with another student or teacher once you have given it some thought on your own. OKI drew a couple of ellipses and I think I understand Keplers laws of planetary motion. But if they are laws, he got it all gured out, right? It is so important to understand the scientic meaning of the words: law, theory, and hypothesis. Before we go on with more physics about gravity, lets take an important aside.
An Important Aside
What is a scientic law? How does it dier from a hypothesis or a theory? How does a theory become a law? These are all great questions that you really need to be able to answer. The earlier in your science studies you understand these dierences and relationships the better. A scientic hypothesis is not just a best guess. Its an idea of how something works or an explanation based on the evidence available. It is a statement limited to a specic situation and must be testable. In other words, it should be something that could be proven wrong. A scientic law is a statement of fact that is believed to be always true, but oers no explanation. The law of inertia is a wonderful example. It is understood that objects at rest will stay at rest unless a force causes them to move. Scientists do not have an explanation for WHY objects cannot begin moving from a state of rest without a force acting on them, but such a thing has never been observed and we believe it to be universally true. Keplers laws of planetary motion t the description of scientic laws well when they were initially stated. In the early 1600s we did not understand that the Sun and planets were exerting forces on each other through gravitation. Kepler put together decades of data and found that for the six known planets, all of them behaved as described by his three statements. His laws oer no explanation for WHY the planets behave this way, thus they are planetary laws. Newtons universal law of gravitation ts this description as well. It does not tell us HOW two dierent masses exert forces on each other, it simply describes it and names it. The question How does a theory become a law? is a trick question. The answer isit cannot! Scientic theories EXPLAIN things. A theory in science provides a big picture understanding and view that helps to explain many dierent phenomena. For example, the atomic theory says that matter is made of discrete units of matter that maintain their identity through physical and chemical change. This atomic theory is very useful in understanding chemical reactions and much more. Therefore, in science, the theory of evolution is not less certain than the law of universal gravitation. They do very dierent jobs. The theory of evolution EXPLAINS HOW speciation occurs through natural selection and Newtons law of universal gravitation states what we observe without explanation. We are still in search of a THEORY of gravitation. There are a few promising hypotheses, however. Theories and laws. Ill try to remember the dierence. What about this universal law of gravitation? I will leave the majority of the teaching of Newtons universal law of gravitation to your traditional textbook or Internet sources. Go read up on it and do a few problems and come back. One place you can do this is the Physics Classroom at http://www.physicsclassroom.com/Class/circles/u6l3a.cfm OK. I solved some problems and Im back. Seems like Newton got it all gured out. In Newtons life (1643-1727) he came to understand that all masses attracted each other with a force
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Figure 2.5: Sir Isaac Newton (16431727) as painted in 1689 by Godfrey Kneller. that was directly proportional to the product of the masses and inversely proportional to the distance between them squared. BUT, neither he nor his contemporaries were able to turn this proportionality into an equality. It is not terribly diicult to think up an experiment to try to measure the constant of proportionality in this equation F = GMm where G is some constant that turns the mathematics from a d2 proportionality to an equation. With equations you can solve problems. First, you might think of taking two objects of mass M and m and placing them d apart. Now all you need to do is measure the force of attraction and solve for G. While this is simple to think of, it is far beyond the ability of simple force scales to measure the incredibly small force of attraction between the two masses, even if the masses are huge. The best scales of Newtons era were not up to the task. Another simple experiment you may think of is to take a known mass m and nd out how much it weighs. This would be the force F of attraction to the mass of the Earth when separated by a distance equal to the Earths radius. During Newtons time the radius of the Earth was well known, but the value of its mass was not known. One equation with two unknowns, the mass of the Earth and the value of G, makes for an unsolvable problem. Newton died with two major aspects of universal gravitation left unexplained: the value of the universal gravitation constant G, and an explanation for HOW gravity reached out through space and exerted a force. After all, if you want to exert a force on a friend you have to physically touch him or throw something at him. For example, it wasnt obvious that the Earth and the Moon were doing either to each other. So how were the Earth and the Moon pulling on each other with gravity? My physics textbook has a value for G, so somebody gured that part out. Cavendish, right? Yes, thats correct. By the end of the 18th century, Henry Cavendish utilized a sophisticated piece of equipment to measure the gravitational attraction between massive lead balls. Comparing this amount of force of attraction to the spheres weight (their attraction to the sphere Earth) he was able to determine the density of the Earth. This allowed others to then determine the mass of the Earth and ultimately (as far as understanding gravity at least) the value of G, the universal gravitation constant. Today that value is known to be 6.67428 1011 m3 kg1 s2 with an uncertainty of about 0.00067 1011 m3 kg1 s2 . Or put another way, about 0.01%. www.ck12.org
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Figure 2.6: In 1798 Cavendish nds a way to measure the incredibly small forces that lead to a determination of the Universal Gravitation Constant.
Wow. They know the value of G really well! No, not very well at all in some respects. To put that in perspective, we know the mass of the electron with 2000 times more certainty, Plancks constant with 2000 times more certainty, and the electrons charge with 4000 times more certainty than we know the Universal Gravitation Constant! http://physics.nist. gov/cuu/Constants/index.html. Another interesting thing about the universal gravitation constant is that we dont have any strong evidence to believe it is necessarily universal or constant. In conjunction with Newtons law of gravitation it does work very well for examining the motions of the planets around the Sun and for getting spaceships to the Moon, Mars, and even the outer fringes of our solar system with great precision. But, there are still some pretty basic questions that can be asked for which we dont know the answers. Such as, has G always been this value from the big bang until now? Is G the same value near the super massive black hole in the center of our galaxy as it is here in a physics lab? What does the value of G really tell us about the fabric of our universe? Um, Ill hold onto those for later. We have a universal law of gravitation, and we know the value of Gat least pretty well. Any luck on how gravity applies a force without touching? Yes, and this question brings us into the 20th century and to the famous physicist Albert Einstein (1879 1955). In 1905 Albert Einstein had a rather remarkable year. Notice in the 1904 picture of Einstein that he is not the iconic old man with unruly hair. This is Albert Einstein at the age of 25, at his sharpest. In 1905 he published three amazing papers. These papers explained the photoelectric eect, explained Brownian motion, and introduced his special theory of relativity. All three are amazing and you may wish to do some studying on any or all of these topics. However, it is the third paper on the special theory of relativity that will forge a connection to gravity for us. In this paper he postulates that the speed of light is a constant in all inertial reference frames and that it is the ultimate speed limit in the universe. The paper postulates that the law of physics are the same (or are invariant) for all observers moving with a constant velocity. Einsteins paper did away with a need for luminous ether, changed concepts of time and space and the concept of simultaneity, but still did not deal with gravity. In 1915 Einstein published his paper on general theory of relativity, in which he postulated that the laws
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Figure 2.7: Albert Einstein at the age of at the time of his most amazing year of work that he published the next year in 1905. of physics are the same for observers moving with constant acceleration (thats why it is more general than the special relativity). In this paper, Einstein introduces the concept that mass bends the fabric of space and time and that this warping of space and time IS gravity. Bending space and timescience ction? And, if I did believe it, how does it account for gravity exerting a force without touching? Not ction, way stranger than science ction, because it really happens. This amazingly complex idea is easy and fun to model. Time for some more homework.
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practice you can easily create two small drops orbiting a larger drop similar to planets orbiting the Sun. And what about time? It doesnt really model that part. But it should help you see how mass can bend the space around it. It turns out time is just another dimension. There are three dimensions of space and one of time in our normal everyday world. When you are in class you are separated from the people to either side, in front and behind you, and in the classroom above you (assuming theres a oor above you) by three dimensions of space. You are separated from the person who uses your desk next by a period by time. In his work on general relativity, Einsteins mathematics led him to believe mass distorted space and time. Im a science student. I want some evidence. Does this really happen?
Figure 2.8: Einsteins letter shows how to look for mass bending light during a solar eclipse. Remarkable claims demand remarkable evidence. Einstein knew that others would be skepticalit is the nature of science! He even oered a few ways for others to test his ideas. One test he suggested was to look at Mercurys orbit. It is so close to the Sun that the way the Sun warps space around it should aect Mercurys orbit. General relativity correctly accounted for some motions of Mercury that were known and could not be explained by Newtonian gravitation. He also suggested utilizing a total eclipse of the Sun to see if the positions of stars located behind the Sun would appear to be shifted because their light had to
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pass so close to our massive Sun. In 1919 the rst attempts were made at making these measurements. These results were inconclusive but subsequent measurements during an eclipse in 1922 matched wonderfully with Einsteins predictions. This science was newsworthy in 1919. Cool. Did general relativity predict any other interesting astronomical occurrences? Boy, it certainly did. It was so amazing that Einstein didnt believe it himself! His mathematics indicated that something was totally wrong according to what was held to be true at that time. It was so remarkable that Einstein introduced a constant to get rid of it and to make the mathematics t the known reality. What was it? What did it indicate? General relativity showed that the universe should either be expanding or contractingthat it could not simply be. It could not exist in a static manner. But I thought that the universe is expandingat least I think Ive heard that. Right. But in the rst half of the 20th century that is NOT what scientists held to be true. Many religions have a moment of creation as part of their theology. The scientic community of the early 1900s did not share that paradigm. The widely held scientic view of the universe was very dierent from what it is today in many ways. One substantial way it was dierent was that most scientists believed that the universe was and always had been very much the way it was seen to be at that time. And how was it seen to be at that time? All the stars that you can see with the naked eye in the clearest, darkest night sky are part of our Milky Way galaxy. In fact, the terms Milky Way and galaxy represented the same celestial bodies in the late 1800s. In fact, even if you have a really nice backyard telescope, all of the stars you can see belong to the Milky Way galaxy. In the mid-1800s that was the extent of our knowledge. Astronomers of the time would have referred to the galaxy and the faint glow of it in our sky as the Milky Way. What we now call our galaxy was considered to be the entire universe. There were a few interesting non-star things in the sky known as nebulae (cloudy spots). You can see a lovely nebula in the constellation Orion in the three stars that make his sword. You can also see a much smaller (smaller in appearance from Earth that is) nebula in the constellation Andromeda known then as the Andromeda nebula. I thought that was called the Andromeda galaxy. It is now. In the late 1800s some very large telescopes were created. When astronomers looked at some nebulae like the Andromeda nebula and the Whirlpool nebula, they were able to observe individual stars. Because such large telescopes were needed to resolve these into individual stars, it meant that these stars were VERY far away. Examining other nebula like the Orion nebula showed they were truly wisps of glowing and reecting gas. We also made observations of our own galaxy that led us to understand that we actually exist in a attened out collection of stars. At this point, we then realized that the universe was MUCH larger than our own cluster of stars and actually contained many far-ung collections of stars. The term galaxy was eventually re-tooled to describe the isolated large clusters of stars and the word universe came to mean all of the known space including these island galaxies. The term Milky Way came to be the name of our galaxy. So three termsMilky Way, galaxy, and universe, which were originally synonymous, came to mean three dierent things as our understanding of the structures in space evolved from the late 1800s into the 1920s. A galaxy is a collection of billions of stars held together by mutual gravitation, the Milky Way is our galaxy, and the universe is ALL of it with some 100 billion individual galaxies each containing billions of stars. So the universe is a lot bigger than we thought, and it contains lots of galaxies. But what does this have to do with gravity? In the early to mid1900s, astronomers turned their attention to these very distant galaxies to try to deterwww.ck12.org
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Figure 2.9: The 1919 Illustrated explains the science of the dayverication of aspects of Einsteins general theory of relativity by British astronomer Arthur Eddington.
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Figure 2.10: The 1919 Solar eclipse. The small white circles were drawn around stars visible during this 1919 solar eclipse.
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Figure 2.11: On a very dark night the Andromeda galaxy (in green box) is barely visible to the naked eye in the constellation Andromeda. It is the only object visible with the naked eye in the northern hemisphere that is not within the Milky Way galaxy.
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Figure 2.12: Photographed through a large telescope using a long exposure the spiral structure of the Andromeda galaxy becomes apparent.
mine how big the universe was. There is some very interesting history of astronomy that Im going to have to leave out. These amazing details are provided at http://cosmictimes.gsfc.nasa.gov/1929/guide/ andromeda_farther.html. The full story involves some fascinating discoveries and early contributions of women in astronomy. The end result is often attributed to Edwin Hubble. One major physics concept that played a key role in Hubbles discovery, as well as later work regarding our universe and galaxies, is the Doppler eect. I think Ive heard of that, but can you review for me? Sure. The standard example is what you observe when a train is coming toward you blowing its horn. As the train approaches, the frequency of the sound you hear is transformed to a higher pitch by the trains motion. As the train passes you, the sound of the horn will drop to a lower pitch as it travels away from you. If you dont have a speeding train nearby, just tune your TV to a NASCAR race. When the coverage cuts to the camera stationed right down along the track you will hear a change. The sound that the engines make shifts frequency as the engines pass the camera. The sound shifts from a high-pitched whine to a deep roar. As the cars race toward you (the camera) the pitch is shifted to a higher frequency. When the car then moves away from you it is shifted to a lower frequency. A microphone riding alongside the car would hear a frequency in between the two. This is a noticeable eect because the speed of the observer is a signicant fraction of the the speed of the sound. As the car rushes toward you, the vibrations causing the roar of the engine are occurring closer and closer to you and thus taking less time to travel to you. Therefore, they arrive at your ears with less time between them, which makes the pitch higher. Of course, the similar argument applies to the car moving away from you. A more detailed explanation can be found at the Physics Classroom, http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/waves/u10l3d.html. This is known as the Doppler eect, and applies to all waves, including electromagnetic waves such as light. We do not observe the Doppler eect with light in every day life because the speeds of the observer and source are a very small fraction of the speed of light. And a much more detailed explanation with history and mathematics can be found at http://www.phy6. org/stargaze/Sun4Adop2.htm. What Hubble concluded from his work and the work of others was that the light arriving from distant galaxies had been Doppler shifted. It had been shifted toward the red end of the spectrum, which meant the galaxies were moving away from us (or vice versa) at speeds that are signicant compared to the speed of the wavewhich in this case is the speed of light! Note, though, that this does not mean that the Earth www.ck12.org
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is at the center of the universe. Imagine the universe as a bread pudding with the raisins representing the galaxies, and pick any raisin to represent our galaxy. As the pudding expands, the distance between the raisin you picked and any other raisin increases just as distant galaxies move away from us. This shows that the observed expansion of the universe does not imply that the milky way is at its center. What he determined was the more distant the galaxy was, the faster it was moving away from us. Every direction he pointed the giant Mount Wilson telescope, every distant galaxy was moving away from us. The conclusion: The universe is expanding!
Figure 2.13: Edwin Hubbles research showed the Milky Way was one of billions of galaxies and that the universe is expanding. Thats what Einsteins general theory of relativity predicted! Good, I see that youve been paying attention. But, at that time it was such a radical departure from what was known to be true, that even Einstein couldnt believe what his own work was telling him. In hindsight, it makes perfect sense. Here on Earth you can throw a ball up in the air. Because its under the inuence of gravity it can either be moving upward and slowing down or it can be moving downward and speeding up. The one thing it cant do is just sit in the air without accelerating. The same thing is true of the universe. Since all the galaxies are pulling on each other with gravity it makes sense that it could either be collapsing in on itself and speeding up as it does so, or expanding outward but slowing its rate of expansion due to gravity trying to pull it all together. Once Hubbles data and conclusions were presented, Einstein proclaimed the addition of the stabilizing constant his biggest mistake. This is the big bang, right? Correct again. If the universe is expanding today, it had to be a bit smaller yesterday. Play the lm backwards in your mind, and eventually the universe had a beginning and took up no space at all. Run it forward in time and you have the Big Bangthe creation of the universe. Youll understand, of course,
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that such a major shift in the understanding of the universe doesnt happen easily or overnight. There were many very bright scientists who tried very hard to argue that the universe wasnt truly expanding. One prominent astronomer, Fred Hoyle, was still arguing against the possibility in the 1950s when he used the term big bang to ridicule the concept that the universe had a beginning and was presently expanding. The name stuck, but unfortunately it is somewhat misleading. How so? A big bang sounds like a loud explosion. Of course, in space there is no sound. Also, an explosion, like a stick of dynamite in a rock quarry, throws energy and matter out into space. The big bang did not throw energy and matter out into space. It is the creation OF space and time and eventually matter condensed out of the energy (but that is matter for another chapter!). So if the universe is expanding, whats it expanding into? Nothing. It is creating more space and time. Its no more or less confusing than to ask where does the time for tomorrow come from. It doesnt exist today, but by the end of tomorrow there will have been one more day in the life of the universe. The dimension of time expanded. Is there other evidence for the big bang besides Hubbles receding galaxies? Lots of evidence. Because the idea of the big bang assumes the universe started very small it also started o with immense heat and energy. Because it has not been expanding for an innite amount of time, there should be some remnants of that energy left over in empty space. In the mid-1960s Arno Penzias and Robert Wilson were working for Bell Laboratories with microwave communication. While doing this work they accidentally discovered that no matter where they aimed their microwave receiver they received a constant background static. It was determined that this signal came from the leftover energy from the big bang and is called background cosmic radiation. This cosmic microwave background radiation (CMBR) tells us the temperature of space is about 2.7 Kelvin. This level of background radiation had been predicted earlier by George Gamow. It is always a great test of theory to PREDICT something and then later nd out that it really exists! Another case of this occurred in the ndings of the Cosmic Background Explorer (COBE) satellite. It was launched in 1989 to look for variations in the background radiation. Earlier examinations from Earth showed the CMBR to be very constant in every direction. This t the theory, but it couldnt be perfectly constant or there wouldnt be clumps of matter (galaxies and stars) like we have now. COBE mapped the entire sky looking for minute variations in the CMBR and found exactly what theories predicted should be therevariations of about one part in 100, 000. Another piece of evidence that should be mentioned is that the general theory of relativity indicates there should be expansion. OR Contraction. Correct. If the ball can be thrown up, it can fall back down. Does this analogy extend to the universe? This is a question still being debated. If the universe could contract then we already have a name for itthe big crunch. There are those that believe this is a possibility and if it is then the universe would be right back where it started and could perhaps have a big bang again. Others believe the mathematics shows the big bang to be a singular event. However, recent ndings make the notion of a big crunch even less likely. What ndings? In 1998 it was discovered that not only is the universe expanding but the rate of expansion is accelerating. That is very exciting and odd. If we return to the analogy of throwing the ball upward, the ball is not only moving upward but it is picking up speed! For the ball to do this, there must be some force continuing to push it upward. The same idea applies to the universe. This force is known as dark energy or Einsteins cosmological constant and it must be pushing outward to cause the universe to accelerate its expansion.
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So Einstein was wrong when he thought he made a mistake? Maybe. But you should recognize he didnt add the constant to address acceleration of expansion. He added the constant to push out against gravity to create a static universea form of the universe that clearly doesnt exist. So, does dark energy exist? Its an idea with lots of support. But it does have its problems. Its not supported or predicted by any bigger theory. It has not been detected in any direct way and it has to make up the majority of the energy in our universe! On the other hand, something has to be causing the accelerated expansion of the universe. So until something better comes along, dark energy is a favorite. Weve come a long way. Can you summarize things up to this point? Ill try. Gravity is a force of attraction between masses. We can describe it very well mathematically with Newtons universal law of gravitation. The universal gravitation constant, G, in the equation is one of the fundamental constants in physics and one of the least well known. Einsteins general theory of relativity explains how gravity is a warping of the fabric of spacetime and also predicts an expanding or contracting universe. The outwardly pushing cosmological constant he added to maintain a static universe may indeed be real and an expression of dark energy, which is causing the universe to accelerate its expansion. There is experimental support for the general theory of relativity and the big bang but currently there is no independent evidence for dark energy. Universal gravitation and general theory of relativity can explain planets orbiting, an expanding universe, spiral galaxies, rocks falling to the ground, my weight, and lots of other things, not just the accelerating expansion of the universe. Well, there is a problem with the spiral galaxies. They dont behave quite the way universal gravitation predicts they should and it doesnt seem to be explained by Einsteins work either.
Figure 2.14: The Whirlpool galaxy beautifully displays its spiral nature while mysteriously hiding exactly how it spins the way it does. Maybe its dark energy again. Good guess, but probably not. The most accepted answer is Dark Matter, but let me explain the problem rst before we jump to an answer. Here is a picture of the Whirlpool galaxy. It was one of the rst galaxies in which scientists resolved individual stars and led us to realize how vast our universe was. Newtons laws and Keplers laws of planetary motion should apply to stars in the galaxy orbiting around the massive
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center (the bright core in the middle) of the galaxy. Remember Keplers laws of planetary motion tell us that planets far from the center should take longer to go around the core than planets near the center. This is his third law: The period squared is directly proportional to the radius cubed. This means that stars far from the center take longer to go around in their orbit AND they are moving more slowly. Note, they take more time to go around because they are going a longer distance, but its not just that. Keplers law says they will be moving more slowly, not just take longer to go around.
Figure 2.15: Vera Rubins work in the mid1970s provided solid observational evidence that galaxies are not moving in accordance with Keplers laws OR possess large quantities of dark matter. In 1975 Vera Rubin determined that the vast majority of stars in several spiral galaxies were all traveling the SAME speed regardless of their distance from the galactic core. This observation means one of two things: Either the stars are not obeying Newtons laws or there is a great deal of matter fairly evenly dispersed between all the stars that we cannot see or detect other than through its gravitational interaction with the visible stars. This matter is not just dust and planets (often referred to as dim matter). Calculations show that in many cases that matter needs to be 50 75 percent of the total mass of the spiral galaxies to account for their orbital mechanics. Interestingly, not all galaxies seem to have the same mix of dark matter to normal matter. Some have hardly any dark matter while some may be made of nearly entirely dark matter. So dark matter really exists? Its very similar to dark energy in that respect. The vast majority of astronomers and physicists accept that it is probable but are really anxious to see some more supporting data, unication with other theories, and explanations of its nature. Dark matter to keep the galaxies spinning right, and dark energy to account for the acceleration of expansion of the universe. Sounds like theyre just making this stu up to account for what they cant explain with normal physics. Precisely! This is the way physics often works. First, observe a phenomenon you cant explain. Second, come up with an explanation. Sometimes the explanation involves things that are already understood and when things get really exciting it involves things no one has ever thought of! Then physicists around the world try to make observations, do experiments, or deal with the mathematics to either lend independent support to or tear down the new idea. Since dark energy is only going into its second decade and dark matter is only working on its fourth, these ideas are in the stage where people are looking for evidence to www.ck12.org
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prove them wrong or for evidence to support them. Oh, now I have lots of questions. You said often. Right. The other way physics often works is now that we have these two relatively new ideas, physicists and astronomers are actively looking for things these theories predict. Sure these two phenomena were made up to describe things we already saw and couldnt explain. But does the presence of dark matter and dark energy predict things we havent seen that we can go look for? Like the general theory of relativity predicted light would be bent by the curvature of space near our Sun! Right again. Finding this prediction to be true provided support for other claims of the theory. What will it take to prove that dark matter and dark energy are correct? It will never be proven correct. Bending of light didnt prove General Relativity correct, it just provided support for the theory. No amount of data, observations and calculations will prove a scientic theory or law to be true. The more data, observations and supporting calculations we have, the more trust we may have in a particular idea and the more we may build upon it. However, it only takes ONE observation of a fact that DOESNT support the law or theory to send physicists scurrying for a new idea or adjustment to the old one. Didnt Vera Rubins observation of the way galaxies were spinning and the 1998 observation of the acceleration of the expansion of the universe show that the physics we were using was wrong? That would be one view. If you go back to the page with the picture of Dr. Rubin, you will see that I said Either the stars are not obeying Newtons laws or there is a great deal of matter fairly evenly dispersed between all the stars that we cannot see or detect other than through its gravitational interaction with the visible stars. You see there are really two choicescome up with a new idea or make adjustments to the old one. In the case of dark matter you either need a lot of rather mysterious matter that doesnt glow with any type of electromagnetic wave (radio, xray, visible light) or block any type of electromagnetic wave, OR you need to adjust other accepted laws of physics. The vast majority of astronomers and physicists have chosen to opt for the mysterious dark matter. The majority. So there are those out there who dont? Correct. There is ongoing scientic debate on whether string theory does or does not predict dark matter, but I wont attempt to (nor am I capable of) explain string theory. However, there are at least two alternate views regarding issues related to gravitation that have received some support and are, at the very least, interesting to examine. MOND is a concept that illustrates a minority view in a very interesting and understandable manner. MOND stands for modication of newtonian dynamics. Developed by Mordehai Milgrom, this theory adjusts Newtons laws of motion to match observation of the way galaxies spin. This is in contrast to assuming there is an abundance of Dark Matter so the dynamics match Newtons laws. An excellent article by Dr. Milgrom explaining the idea of MOND may be found at http://www.astro. umd.edu/~ssm/mond/sad0802Milg6p.pdf. Ill go read the article, but what does MOND actually say? Do go read the article, but essentially what MOND does is claim that when acceleration is less than some minimum value then the force on an object is no longer equal to mass times acceleration (Newtons second law) but equal to mass times acceleration squared. Making this assumption allows many things (not all, mind you) to work correctly without the need for dark matter. And you said at least two alternate views? John Moat of the University of Toronto has proposed a Non-Symmetric Gravitational Theory. Here Newtonian dynamics is left unchanged, but general relativity is altered from the way Einstein had it.
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If non-symmetrical gravitation theory is true it also avoids dark matter and accounts for the galactic rotation curves. Dr. Moats book Reinventing Gravity explains this at a popular level that you might nd interesting to read. All right then. We either need dark matter, MOND, non-symmetric gravitation, or something else for explaining certain phenomenon, like the mechanics of spiral galaxies and dark energy or something else to explain the accelerating expansion of the universe. Ill wait and see which idea(s) come out on top. Wonderful, me too. And remember, it wont be that one gets proven correct, it will simply be that one theory is capable of explaining more phenomena and is supported by more observations. I hesitate to ask this, but what else is being searched for related to gravitation? Well, since you asked.In 1918 Einstein predicted that when massive objects (neutron stars, quark stars, black holes, supernova) explode, spin, or collide they should create ripples through the space-time fabric. These ripples are dubbed gravity waves. As of yet, physicists have had no luck in nding them. Once we do nd them (if they exist) it is reasonable to assume they would carry information with them about the object that generated them. NASA will soon (hopefully!) be launching LISA to search for these gravity waves. You can check out http://lisa.nasa.gov/ for all the details of how LISA will accomplish this and what it hopes to discover. Meanwhile, here on Earth, LIGO is looking for the same phenomenon. Details regarding LIGO can be found at http://www.ligo.caltech.edu/. The other holy grail related to gravitation is the graviton. The graviton is the hypothetical particle that may carry the force of gravity. This is in the same sense that the photon is the particle that transmits electromagnetic radiation. In many sources you will see the word mediatethe graviton would mediate the force of gravity. It seems that actually detecting a graviton will be far in our future if indeed it is ever possible. Detection and analysis of gravity waves may eventually allow more concrete knowledge of whether gravitons actually exist or not. Now Id like to suggest some activities or assignments for you to do to assess your understanding of portions of the content of this chapter. 1. Create two ellipses in the manner described at the beginning of this chapter and use them to describe and explain Keplers laws of planetary motion. 2. Explain why Newtons universal law of gravitation is a law and not a theory or hypothesis. 3. Go to http://imagine.gsfc.nasa.gov/docs/science/know_l1/dark_matter.html and read about dark matter. Read the article regarding MOND linked earlier in this chapter. Write an essay explaining which theory you believe is most likely to be found valid. 4. Create the coee-can-and-soap-lm universe explained in this chapter. Describe the experiments you were able to conduct and explain how this models aspects of the general theory of relativity. Explain in what ways this is NOT a good model of Einsteins notion of gravitation. 5. Go to http://cosmictimes.gsfc.nasa.gov/1929/guide/andromeda_farther.html and read the details of what preceded Hubbles determination that distant galaxies are receding from us. Click on the link at the bottom regarding Harvards Computers and read the four biographies. Create a timeline showing the discoveries in these ve dierent articles that lead to an understanding of an expanding universe. In your own words, explain the role that women played in uncovering the big bang.
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Image Sources
(1) Unknown. 1919 Solar Eclipse. Public Domain. (2) Unknown. Johannes Kepler. Public Domain. (3) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (4) Rembrandt. wiki/File:Rembrandt_Harmensz._van_Rijn_013.jpg . Public Domain. (5) Unknown. Einsteins Letter. Public Domain. (6) Andrew Jackso. Creating an Ellipse. CC-BY-SA. (7) Unknown. Albert Einstein. Public Domain. (8) Akira Fujii. Andromeda Galaxy. Public Domain. (9) Godfrey Kneller. Sir Isaac Newton (16431727). Public Domain. (10) Unknown. Edwin Hubble. Public Domain. (11) Unknown. Photograph of Andromeda galaxy. Public Domain. (12) Andrew Jackson. Creating an Ellipse, part 2.. CC-BY-SA. (13) Unknown. Vera Rubin. Public Domain. (14) Unknown. Cavendish. Public Domain. (15) Unknown. The Whirlpool Galaxy. Public Domain. (16) Arthur Eddington. Image from the 1919 Illustrated London News. Public Domain.
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3.1 Introduction
This chapter is a short non-mathematical course introducing high school physics students and interested non-scientists to the physics of the atomic nucleus and to phenomena associated with nuclear ssion. You can also access a summary of this chapter on David Sterns website, http://www.phy6.org/stargaze/ SnucEnerA-0.htm , as well as the entire chapter at http://www.phy6.org/stargaze/SnucEnerA-1.htm .
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2. Chemical properties of atoms are determined by electrical forcesfrom lightweight, negatively charged electrons, balanced by an equal number of much heavier protons with an equal but positive electric charge. Atoms also contain neutrons, which are similar to protons, but without electric charge. Whole atoms, with equal numbers of both, have zero net electric charge. Certain chemical molecules however (acids, bases, and salts) are formed by some atoms borrowing an electron from others with which they are combined. Because water weakens electrical forces at molecular dimensions, when such compounds are dissolved in water, the electrical components missing an electron or having an extra borrowed one (ions) may sometimes temporarily separate. Such solutions (e.g., sea water) therefore conduct electricity and their ions may sometimes be separated by an electric current (for more, see reference #2). Ionic compounds melted by heat (e.g., molten salt) and compounds dissolved in them may also be separated by electric current. In addition, ions form in rareed gases when suicient voltage is applied (and in other ways). They carry electric currents in uorescent light xtures (helped by free electrons), also in the ionosphere and in more distant space. 3. Electrons may also be boiled o a hot object in a vacuum (#3). Other methods allow the creation in a vacuum of free positive or free negative ions (atoms that have lost one or more electrons, or have attached extra electrons). Any of these may be accelerated in the laboratory by accelerators to velocities close to that of light, and given high energies. Much of our information about atoms comes from studies of collisions of such fast particles with atoms. 4. The element with the lightest atom is hydrogen, and its positive part is known as a proton, 1836 times heavier than the electron. The atomic weight of other atoms gives the approximate number of times their atom is heavier than that of hydrogen, e.g., 4 for the main component of helium, 12 for that of carbon, 14 for nitrogen, 16 for oxygen, and so on, up to 238 for the main variety of uranium, the heaviest atom found in nature.
Figure 3.1: The Helium Nucleus The nucleus of helium has the positive charge of 2 protons, although it is 4 times heavier. Similarly carbon has only 6 times the charge. That suggests these nuclei contain an equal number of uncharged protons,
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known as neutrons. The free neutron (discovered by Chadwick in 1931) is ejected from certain nuclear collisions (see further below), but is unstableafter an average of about 10 minutes it becomes a proton, electron, and a very light uncharged neutrino. One gram of hydrogen, 4 grams of helium, 12 of carbon etc. all contain NA = 6.022 1023 atoms, a constant known as Avogadros number. That too is the number of molecules in 2 grams of hydrogen (molecule H2 ), 18 grams of water (molecule H2 O, 44 grams of carbon dioxide (molecule CO2 ) and so forthnumbers formed by adding atomic weights of a component to give the molecular mass. 5. To denote an element in nature, an abbreviated symbol is used, e.g., H for hydrogen, O for oxygen, C for carbon, U for uranium, Na for sodium (Natrium), Pb for lead (Plumbum), Cl for chlorine, Fe for iron (Ferrum) etc. Actually, most atoms in nature have several varieties (isotopes), diering in weight by very close to the weight of a nucleon (i.e., proton or neutron). To denote a specic isotope, a superscript giving its atomic weight is added to its symbol. For instance, chlorine in nature is a mixture dominated by approximately 75% 35 Cl and 25% 37 Cl. Hydrogen (H) has 3 known isotopes: Ordinary hydrogen 1 H, heavy hydrogen 2 H (also known as deuterium D) forming 1/6000 of atoms in nature, and tritium 3 H, which is unstable, must be produced articially, and decays with an average time of 12.5 years (half life, time after which only half its atoms are left). It turns into a helium isotope 3 He as it emits an electron and one of its neutrons (see 7 below) becomes a proton. 6. Apart from the electrons, the mass of the atom is concentrated in a very compact atomic nucleus. 7. The nucleus of the most common isotope of helium has twice the positive charge of the proton, but close to four times the mass. It turns out it contains two protons and two neutrons, particles similar to protons but slightly heavier and with no electric charge. Light atoms have about an equal number of protons and neutrons, e.g., 6 + 6 in 12C, 8 + 8 in 16 O. In heavier atoms neutrons have the majority, which increases as atomic weight rises, e.g., 238 U has 92 protons and 146 neutrons. Isotopes of the same element have the same number of protons (which equals the number of electrons and determines the chemical properties) but dierent numbers of neutrons. This imbalance (further discussed below) plays a crucial role in the release of nuclear energy by the ssion chain reaction. 8. Atomic nuclei may be unstablein particular, in very heavy elements and in isotopes whose number of neutrons diers signicantly from their number in the most prevalent isotope. Unstable nuclei may undergo radioactive decay to a more stable state. Most radioactive nuclei do so by emitting one of three kinds of nuclear radiation denoted for historical reasons by the rst 3 letters of the Greek alphabet(, , ) or (alpha, beta, gamma) radiation. Alpha () particles are nuclei of helium, and emitting them changes an atom to one with two fewer protons and two fewer neutrons (the alpha particle, after being slowed down by collisions, combines with two electrons of its surroundings to become regular helium, while the emitting atom sheds two electrons, which keep the surrounding material neutral). Alpha particles have a very short range in matter and can hardly penetrate skin. However, they cause great damage if ingested into the bodyas in the case of Alexander Litvinenko, a Russian oicer given asylum in London, who died in November 2006 after being poisoned with emitting polonium. Beta particles are fast electrons or positrons (the anti-particle of the electron) emitted when a neutron is converted into a proton or a proton is converted into a neutron, respectively. This usually involves neutrons inside an unstable nucleus. However, free neutrons produced in high-energy collisions in the lab (from accelerated particles, also by natural alpha particles hitting beryllium nuclei) also undergo such conversion, with a half-life of about 10 minutes, producing a proton, an electron and an uncharged, almost massless neutrino or its twin anti-neutrino, either of which can pass through matter almost unhindered. Gamma rays are similar to xrays, a form of electromagnetic radiation (see next item below) similar to light or radio waves. Just as visible light can be emitted at well-dened energies by atomic electrons in excited atoms jumping from one energy level to a lower one, gamma rays arise from nuclei passing from www.ck12.org
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an excited energy level to another onepossibly to the lowest level, the stable ground state. 9. The word radiation should be used with caution. Physicists usually apply it to electromagnetic (EM) radiation, a family of disturbances propagating through space and including radio waves, microwaves, light (visible, infra-red, and ultra-violet), xrays, gamma rays, and ranges between the named ones. These dier in wavelength and are described qualitatively in (#4). Nuclear radiation emitted from unstable nuclei may be electromagnetic (gamma rays) or consist of particles with mass (alpha and beta rays), perhaps accompanied by gamma rays. Articial isotopes may in addition emit neutrons and positrons (positive electrons). Some people do not realize the dierence between nuclear radiation and electromagnetic radiation! Colloquially, we nuke food in a microwave oven, when in fact atomic nuclei are not involved, only very short wave radio waves, whose energy is absorbed by water molecules in the food and heats it. This discussion of nuclear power involves mostly nuclear radiation, so here (only here!) the unqualied word radiation implies nuclear radiation. 10. In an atom, negative electrons surround the positive nucleus and are held by electric attraction, similar to the way planets are held by the gravity of the Sun. A big dierence exists however, because Newtons laws are modied on the atomic and subatomic scale of distances, to follow quantum mechanics. In a way, matter behaves like sand: On a large scale, it ows like a uid, but its small-scale behavior depends on the existence of individual grains. The graininess, which rules quantum phenomena, is determined by h, a constant of nature named Plancks constant after its discoverer. For more about quantum phenomena, see (#5) and the 7 Web les linked from it (Q2 ... Q8 htm). A fundamental equation containing h involves light (or any other EM radiation). A frequently heard statement is that light can be both a wave and a particle. Basically, when EM radiation spreads, it does so like a wave with wavelength L (also denoted by lambda the Greek letter L), spreading with velocity c (the speed of light, 300, 000 km/sec). As the wave passes a point in (empty) space, a total wave train of length c must go each second through it, chopped into up-down oscillations (of the electric or magnetic force, but that is not important here) of length L each, so the total number of up-down excursions each second, the frequency f of the wave, is: f = c/L (also denoted by nu , the Greek N). The wavelength can be measured, and the wave describes all optical phenomena. However, when an EM wave interacts with matter and gives up its energy, it was found that it happens only in discrete lumps of energy or photons, each of which contains energy E = h f with h equal to Plancks constant. Max Planck in Germany (Nobel Prize, 1918) proposed that equation in 1900 to explain the color distribution emitted from hot objects, but its signicance in atomic processes was recognized after Einsteins 1905 explanation of the ejection of electrons from metal by light of dierent colors (photoelectric eect). That was what earned Einstein his 1921 Nobel Prizenot his 1905 discovery of relativity! Photons are localized to perhaps just the atom which absorbs the energy, and not spread over all space like a wave; however they require a quantum mechanical description. For more, see #4 and the web pages under #5 above. As mentioned earlier, beta particles are fast electrons or positrons emitted when a neutron is converted into a proton or a proton is converted into a neutron, respectively. Because of quantum rules, an electron in an individual atom of a gas can only move in certain well-dened orbits and no otherslike a wave with well dened stable patterns, e.g., sound in a musical instrument. When an atom is excited (e.g., by electrical forces in uorescent tubes or in sodium vapor lamps of street lights), an electron may be moved to a higher energy level; then, as it returns to a lower level, it emits well-dened frequencies of light (see #6 for examples), sensed (when visible) as specic colors, and each frequency represents (by the above equation) the energy dierence between two states of the atom. All
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such electrons end in the ground state of lowest energy, which is stable. Because of the existence of the ground state (which is determined by quantum laws), the electron is in no danger of moving further and falling into the atoms nucleus.
Tidbits
And by the wayPractically all helium on Earth (as used in party balloons, for instance) is usually extracted from natural gas, and has originated as particles emitted by uranium, thorium, or some of their daughter products. As evidence, helium from the Sun contains a small amount of the isotope 3 He one neutron, two protons), but terrestrial helium is almost pure 4 He.
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electromagnetic radiation A family of waves propagating in space, representing oscillating electric and magnetic forces, e.g., light, radio. electron Light elementary particle, negatively charged, found in all atoms. energy level One of the energies at which, according to quantum laws, atoms or nuclei may be found. excited state of atom A state of an atom with more energy than the lowest ground state. excited state of atomic nucleus A state of the atomic nucleus with more energy than the stable (or most stable) ground state. frequency of EM wave Number of oppositely directed excursions of the electric or magnetic force at a point in space where the wave passes. gamma rays Electromagnetic radiation of very short waves, emitted by nuclei. ground state The lowest energy state of an atom or nucleus. half life For a radioactive element, the time needed for half of it to decay. ion Atom or molecule which has lost one or more electrons, or attached extra ones. isotope Variety of a chemical element with a certain number of protons and neutrons. molecule A chemical combination of two or more atoms. molecular weight The sum of atomic weights of a molecule. neutrino Uncharged and nearly massless elementary particle; may carry energy. neutron Uncharged nucleon, similar to proton. nuclear radiation Waves or particles emitted by unstable atomic nuclei. nucleus (of atom) Core of an atom, electrically positive and with most of the mass. photon Quantity of energy associated with the emission or absorption of an electromagnetic wave. Plancks constant A physical constant appearing in equations of quantum physics. proton An elementary positive particle; neutrons and protons form the atoms nucleus. quantum mechanic Rules of mechanics on the atomic and nuclear scale. radiation General name for either electromagnetic or nuclear radiation. 3. Very highenergy ions from space (cosmic radiation) arrive at the top of the Earths magnetosphere, collide with atoms and splash out fragments, some of which are neutrons. Neutrons do not feel magnetic forces, but electrons and protons can get trapped, though those splashed from the atmosphere always return and hit the atmosphere again. Is this a credible explanation to the radiation
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belt trapped in the magnetic eld of the Earth? [Yes. Particles from the atmosphere always return and are absorbed by the atmosphere, but neutrons may decay in ight and yield energetic protons (also electrons), which could appear on a magnetically trapped orbit. The original Van Allen belt is believed to originate that way.] 4. A certain radioactive isotope has a half-life of 2 days. How long approximately does it take until only 1/1000 of it remains in a given sample? [About 20 days, or 10 half-lives, because (1/2)10 = 1/1024] 5. Hydrogen (forming H2 molecules) weighs about 90 grams per cubic meter. How many molecules of hydrogen are in one cubic micron (a micron is the millionth part of the meter)?If NA is Avogadro's number 6.022 1023 then 2 grams hydrogen contains NA molecules, and 90 grams contain 45 NA . A cubic micron is 1018 cubic meters, so the number is: [N = 45(6.022 1023 )1018 = 271 105 = 2.71 107 or about 27 million molecules.]
Nuclear Fusion
The binding energy of helium is appreciable, and seems to be the energy source of the Sun and of most stars. The Sun has plenty of hydrogen, whose nucleus is a single proton, and energy is released when 4 protons combine into a helium nucleus, a process in which two of them are also converted to neutrons. The conversion of protons into neutrons is the result of another nuclear force, known as the weak force (the word nuclear is assumed here). The weak force also has a short range, but is much weaker than the strong force. The weak force tries to make the number of neutrons and protons in the nucleus equal; these two particles are closely related and are sometimes collectively known as nucleons. The protons combine to helium only if they have enough velocity to overcome each others repulsion and get within range of the strong nuclear attraction, which means they must form a very hot gas. Hydrogen hot enough for combining to helium requires an enormous pressure to keep it conned, but suitable conditions exist in the central regions of the Sun (core), where such pressure is provided by the enormous weight of the layers above the core, created by the Suns strong gravity. The process of combining protons to form helium is an example of nuclear fusion. Our oceans have plenty of hydrogen, and helium does not harm the environment, so it would be great if physicists could harness nuclear fusion to provide the world with energy. Experiments in that direction have so far come up short. Suiciently hot hydrogen will also be ionized, and to conne it, very strong magnetic elds have been used, because charged particles (like those trapped in the Earths radiation belt) are guided by magnetic eld lines. Fusion experiments also rely on heavy hydrogen, which fuses more www.ck12.org
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easily, and gas densities have been kept moderate. In spite of all such tricks, though fusion energy has been released, so far more energy is consumed by the apparatus than is yielded by the process.
Figure 3.2: The Binding Energy of Nuclei The net binding energy of a nucleus is that of the nuclear attraction, minus the energy of the repulsive electric force. As nuclei get heavier than helium, their net binding energy per nucleon (deduced from the dierence in mass between the nucleus and the sum of masses of component nucleons) grows more and more slowly, reaching its peak at iron. As nucleons are added, the total nuclear binding energy always increasesbut the total energy of electric forces (positive protons repelling other protons) also increases, and past iron, the second increase outweighs the rst. One may say 56 Fe is the most tightly bound nucleus (see #10-b). To reduce the energy of the repulsive electric force, the weak interaction allows the number of neutrons to exceed that of protonsfor instance, in the main isotope of iron, 26 protons but 30 neutrons. Of course, isotopes also exist in which the number of neutrons diers, but if these are too far from stability, after some time nucleons convert to a more stable isotope by beta emission radioactivityprotons turn into neutrons by emitting a positron, the positive counterpart of the electron, or neutrons become protons by emitting electrons (neutrinos are also emitted in these processes). Among the heaviest nuclei, containing 200 or more nucleons, electric forces may be so destabilizing that entire chunks of the nucleus get ejected, usually in combinations of 2 protons and 2 neutrons (alpha particles, actually fast helium nuclei), which are extremely stable. The curve of binding energy (drawing) plots binding energy per nucleon against atomic mass. It has its main peak at iron and then slowly decreases again, and also a narrow isolated peak at helium, which as noted is very stable. The heaviest nuclei in nature, uranium 238 U, are unstable, but having a lifetime of 4.5 billion years, close to the age of the Earth, they are still relatively abundant; they (and other nuclei heavier than iron) may have formed in a supernova explosion (#8) preceding the formation of the solar system. The most common isotope of thorium, 232 T, also undergoes particle emission, and its half-life (time over which half a number of atoms decays) is even longer, by several times. In each of these, radioactive
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decay produces daughter isotopes, that are also unstable, starting a chain of decays that ends in some stable isotope of lead.
Tidbits
The book The Curve of Binding Energy by John McPhee is actually the story of nuclear physicist Theodore Taylor and his diverse side-interests.
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(b) 4 gram helium contain A atoms, so the energy released is, E = (6.022 1023 )(2.38 107 )(1.60 1019 )joule Add exponents 23 + 7 19 = 11 Multiply coeicients (6.022)(2.38)(1.60) = 22.93 = 22.93 1011 joule = 2.293 1012 joule (c) 1gram TNT = (3.8)(4184) = 1.59 104 joule 2.293 1012 = 1.442 108 gram 1.59 104 = 144.2ton TNT 6. Let us compare the mass loss due to either process through an area of 1 square metre at the Earths orbit, perpendicular to the ow of sunlight, during one second. Working in metres, seconds and kilograms, c = 3 108 meter/sec, and the energy ow is 1300 joule/sec. If m is the mass lost during that time through the chosen area (by conversion to radiant solar energy) E m= 2 c 1300 = 9.1016 = 1.444 1014 kilograms The solar wind passing through the same area includes all the matter contained in a column of cross section 1 meter2 and of length v = 400 kilometers or 4 105 metres. One cubic metre contains 106 cubic centimeters and the mass of 107 protons. The ow through the area is therefore 4 1012 protons, with a mass 6.69 1015 kilograms. The loss due to sunlight is therefore greater by about a factor of two. Still, it is remarkable how close these two numbers are to each other - one dictated by processes in the innermost core of the Sun, the other by processes in its outermost layer. Coincidence, you say? 7. 9.39535\times 10{8} 8. If m is the mass of the neutron, E_0 = mc2 = 9.39535108 evT heKineticenergyis,E1 = ( )2 mv2 1 mv2 v1 1 2 2 = 0.5 mc2 = 0.5 c mc ( )2 4 = 0.5 1.1310 8 310
= 0.5(0.376666 104 )2 = 0.5(0.1418777 108 ) = 0.070939 108 T ocalculateE_1, wemultiplybythevalueo f E_0E1 = (9.39535 108 )(0.070939 108 ) = 0.6665eV
This is less than 1 eV! Radiation belt particles have energies of the order of MeV, and even electrons of the polar aurora have of the order of 10,000 eV (thermal energy of air molecules in your room is about 0.03 eV). Gravitational energy is therefore completely negligible by comparisonor in other words, the electromagnetic forces on particles in space tend to be much, much bigger than their gravitational forces. www.ck12.org
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Figure 3.3: Typical Nuclear Fission Energies of atomic and nuclear processes are measured in electron volts (eV), the energy acquired by an electron or proton (electric charges of the same magnitude) going through a voltage drop of one volt. The eV is a unit appropriate for atomic processes, associated with the chemical binding energy of electrons. For nuclear processes, a more appropriate unit is the MeV, million electron volt. Each of the two ssion fragments carries a positive charge, and their mutual repulsion typically releases 161 Mev (#9), compared to typical energies of 2 MeV for rays and 4 MeV for particles (further details in #10). This mode is known to occur spontaneously in articial elements heavier than uranium. However, the absorption of a neutron by a suitable uranium nucleus235 U or 233 Ucan also trigger its ssion. A proton aimed at a nucleus, even if headed straight toward it, needs to be accelerated to a considerable energy to overcome the repelling electric force and get close enough to be captured by the strong nuclear force. A neutron, on the other hand, is not repelled and can reach its target, even if it moves quite slowly, e.g., a thermal neutron whose energy is comparable to that of molecules in ordinary matter or in air, about 0.03 eV. Imagine the nucleus as a target of a certain size, then the nuclear cross section is the area a projectile must hit to produce a certain reaction (it is also proportional to the likelihood of the projectile sticking to the nucleus). Nuclear cross sections are measured in barns, where 1 barn is equivalent to a target size of 1024 cm2 (big as a barn for nuclear physicists). The cross section for a neutron to hit a nucleus varies from one isotope to another, and with the energy of the neutron (similarly for other particles undergoing collision). For instance, the chance of a thermal neutron sticking to a nucleus of heavy hydrogen (the 2 H isotope or deuterium) is rather small, because that type of hydrogen already has an extra neutron. As a neutron reaches its target nucleus, one may visualize the nuclear attraction speeding it up, so that it hits with appreciable energy, agitating the target nucleus.
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The eects of this extra energy may vary. The target nucleus may simply emit it as a ray photon (see end of #11), or it may undergo some internal change, e.g., the neutron may become a proton, emitting an electron (radioactivity). But with 235 Uan isotope forming about 0.7% of natural uraniumthe result is usually nuclear ssion, splitting the nucleus into two fragments. The products may vary, but typically the ratio of the masses of the two fragments is close to 2 : 1. Nuclear ssion was identied in Germany in 1939 by Hahn, Strassman, and Lise Meitner. (That was in Nazi GermanyHahn was awarded the Nobel Prize in 1944, and his long-time associate Meitner, who was Jewish, was lucky to escape to Sweden). Very soon, physicists all over realized that the process could provide usable energy. Not only did it release appreciable energy per nucleus, but more important, it also released additional neutrons, making possible a self-sustaining chain reaction.
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water), or into an array of holes in a core of carbon bricks (drawing). Neutrons released from a ssion in one rod soon wander into the moderator and are slowed down there to thermal speed, and after a while (unless they escape or are captured) they reach another rod and initiate another ssion event there. The extracted energy appears as heat which is passed to steam: The enormously fast ssion fragments keep colliding with the moderator, and ultimately spread their energy around. If the moderator is water (heavy or light) it is kept under pressure to raise its temperature, because energy extraction from hot steam gets more eicient the higher the temperature is. In a solid moderator, pipes carry a uid to remove the heat by superheated water or by another uid. Even liquid sodium metal has been used (in fast breeder reactors)an extremely tricky substance that bursts into ames if allowed access to air. Other pipes linked to the heat removal system carry high pressure steam into ordinary steam turbines (similar to those in conventional power stations) that turn electric generators. The cooled-down expanded steam is then turned back to water in cooling towers (often drawn as ominous symbols of nuclear power, though most any steam-driven power station has them) and are recycled to the reactor to pick up more heat. Third, as fuel is consumed, ssion fragments accumulate. These are often ercely radioactive or hot (letting go of 2 3 neutrons makes them more stable, but instability remains), and disposing of them is a major challenge. They may remain hot for years and even centuries, and need to be stored out of contact with life and with ground water. Because radioactivity releases thermal energy (heat), initially they also need to be cooled. Ideally, before being stored, a fuel rod needs to be reprocessed. It still contains some useful fuel, which can be reused. Also, some radioactive isotopes produced by ssion can be separated and used as radiation sources in medicine or research. Others are pure waste and need to be stored out of contact with the natural environment. Natural uranium was used in the earliest reactorsbut because it is used up rapidly, enriched uranium is preferred, in which the fraction of 235 U is increased by an enrichment process. The chemistry of dierent isotopes is practically the same, so non-chemical separation must be used, with gaseous compounds such as UF6 (uranium hexauoride). In such a gas, molecules with 235 U are about 1% lighter than those with 238 U, and therefore, at a given temperature they move faster and diuse more rapidly through porous partitions. Alternatively, a specially designed centrifuge, with a rapidly spinning shaft, may spin the gas and cause heavier molecules to be concentrated in the outer layers.
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In either case, because the separated isotopes are so close in mass, the dierence in concentration is very small. Therefore, uranium separators must be connected in a cascade of many units feeding each other, with the enriched fraction advancing to the next level and the depleted fraction recycled to an earlier one. (Completely depleted uranium is sometimes used for armor-piercing ammunition because it is very dense and at bullet-speeds packs a lot of kinetic energy.) Usually most of reactor fuel still consists of the more abundant isotope 238 U. Neutron absorption makes this isotope unstable and after some nuclear changes it turns into plutonium 239 Pu, an articial element with 94 protons. Plutonium is also a suitable nuclear fuel, and part of the energy released in a nuclear reactor comes from the ssion of plutonium produced there. Reprocessing nuclear fuel is a diicult task because spent fuel is too radioactive for humans to handle directly. All devices involved in reprocessingincluding those that pull out used fuel rods and transport themare operated by remote control, and when discarded many must be stored safely (like the spent fuel) for long periods. One reason partially spent fuel must be removed from reactors and reprocessed is that some ssion products absorb neutrons and thus reduce eiciency (poison the reactor). Currently, the United States has stopped reprocessing spent fuel fresh from power stations, and allows it to cool down in pools located near reactors, but reprocessing is about to be resumed. France which gets most of its electric energy from ssion, Russia and other countries, do maintain successful reprocessing centers.
Tidbits
Nuclear reactors were recognized early as ideal power sources for large submarines, because they needed no air and required only infrequent refueling. Fission reactors were also designed for powering spacecraft. The United States launched SNAP 10-A in 1965, but it was shut down after 43 days due to malfunctions. Soviet Russia launched many reactors (#13), which were later detached and boosted to a higher orbit, with a lifetime of centuries. That program ended when the reactor on Cosmos 954, powering an ocean-surveillance radar, failed to detach. The satellite with its reactor crashed on 24 January 1978 into a frozen lake in Canada, creating strong protests and ending the use of reactors in space. In addition, the radioactive heat produced by plutonium is used in radioisotope thermal generators (RTGs) to power space probes to the outer parts of the solar system, too far from the Sun for solar cells to generate suicient power. RTGs gradually lose power after 20-30 years, and, of course, they never return to the Earths neighborhood. Nazi Germany also tried to develop nuclear energy during World War II, on a much more limited scale than the allied powers. However, graphite was regarded as unsuitable, as samples tested for moderator were not pure enough and absorbed too many neutrons. Heavy water was chosen instead, a by-product of hydroelectric power stations in Norway. However, the Norwegian underground eectively sabotaged its production there.
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cross section (for nuclear interaction), delayed neutrons, enrichment (of uranium), ssion (nuclear), ssion fragments, fuel rods, graphite, heavy water, isotope separation by centrifuges, isotope separation by porous partitions, photon, plutonium, poisoning of a nuclear reactor, prompt neutrons, reprocessing of nuclear fuel, thermal neutron.
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isotope separation by porous partitions Separation of an isotope by gas ow through porous partitions. photon A packet of energy in which an electromagnetic wave is absorbed. plutonium An articial element with 94 protons, common nuclear fuel. poisoning of a nuclear reactor The accumulation of neutron-absorbing ssion fragments, reducing or stopping ssion in a reactor. prompt neutrons Neutrons emitted promptly from nuclear ssion, about 98%. reprocessing of nuclear fuel Chemical separation of ssion product from unburned nuclear fuel and articial isotopes. thermal neutron A neutron slowed down by a moderator to thermal energies.
Figure 3.5: Nuclear Power Station Control is maintained by control rods of a material such as the metal cadmium, which has a high absorption cross section for neutrons. The rods are automatically pushed deeper into the reactor to reduce the rate of ssion, or pulled out to maintain or increase it. Delayed neutrons allow for the control. About 98% of the neutrons released in ssion are prompt www.ck12.org
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neutrons, released very quickly, faster than the reaction time of automatic control machinery. However, 2% are delayed neutrons, which provide a very narrow margin for reactivity control. Reactors need to stay on the 2% margin between a zzle and runaway ssion. It is a very small margin, and because of its narrowness, any power reactor has multiple independent safety devices In case of an emergency, an emergency shutdown (a scram) automatically pushes or drops the rods in all the way, as well as extra rods for emergency use, usually withdrawn. The chain reaction then stops immediately, but not the radioactive decay of ssion fragments. The energy these release is much less than that of the ssion process, but in the hours after shutdown enough heat is still produced to melt or damage parts of the reactor (nuclear meltdown) so the ow of cooling water must be maintained. On 28 March 1979 the power reactor at Three Mile Island in Pennsylvania encountered a problem and shut down automatically, but because operators misinterpreted the behavior of the reactor and shut down safety controls that provide cooling in an emergency, it suered a partial meltdown. In the United States and in most countries, reactors are enclosed in a thick concrete containment building so that even if meltdown occurred and contaminated ssion products escaped the reactor (not the case at Three Mile Island), they are kept from spreading. Operator error was also the cause of a reactor accident at Chernobyl on 25 April 1986. One of the reactors in a power station supplying Kiev, the capital of the Ukraine, went prompt critical, with its chain reaction sustained by the uncontrollable prompt neutrons alone. It had a graphite core, and the sudden heat release blew o the top of its enclosure. The core then caught re, generating a smoke plume laced with radioactive ssion products, contaminating a wide area around the station, which was evacuated (and remains so), and also spreading radioactive contamination over parts of Europe.
Breeder Reactors
Chain reactions are possible because a ssion releases more than one new neutron. The fact that the number is typically 2.3 makes possible a breeder reactor, in which each ssion not only provides a neutron to continue the chain, but also an extra neutron to be captured by ordinary 238 U, turning it into plutonium to replace the used-up fuel. Such a reactor could, in principle, use almost all its uranium as fuel. Thorium 232 T could similarly be used to breed 233 U, another possible nuclear fuel; India in particular is interested in such a process, as it has large thorium deposits. The rst commercial power reactor, a relatively small one, started operating in 1957 near Shippingport, outside Pittsburgh, Pennsylvania. It originally used a conventional fuel cycle based on 235 U and sloweddown (thermal) neutrons. In 1977 it was however restructured to successfully breed thorium into 233 U, http://www.phy6.org/stargaze/Sthorium.htm. Power generation ended in 1982, after a run of 25 years, and the reactor was successfully decomissioned and buried in a distant site in Washington State Breeder reactors based on uranium are diicult to design and maintain, because the conversion of 238 U to plutonium is more eicient with fast neutrons (also used in nuclear bombs). They cannot be cooled by water (which slows down neutrons) but operate at high temperatures and are cooled by a metal above its melting point, e.g. liquid sodium. Some such fast breeders were built and ran successfully, but so far very few have been used for power generation.
Tidbits
The uranium mines of Gabon, Africa, have been supplying the French power system with nuclear fuel. In 1972, it was discovered that some uranium deposits from Oklo, Gabon, were slightly depleted in 235 U and contained an unusual variety of isotopes that might have come from nuclear ssion. It is believed that about 1.5 billion years ago, when the concentration of 235 U was higher (its half-life is about 0.8 billion years),
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a natural ssion process was sustained in some of the deposits for a long time (#14). It was caused by water leaking into the deposit and forming a natural moderator. The process was probably cyclicalheat generated by ssion would drive out the water and stop the reaction until fresh water entered again.
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A partial meltdown in 1979 of a nuclear power reactor at Three Mile Island, near Harrisburg, Pennsylvania.
Figure 3.6: This painting by Gary Sheehan illustrates the worlds rst self-sustaining nuclear chain reaction that took place on a squash court beneath Stagg Field on the University of Chicago campus. The rst controlled nuclear reaction was achieved on 2 December 1942. The reactor was a near-spherical pile of pure graphite (carbon) bricks, in which cans of uranium oxide were embedded at xed intervals, and holes were also left for control rods (one of which is being manipulated by the man standing in the center of the image). It was located in a closed space under stadium seating (torn down since then) at the University of Chicago, and the project was led by the Italian physicist (and Nobel laureate) Enrico Fermi. After a successful chain reaction was achieved (kept at a low level, since no cooling was provided), Arthur Compton, one of the leaders of the project, reported by telephone to James Conant in Washington, chairman of the National Defense Research Committee. The project was secret, so he had to improvise. He said (from #15, abbreviated): Youll be interested to know that the Italian navigator has just landed in the new world. Conant replied: Were the natives friendly? Compton: Everyone landed safe and happy. More than 60 years have passed since then and nuclear energy has had an enormous impact. It now supplies most of the electricity in France, and great amounts in the United States, Germany, the United Kingdom, Spain, Russia, and other countries. It can light and heat our homesbut is also capable of frightening destruction, and nuclear waste needs to be held safe for thousands of years. Handle with care. Note: This material was in part taken from the Web collection From Stargazers to Starships, listed at #1 and #12 below. Additional information may be found there.
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Image Sources
(1) CK-12 Foundation. Typical Nuclear Fission. CC-BY-SA. (2) CK-12 Foundation. Nuclear Power Station. CC-BY-SA. (3) Gary Sheehan. Nuclear Chain Reaction Painting. (4) CK-12 Foundation. The Helium Nucleus. Public Domain. (5) CK-12 Foundation. The Binding Energy of Nuclei. Public Domain. (6) CK-12 Foundation. Fuel Rods inside a Neutron Moderator. CC-BY-SA. (7) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf
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4.1 Visual Overview for The Standard Model 4.2 In the Beginning
I can see no escape from the conclusion that [cathode rays] are charges of electricity carried by particles of matter. What are these particles? Are they atoms, or molecules, or matter in a still ner state of subdivision? 1897 Experiments, J. J. Thomson And so it begins, the modern search for the building blocks of matter. What are we made of? What are the smallest constituents of all matter? What do they all have in common? What is dierent? What holds all the matter together? Where did we come from and where are we going? The search for the building blocks goes back to the days of Aristotle and has always had one goal: to simplify our understanding of nature. Aristotle believed that there were four elements that comprised nature: earth, water, air, and re. Democritus, a contemporary of Aristotle, stated that matter could be cut into smaller and smaller halves until you could cut the piece no smaller and it became indivisible. Our present word atom comes from Democritus use of the Greek word for indivisible, atomos. Aristotles theory of the four elements survived until the 18th and 19th century when these four elements were replaced by our modern chemical elements. In the beginning there were a couple dozen elements, but this number soon grew to nearly 100. It appears that science went from a simple model (four building blocks) to a much more complex model (nearly 100 building blocks). Would 100 building blocks all be fundamental? Another change was about to occur with the discovery of the atom and the idea of the indivisible nature of matter returned. The atom was made up of three building blocks and it appeared that a simpler model was restored. This is where our chapter truly beginswith the discovery of these three fundamental particles.
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Figure 4.3: Cathode Ray Tube As a result of Maxwells work in the 1860s it was known that all electromagnetic waves, including visible light, travel at a speed of 3 108 m/s in a vacuum. Experimentation with cathode rays showed that their direction of travel could be altered by placing the tube in a magnetic eld. With these two ideas in mind, J. J. Thomson began his experimentation on the mystery of the cathode rays. In 1894, he decided to experimentally determine the velocity of the cathode rays. The measured velocity could then be compared to the speed of an electromagnetic wave, which could help possibly determine something about its structure. Through the use of mirrors and the cathode rays, Thomson was able to determine the velocity of the rays to be approximately 200, 000 meters per second, which is signicantly less than the speed of light. So, it appeared that cathode rays were not electromagnetic waves, but actually small particles. This result was not widely celebrated by the scientic community, but it did lead to further experimentation by other scientists. The rays are inuenced by a magnetic eld and they travel much more slowly than an electromagnetic wave. From this experimental evidence, one might conclude that the rays are particles. Thomson did not stop at this point. He continued to use electric elds and magnetic elds to determine how much they inuenced the motion of the rays. The rst conclusion that he reached through this line of inquiry was that the rays must be particles or, as he called them, corpuscles. Thomson found that the mysterious stream would bend toward a positively charged electric plate. Thomson theorized, and was later proven correct, that the stream was in fact made up of small particles, pieces of atoms that carried a negative charge. These particles later became known as electrons. Thomson was unable to determine the mass of the electron, but he was able to determine the charge-to-mass ratio, or q/m. He knew the q/m for the hydrogen ion and it was much smaller than the q/m for the cathode rays. He assumed that the mass of the particle was much smaller than the mass of the charged hydrogen atom. Thomson went on and made a bold speculative leap. Cathode rays are not only material particles, he suggested, but in fact the building blocks of the atom: they are the long-sought basic unit of all matter in the universe. 1897 Experiments, J. J. Thomson. www.ck12.org
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Based on Thomsons belief that the atom is divisible and consists of smaller blocks, namely the electron, he then developed a model for the atom. His nding has been called the plum pudding model in which the atom is represented as a positively charged ball with negatively charged particles inside. This model was the accepted explanation for the structure of the atom until Ernest Rutherford and his gold foil experiment in 1911.
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Figure 4.5: Images showing the expected and the actual results from Rutherfords gold foil experiment.
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Marsden. The alpha particles that came close to the nucleus had been deected through varying angles, but the majority of alpha particles passed relatively far away and therefore experienced no deection at all. Over the next 10 years, Rutherford and many other physicists continued to explore the components of the atom. It was widely accepted that positively charged particles were contained within the nucleus. It was believed that the positive charge of any nucleus could be accounted for by an integer number of hydrogen nuclei. Rutherford was the rst to refer to these hydrogen nuclei as protons in 1920.
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have any amount of energy you wantbig waves have a lot of energy, small waves have very little. And if light is a wave, then the brightness of the light aects the amount of energythe brighter the light, the bigger the wave, the more energy it has. The dierent colors of light are dened by the amount of energy they have. If all else is equal, blue light has more energy than red light with yellow light somewhere in between. But this means that if light is a wave, a dim blue light would have the same amount of energy as a very bright red light. And if this is the case, then why wont a bright red light produce a current in a piece of metal as well as a dim blue light? In 1905, Einstein used Plancks revolutionary idea about the quantization of energy and applied it to the photoelectric eect. Although it was universally agreed that light was a wave phenomenon, he realized that the only way to explain the photoelectric eect was to say light was actually made up of lots of small packets of energy called photons that behaved like particles http://www.lon-capa.org/~mmp/kap28/PhotoEffect/photo.htm Photoelectric [Eect Applet]. Einstein was able to explain all the observations of the photoelectric eect. The ejection of an electron occurs when a photon hits an electron and the photon gives its entire energy to the electron. If the photon has suicient energy to transfer to the electron, the electron may be ejected from the atom and a current will start. If the photon does not have enough energy, then the electron will not be supplied enough energy and no current will be produced. The amount of energy each photon can transfer is dependent upon the frequency (color) of the light and not on its brightness. The energy of a photon is determined by Plancks relationship, E = h f . So, no matter how bright the red light may be, the frequency of the red light will not provide it with enough energy to ever eject a photon, no matter how bright or how long that red light shines on the metal. Whereas dim blue light will eject electrons, because the frequency of blue light is large enough to provide enough energy to the photon to eject the electron. With the discovery of the three fundamental particles of the atom and the development of the idea of the photon, it appeared that by 1932 the building blocks of matter had been rediscovered. The hundred dierent building blocks of matter had been replaced by a much simpler view of the physics world. This elegant picture of the physical world did not last for long, though. As technology improved and more questions were posed and eventually answered, many new and rather strange observations were made. The rst and perhaps most bizarre discovery happened right after the neutron was discovered in 1932 and it represented an entirely new type of matter.
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Figure 4.9: An actual cloud chamber picture from Carl Andersons experiment. converted into mass (the pair of particles). Interestingly enough, the reverse is true as well. If an electron and a positron collide, their mass is converted into energy. This process is true for any matter-antimatter pair and is called pair annihilation. With the discovery of the antielectron, the search for other antimatter particles heated up. It seemed reasonable that if the electron had an antiparticle, so too should a proton and a neutron. The methods for probing the reality of subatomic particles began with experiments as simple as those with which the electron, proton, and neutron were discoveredring beams of light or electrons at various substances and then making very precise observations and drawing as many conclusions as possible. Physicists of the early 20th century were able to make some amazing discoveries about the structure of the atom. However, from our point of view, their technology was limited, but they did the best with what they had to work with. In order to discover these new particles, a way to produce controlled, reliable high - energy experiments was needed. This led to the creation of particle accelerators and detectors.
Cosmic Rays
With the discovery of radioactivity in the late 1800s, measurement and detection of this radiation became a driving force in physics. It was soon found that more radiation was being measured on the Earth than was predicted. In an eort to nd the source of this radiation, Victor Hess in 1912 carried detectors with him in a hot air balloon to a height of 5000 meters (without the aid of a breathing apparatus). At this height he was able to discover cosmic rays, which shower Earth from all parts of the universe at incredibly high speeds. Others soon discovered that the rays were actually charged particles, such as alpha particles and protons. As it turns out, these charged particles that zoom through space began their journeys from the Sun, supernovae, and distant stars. Most of the primary cosmic rays are protons or alpha particles traveling at very high speeds. When they hit another nucleus in our atmosphere and stop, many more particles are www.ck12.org
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knocked downward, creating a cascading eect called a shower. When these reactions and the particles that they produced were rst analyzed it was quickly discovered that nothing like this had been seen on Earth before. Thus began a urry of research to discover more about these particles from outer space. Up until the 1950s and the development of particle accelerators, cosmic rays were the primary source of highenergy particles for physicists to study. Carl Anderson not only discovered antimatter through his cosmic ray research, but he went on to discover a particle that had a unit charge with a mass between the electron and proton. Muons were later shown not to have any nuclear interactions and to be heavier versions of electrons. In 1947, Cecil Powell discovered another particle that did interact with nuclei. The mass of this new particle was greater than the muon and it was soon determined that the particle would decay into a muon. This new particle was given the name pi-meson, or pion. A few months later, new particles with masses in between the pion and the proton were discovered. The kaon was a strange new particle that was always produced in pairs, had a relatively long lifetime, and decayed into pions and muons. Although a number of exciting new particles were discovered with the cosmic rays, there were limitations to this type of research. Interesting events happen very rarely and when they do it is very diicult to catch them in a particle detector. Researchers have no control over when or where the cosmic ray shower will occur, making it very diicult to perform experiments. The other problem that was quickly becoming apparent was that all the low energy events seemed to be well researched and that the interesting events were the highenergy events. The problem with the highenergy events was that they were incredibly rare. So, the lack of control over when and where these events would occur and the infrequent highenergy cosmic ray events posed a problem for researchers. Physicists needed to come up with a solution to these problemsnamely, to create controlled highenergy experiments in a laboratory-type setting.
Particle Accelerators
Particle accelerators were designed to study objects at the atomic scale. Particle accelerators allow for millions of particle events to occur and to be studied without waiting for the events to come from the sky. Accelerators do for particle physicists what telescopes do for astronomers. These instruments reveal worlds that would otherwise be left unseen. Vacuum tubes and voltage dierences accelerated the rst electrons and then the Cockcroft-Walton and Van de Graa machines were invented using the same principles only on a grander, more complex scale. A modern example of this type of device is the linear accelerator, such as the Stanford Linear Accelerator (SLAC). In order to achieve high energies, all linear accelerators must be very long. For example, the Stanford Accelerator is nearly 2 miles long and actually crosses under a highway in California. SLAC is able to achieve energies of up to 50 GeV. An electron volt (eV) is a unit of energy that is equivalent to 1.6 1019 J. A GeV is equal to 109 eV. The need for such great length to achieve the high energy is a major limitation with this type of accelerator.
Figure 4.11: Stanford Linear Accelerator Center, Palo Alto, CA The great breakthrough in accelerator technology came in the 1920s with Ernest O. Lawrences invention of the cyclotron. In the cyclotron, magnets guide the particles along a spiral path, allowing a single electric eld to apply many cycles of acceleration. The rst cyclotrons could actually t in the palm of your hand and could accelerate protons to energies of 1 MeV. Over the next decade or two, unprecedented energies www.ck12.org
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were achieved (up to 20 MeV), but even the cyclotron had its limitations due to relativistic eects and magnet strength. Fortunately, the same type of technology that allows for a cyclotron to work also works in the next version of the accelerator, a synchrotron. The synchrotrons circular path can accelerate protons by passing them millions of times through electric elds allowing them to obtain energies of well over 1 TeV. The rst synchrotron to break the TeV energy level was at Fermilab National Accelerator Laboratory (Fermilab). The Tevatron at Fermilab is nearly 4 miles in circumference and can accelerate particles to 1 TeV in each direction around the ring.
Figure 4.13: Jeerson National Accelerator Laboratory, Newport News, VA The last advancement in accelerator technology involved the collision of the accelerated particles. Up until the 1970s, all accelerators were xed target machines. This means that the very energetic particles collide with a stationary target and all the newly produced particles continue moving in the same direction as the
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debris, the new particles and energy, which comes from the collision. As a result, not all of the mass-energy that derives from the accelerated particles is available to be converted into new particles and new reactions. Some of the mass - energy is lost into the target and not all of it is transferred into the particle collisions. Early in the 1960s, physicists had learned enough about accelerators to build colliders. In a collider, two carefully controlled beams pass around the synchrotron in opposite directions until they are made to collide at a specic point. Although colliders are signicantly more challenging to build, the benets are great. In a collider, the accelerating particles moving in opposite directions are brought to a point for the collision and because they are traveling in opposite directions their collision energy is greater than a xed target collision and the net momentum is zero. This means that all their energy is now available for new reactions and the creation of new particles. For example, although the Tevatron at Fermilab can only accelerate the protons and antiprotons to energies of 1 TeV, the energy that is involved in each proton-antiproton collision will approach 2 TeV. Why the need to achieve such high energies? Highenergy physicists know that it takes particles with energy about 1 GeV to probe the structure inside of a proton. In order to get to the even smaller parts of matter, higher energy is needed. Also, higher energies would allow for more massive particles to be created. Currently, the Fermilabs Tevatron has enough energy to produce the top quark ( 170 GeV). If particle physicists want to learn more about the building blocks of matter they need more energy. Over the past decade in Geneva, Switzerland, they have been trying to accomplish just thatto build the worlds largest particle accelerator. In 2009, the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) is scheduled to go online. The LHC is 27 km in circumference and will accelerate particles to energies approaching 7 TeV. This means that at the collision point the energy will be up to 14 TeV and the potential for new particle discoveries are endless.
Figure 4.14: Section of the Large Hadron Collider tunnel, CERN, Geneva, Switzerland
Particle Detectors
The rst particle detectors resembled the ones used by Rutherford in his famous gold foil experiment. The detectors involved the emission of light when charged particles hit a coated screen. Other methods for detecting radiation were soon developed, such as electroscopes (that could tell if a charged particle was present) and Geiger counters (which counted how many charged particles were present). All of these detectors could only tell if a charged particle was present and/or provide a rough approximation to how www.ck12.org
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many charged particles were present. They were all incapable of providing any specic information about the properties of the charged particles. Then a breakthrough came in 1912 when the cloud chamber was invented. The cloud chamber involved producing a vapor that remained in a supersaturated state. C. T. R. Wilson, a Scottish physicist, developed a cloud chamber based on his studies of meteorology and his research into the atmosphere and cloud formation. It was well known that an electrical charge could cause condensation in this kind of supersaturated state. Wilson was eager to nd out if he could produce a similar eect with Xrays. In 1896, he performed an experiment and found that, like electricity, Xrays could induce condensation in the supersaturated vapor. In 1912, he incorporated all of his ideas into a device that he called a cloud chamber. He found that radiation from a charged particle left an easily observable track when it passed through the cloud chamber. The track was a result of the interaction between the charged particles and the air and molecules within the container. This interaction resulted in the formation of ions on which condensation occurred. This provided a plain view of the path of the radiation and so gave a clear picture of what was happening. The events could then be viewed by taking a photograph of them. When used, the cloud chamber is placed between the poles of a magnet. The magnetic eld causes particles to bend in one direction or another, depending on the electrical charge they carry. The magnetic eld B, the velocity v, the radius of the circular orbit R, the mass m, and the charge q are related by the formula: R = mv . qB The kind of particles that have passed through the chamber can be determined by the types of tracks they leave. Although the cloud chamber had many useful applications, it was replaced by the bubble chamber that was invented in 1953 by Donald Glaser. The bubble chamber is a more sophisticated version of the cloud chamber. Glasers idea was to use a liquid, like liquid hydrogen, as a detecting medium because the particles in a liquid are much closer together than are those in a gas. Glasers bubble chamber is essentially the opposite of a cloud chamber. It contains a liquid that is heated beyond its normal boiling point. If the liquid is kept under pressure it will not boil. Instead, it will remain in a superheated state. Particles released from the radioactive source will travel through the bubble chamber and interact with atoms and molecules in the liquid. This interaction will result in the formation of ions, atoms, or molecules that carry an electrical charge. The ions act as nuclei on which the liquid can begin to boil. The path taken by the particle as it moves through the bubble chamber is marked by the formation of many very tiny bubbles, formed where the liquid has changed into a gas. At this moment, the camera records the picture. Bubble chambers were widely used to study nuclear and particle events until the 1980s. For a long time, bubble chambers were the most eective detectors in particle physics research. Bubble chambers were very eective, but they did require a picture to be taken and then analyzed. With the improvement in technology, it became desirable to have a detector with fast electronic read-out. Bubble chambers, thus, have largely been replaced by wire chambers, which allow particle numbers, particle energies, and particle paths to be measured all at the same time. The wire chamber consists of a very large number of parallel wires, where each wire acts as an individual detector. The detector is lled with carefully chosen gas, such that any charged particle that passes through the tube will ionize surrounding gaseous atoms. The resulting ions and electrons are accelerated by an electric potential on the wire, causing a cascade of ionization, which is collected on the wire and produces an electric current. This allows the experimenter to count particles and also determine the energy of the particle. For high - energy physics experiments, it is also valuable to observe the particles path. When a particle passes through the many wires of a wire chamber it leaves a trace of ions and electrons, which drift toward the nearest wire. By noting which wires had a pulse of current, an experimenter can observe the particles path. The wire chamber became one of the main types of detectors in modern particle accelerators. They were much more eective at collecting information about the particle events and in storing them to be analyzed at a later time. A bubble chamber could only produce one picture per second and that picture could not be stored in a computer. A typical wire chamber could record several hundred thousand events per second,
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which could then be immediately analyzed by a computer. The ability to collect hundreds of thousands of events and allow those events to be quickly analyzed and stored on a computer led to the creation of the magnicent modern particle detectors.
Figure 4.15: Schematic of the Compact Muon Solenoid Detector, CERN, Geneva, Switzerland The Compact Muon Solenoid (CMS) is one of the two major detectors of the LHC (the other one is called ATLAS). Each of these detectors is quite similar in their general features and in their ability to collect and quickly analyze millions of particle events per second. CMS is 21 m long, 15 m wide, and 15 m high and it weighs 12, 500 tons. The huge solenoid magnet that surrounds the detector creates a magnetic eld of 4 Teslas, this is about 100,000 times the strength of the Earths magnetic eld. CMS is an excellent example for illustrating the construction of a modern particle detector. The various parts are shown in Figure 13 with a brief description following.
Tracker
Purpose is to make a quick determination of particle momentum and charge. The tracker consists of layers of pixels and silicon strips. The pixels and strips cover an area the size of a tennis court. 75 million separate electronic read-out channels, 6, 000 connections per square centimeter. The tracker records the particle paths without disturbing the energy or motion of the particle. Each measurement that the tracker takes is accurate to 10 m , a fraction of the width of a human hair. The tracker can re-create the paths of any charged particle; electrons, muons, hadrons, and short-lived decay particles.
Electromagnetic Calorimeter
Purpose is to identify electrons and photons and to do it very quickly (25 ns between collisions). Very special crystals are used that scintillate, momentarily uoresce, when struck by an electron or photon. These high-density crystals produce light in fast, short, well-dened photon bursts that is proportional to the particles energy. The barrel and the endcap of the detector are made up of over 75, 000 crystals. www.ck12.org
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Hadron Calorimeter
Purpose is to detect particles made up of quarks and gluons, for example protons, neutrons, and kaons. Finds a particles position, energy, and arrival time. Uses alternating layers of brass absorber plates and scintillator that produce a rapid light pulse when the particle passes through. The amount of light measured throughout the detector provides a very good measurement of the particles energy. There are 36 barrel wedges, each weighing 26 tons.
Muon Detector
The purpose of the muon detector is to detect muons, one of the most important tasks of CMS. Muons can travel through several meters of iron without being stopped by the calorimeters, as a result the muon chambers are placed at the very edge of the detector. Due to the placement of the muon chambers the only particles to register a signal will be a muon. The muon chambers have a variety of detectors that help track these elusive particles.
Computing
One billion proton-proton interactions will take place inside the detector every second. A very complex trigger system will be set up in the computers to eliminate many of the events that are not interesting to the physicists. Only less than 1 percent of all interactions will be saved to a server. Nearly 5 petabytes, a million gigabytes, of data per year will be saved when running at peak performance. To allow for the storage of all this data, a worldwide grid has been created that uses tens of thousands of regular computers. This distribution of the data allows for a much greater processing capacity than could ever be achieved by a couple of supercomputers. The other benet is now that the data are capable of being stored all over the world; physicists do not need to be at a central location (for instance CERN), in order for them to analyze the particle events coming from CMS.
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laws, he was even able to predict its properties. The neutrino must be neutral and the neutrinos rest mass must be very, very small. Although many scientists did not expect it to take long for the neutrino to be detected, it took over 25 years for their existence to be conrmed. In 1956, Clyde Cowan and Frederick Reines nally detected neutrinos using radiation coming from the Savannah nuclear reactor. The properties of the neutrino were conrmed through the study of the results of this experiment. An interesting note is that the reason it took so long for the neutrino to be detected, and continues to be quite elusive to detect to this day, is due to the fact that the neutrinos interaction with other particles is so weak that only one of a trillion neutrinos passing through the Earth is stopped.
Hadrons
With the explosion of new particles being detected from the 1950s to present times, it might appear that once again the simplied model of the early 1900s has become more complicated. It got so bad when over 150 new particles were identied, that physicists started referring to it as the particle zoo. It isnt quite as bad as that, though. Just like zookeepers build order in their zoos by grouping the animals based on biological categories like genus and species, particle physicists started looking for a way to group all the particles into categories of similar properties. The observed particles were divided into two major classes: the material particles and the gauge bosons. Well discuss the gauge bosons in another section. Another way to divide the particles was through the interactions in which they participated. The material particles that participate in the strong force are called hadrons and particles that do not participate in the strong force are called leptons. The strong force is one of the fundamental forces of nature. A discussion of the properties of the leptons may be found later in this chapter. Most of those 150+ particles are mesons and baryons, or, collectively, hadrons. The word hadron comes from the Greek word for thick. Most of the hadrons have rest masses that are larger than almost all of the leptons. Hadrons still are extremely small but, due to their comparatively large size particle, physicists think that hadrons are not truly elementary particles. Hadrons all undergo strong interactions. The dierence is that mesons have integral spin (0, 1, 2 . . .), while baryons have half-integral spin (1/2, 3/2, 5/2 . . .). The most familiar baryons are the proton and the neutron; all others are short-lived http://hyperphysics.phy-astr.gsu.edu/Hbase/particles/baryon.html#c1 [Table of Baryons]. The most familiar meson is the pion; its lifetime is 26 nanoseconds, and all other mesons decay even faster http://hyperphysics.phy-astr.gsu.edu/Hbase/particles/meson.html#c1 [Table of Mesons].
Quarks
The rapid increase in the number of particles soon led to another question: Is it reasonable to consider that all of these particles are fundamental? Or, is there a smaller set of particles that could be considered fundamental? To many physicists the idea of something even smaller making up hadrons seemed to be reasonable as experimental evidence supported the notion that the hadrons had some internal structure. In 1964, the most successful attempt to build the hadrons, the quark model, was developed by Murray Gell-Mann and George Zweig. The original quark model started with three types, or avors, of quarks (and their corresponding antiquarks). The rst three quarks are currently called up (u), down (d), and strange (s). Each of these quarks has spin 1/2, andthe most radical claim of the modela fractional charge when compared to the elementary charge of an electron. The fractional charge of the quark should make the quarks easy to nd, but that has not been the case. No single quark has ever been detected in any particle experiment.
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Regardless, the quark model has been very successful at describing the overall properties of the hadrons. In order to make a hadron, the quarks must be combined in a very specic way. The baryons are all made up of three quarks (the antibaryons are made up of three antiquarks). As an example, the proton is made up of two up quarks and one down quark and a neutron consists of two down quarks and one up quark. The mesons are all made up of one quark and one antiquark. For example, the positive pion is made up of one up quark and one anti-down quark. To make a particle out of quarks, or to determine the quarks of a known particle, it is simply a matter of checking the particle and quark properties in a chart and using some simple addition [make a Hadron Applet]. In 1974, a new particle was discovered that could only t the quark model if a fourth quark was added. The quark was given the name charm (c). In 1977, a fth quark was added, bottom (b), and nally in 1995 the existence of a sixth quark was conrmed, top (t). The six quarks of the quark model have all been veried and supported by experiments, but the existence of more quarks is still an open question in particle physics. Table 4.1: Flavor Down Up Strange Charm Bottom Top Symbol d u s c b t Charge 1/3 +2/3 1/3 +2/3 1/3 +2/3
Leptons
At almost the same time that the quark model was being developed another group of particles appeared to have a similar symmetry with the quarks. These particles, called leptons (Greek for light), appeared to be fundamental and seemed to match up in number to the quarks. Leptons are particles that are like the electron: they have spin 1/2, and they do not undergo the strong interaction. There are three avors of charged leptons: the electron, the muon, and the tau. They all have negative charge, and with the exception of the tau, are less massive than hadrons. The electron is the most stable and can be found throughout ordinary matter. The muon and the tau are both short-lived and are typically only found in accelerator experiments or cosmic ray showers. Each charged lepton has an associated neutral lepton partner. They are called the electron neutrino, the muon neutrino, and the tau neutrino. Neutrinos have almost zero mass, no charge, interact weakly with matter, and travel close to the speed of light. Each of these six particles has an associated antiparticle of opposite charge, bringing the total number of leptons to twelve. Table 4.2: Flavor Electron Electron neutrino Muon Muon neutrino Tau Tau neutrino Symbol e e Rest Mass ( MeV/c2 ) .511 0 105.7 0 1784 0
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Conservation Laws
Conservation laws apply in the particle world just as much as they apply in the macroscopic world. The conservation of momentum, mass-energy, angular momentum, and charge are all required by the particle events that have been discovered over the past 100 years. The importance of these conservation laws allowed for the prediction of the neutrino, as we saw earlier in this chapter. Any reaction that occurs must satisfy these laws. Look at the following two possibilities for beta decay: n p + e + e v n p + e+ + e v Which of the two decays will actually occur? What conservation law(s) does the other decay violate? The conservation of mass-energy is a little tricky. Due to Einsteins principle of mass-energy equivalence, mass may be converted into energy and vice versa. Because energy can be converted into mass, when two moving particles collide it is possible that the incident kinetic energy will be converted into mass during the collision. In this case, the masses of the product particles may be greater than the masses of the incident particles. So, it is very diicult to determine if mass-energy is conserved in a particle interaction, because there is no way of knowing just how much kinetic energy each particle has to start with and how much of that energy is converted into mass. Although, typically when a particle decays into other particles, it can be shown that the sum of the masses of the product particles will be smaller than or equal to the rest mass of the particle that decayed. As more and more particles were discovered and more and more particle events analyzed it became increasingly clear that more conservation laws were necessary to help explain what was seen, and maybe more importantly, what was not seen. One of the most important of these is the conservation of baryon number. Each of the baryons is assigned a baryon number B = +1, antibaryons a baryon number B = 1, and all other particles a value of B = 0. In any reaction the sum of the baryon numbers before the interaction or decay must equal the sum of the baryon numbers after. No known decay process or interaction in nature changes the net baryon number. For example, suppose a positive pion collided with a neutron, which result could not happen? + + n p + 0 pi+ + n + + + + Because the baryon number in the rst interaction is +1 before and +1 after, this interaction could occur. But, the second interaction has a +1 baryon number before and a baryon number of zero after, so this interaction cannot take place. The decay of a proton could not proceed by the following event, because the baryon number is not conserved. p + + As a matter of fact, because the proton is the baryon of smallest mass it may not decay at all. Conservation of baryon number would require that any product of proton decay to have greater mass than the proton, and this would not be allowed due to conservation of mass-energy. As physicists continue to explore the particle world new discoveries may be made and new conservation laws may be created to allow for the decay of a proton, but for now a proton is considered stable. Also, there is not a conservation of meson number. Mesons can be involved in any particle event as long as they do not violate the other conservation laws. www.ck12.org
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There is a conservation law for leptons, but it is slightly more complicated than for the baryons. To rst see how the lepton number is conserved; let us look at this variation of beta decay: n p + e + ve This event has never been observed, but according to all the other conservation laws there is no reason that it could not be. Conservation of lepton numbers require that all leptons and corresponding neutrinos be assigned a lepton number of +1, the antileptons and antineutrinos a lepton number of 1, and all other particles a lepton number of 0. Looking at the example above, the lepton number before the event is 0 and the lepton number after the event is +2, so lepton number is not conserved. How could you conserve lepton number and make a valid reaction in the decay shown above? A look at the following decay shows that there is a little bit more to the conservation of lepton number: n p + e + v Following the rules of lepton number conservation, the preceding example could be observed, but it never has been. There must be something more to the conservation of lepton number and that is each lepton and neutrino partner are assigned its own specic number. So, there is a separate conservation of electron lepton number, muon lepton number, and tau lepton number. Because there are actually three lepton numbers that need to be conserved, the above example will not happen. If this reaction were to take place, electron lepton number and muon lepton number are both not conserved. The decay begins with an electron lepton number of 0 and ends with an electron lepton number of +1; also it begins with a muon lepton number of 0 and ends with a muon lepton number of 1. Clearly, this decay cannot proceed because it violates not one, but two lepton conservation laws. A summary of the lepton numbers is shown in the table below (Note: all of the anti leptons have a lepton number of 1) Table 4.3: Lepton e e Conserved Le Le L L L L Quantity Lepton Number +1 +1 +1 +1 +1 +1
Fundamental Interactions
There are four fundamental forces within all atoms that dictate interactions between individual particles and the large-scale behavior of all matter throughout the universe. They are the strong and weak nuclear forces, the electromagnetic force, and gravity. Gravitation is a force of attraction that acts between each and every particle in the universe. Gravity is the weakest of all the fundamental forces. However, the range of gravity is unlimitedevery object in the universe exerts a gravitational force on everything else. The eects of gravity depend on two things: the mass of two bodies and the distance between them. In more precise terms, the attractive force between any two bodies is directly proportional to the product of the masses and inversely proportional to the square of
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the distance between the bodies. It is always attractive, never repulsive. It pulls matter together, causes you to have a weight, apples to fall from trees, keeps the Moon in its orbit around the Earth, the planets conned in their orbits around the Sun, and binds together galaxies in clusters. The electromagnetic force determines the ways in which electrically charged particles interact with each other and also with magnetic elds. Like gravity, the range of the electromagnetic force is innite. Unlike gravity, electromagnetism has both attractive and repulsive properties that can combine or cancel each other out. Whereas gravity is always attractive, electromagnetism comes in two charges: positive and negative. Two positive or two negative things will repel each other, but one positive and one negative attract each other. The same rule applies for magnets, as well, and can be easily demonstrated when two magnets are placed near each other. A north pole near a north pole will cause a repulsive force and a north pole placed near a south pole will cause an attractive force to develop. The electromagnetic force binds negatively charged electrons into their orbital shells, around the positively charged nucleus of an atom. This force holds the atoms together. The strong nuclear force binds together the protons and neutrons that comprise an atomic nucleus and prevents the mutual repulsion between positively charged protons from causing them to y apart. The strong force is the strongest of the fundamental forces, but it is also very short range, limited to nuclear distances. It is also responsible for binding quarks into mesons and baryons. An interesting feature of the strong force is that the strength of the force behaves like a rubber band. It actually gets stronger as the quarks move apart, but just like a rubber band, it will eventually break apart when stretched too far. Unlike a rubber band, when the strong force breaks, new quarks are actually formed from the newly released energy. This process is called quark connement. There has never been an experiment that has found a quark in isolation.
Figure 4.19: Quark connement The weak nuclear force causes the radioactive decay of certain particular atomic nuclei. In particular, this force governs the process called beta decay, whereby a neutron breaks up spontaneously into a proton, an electron, and an antineutrino. It operates only on the extremely short distance scales found in an atomic nucleus. According to modern quantum theories, forces are due to the exchange of force carriers. The various fundamental forces are conveyed between real particles by means of particles described by physicists as virtual particles. Virtual particles essentially allow the interacting particles to talk to one another without exchanging matter. The forcecarrying particles, or bosons, for each of the forces are as follows: electromagnetic forcephotons; weak nuclear interactionvery massive W and Z particles; and the strong nuclear interactiongluons. Although it has not been possible to devise a completely satisfactory theory of gravitation, it too should have an exchange particlethe graviton (which has not yet been www.ck12.org
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discovered).
Figure 4.20: The Standard Model of Particle Physics The standard model currently has sixteen particles. Twelve of the particles are fermions, or matter particles and they are the six quarks and six leptons. Each elementary particle also has an antimatter partner. The remaining four particles are called bosons and are the exchange particles through which the four fundamental interactions are transmitted. The hypothetical exchange particle for gravity, the graviton, does not currently have a place in the standard model. Every phenomenon observed in nature can be understood as the interplay of the fundamental forces and particles of the standard model. Interesting to note, that although the standard model does a terric job at explaining all the matter and forces that occur in nature, nearly 85% of all matter that makes up the universe has still not been discoveredthe elusive dark matter. But physicists know that the standard model is not the end of the story. It does not account for gravity and the mysterious dark matter. The standard model also requires the existence of a new particle, known as the Higgs boson. The existence of this particle is essential to understand why the other building blocks (the quarks, the leptons, and the gauge particles) have mass. The Higgs has not yet been seen in any experiment. As the experiments become grander in scale and the discoveries multiply, will the standard model be supported or does a new model need to be developed? The standard model raises almost as
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many questions as it answers. Today physicists all over the world are searching for physics beyond the standard model that may lead to a possibly more elegant theorya theory of everything.
Image Sources
(1) CK-12 Foundation. Quark Combinations for Various Hadrons. CC-BY-SA. (2) CK-12 Foundation. Thomsons Plum Pudding Model. CC-BY-SA. (3) Unknown. Section of the Large Hadron Collider tunnel, CERN, Geneva, Switzerland. CC-BY-SA. (4) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (5) CK-12 Foundation. The Standard Model Cribsheet #2. CC-BY-SA. (6) Quark connement. CC-BY-SA. (7) Unknown. Stanford Linear Accelerator Center, Palo Alto, CA. Public Domain. (8) CK-12 Foundation. The Beta Decay Dilemma. CC-BY-SA. (9) CK-12 Foundation. The Standard Model Cribsheet #1. CC-BY-SA. (10) CK-12 Foundation. Rutherfords gold foil scattering experiment. CC-BY-SA. (11) CK-12 Foundation. Cosmic Ray Shower. CC-BY-SA. (12) The Standard Model of Particle Physics. CC-BY-SA. (13) Unknown. Cloud Chamber. GNU Free Documentation License. (14) Unknown. Fermi National Accelerator Laboratory, Batavia, IL. Public Domain. (15) CK-12 Foundation. Cathode Ray Tube. CC-BY-SA. (16) CERN. Schematic of the Compact Muon Solenoid Detector, CERN, Geneva, Switzerland. CC-BY-SA. (17) Jeerson National Accelerator Laboratory, Newport News, VA. Public Domain. (18) CK-12 Foundation. Rutherfords Planetary Model of the Atom. CC-BY-SA. (19) CK-12 Foundation. Simulation of a Higgs Boson Event in CMS Detector. Public Domain. www.ck12.org
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(20) CK-12 Foundation. Expected and actual results from Rutherfords gold foil experiment. CC-BY-SA. (21) CK-12 Foundation. The Photoelectric Eect. CC-BY-SA.
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The amount of charge on a particle is described using a unit called a coulomb. When the electron was believed to carry the smallest size charge, (1.602 1019 C), physicists created a unit of energy to match the electrons charge. It is called the electron voltabbreviated eV. An eV is equal to 1.602 1019 joules. Instead of saying a particle carries an energy of 1.602 1019 J or 3.204 1019 J, physicists now can say a particle carries an energy of 1 eV or 2 eVs respectively. One of the nice aspects of the electron volt is that it also relates the energy gained by an accelerating particle to the potential dierence it crosses. This is the mechanism that a linear accelerator uses to accelerate a charged particle. One particle with a charge equal to an electron, changes its kinetic energy by 1 eV when it accelerates between two plates connected to a 1 volt potential dierence, shown in Figure 1. Colliders are large machines designed to smash small charged particles such as protons, electrons, and the nucleus of atoms at extreme speeds. The colliders send particles into each other or into a stationary target. These moving particles have kinetic energy, KE = 1 mv2 , where KE is the kinetic energy of the particle, m 2 www.ck12.org
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Figure 5.1: A particle with a net electric charge equal to one electron gains one of energy after crossing the metal plates. It gains instead of loses the energy because the plate (on the right) opposite the electron is oppositely charged and attracts the negatively charged electron. is the particles mass in kilograms, and v is the particles velocity. An object has kinetic energy as long as it has velocity. One of the ways colliders are classied is by the kinetic energy of the collisions. Because the particles are very small with masses typically in the range of 1028 kg the kinetic energies are measured in eV s. But the collisions are millions or billions of eV s not just 10 or 20. The collisions energies are listed using the prexes listed below. Table 5.1: Prex MeV GeV T eV Pronunciation Mega eV s Giga eVs Tera eV s Number Millions of eV s Billions of eV s Trillions of eV s Math Expressions 106 eVs 109 eVs 1012 eVs
Einstein showed that a particle at rest has a rest energy given by E = mc2 , where m is the mass of the particle measured in kilograms, c is the speed of light, 3.00 108 m/s, and E is the rest energy. The rest energy is measured in the standard S.I. unit of joules. If an object of mass, m, was annihilated (destroyed), then this formula would describe how much energy would be released. This equation shows that the mass and energy are equivalent: It allows physicists to quantify the mass of an object in terms of energy. Example: The mass of a proton is 1.67 1027 kg. What is the energy associated with protons mass in units of joules and eV s? Solution: We use E = mc2 with the speed of light c = 3.00 108 m/s. Then E = (1.67 1027 kg)(3.00 108 m/s)2 E = 1.50 1010 J E = 938, 000, 000 eV
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Using the prexes shown above, this is typically written as 938 MeVs.
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universe cooled down, the quarks began to regroup into protons and neutrons. Today, the universe in our location is too cool for quarks to oat around by themselves. The collider will do two things to solve this. First, it will accelerate protons or electrons to such high speeds that the energy of the charges at impact will be converted into thermal energy. Second, the energy of the particles at impact will be converted into new particles. To generate these high levels of energy, charged particles are accelerated into each other at speeds near the speed of light. Nothing can start out slower than the speed of light and then accelerate to a speed faster than the speed of light. However, large electric elds are used to accelerate charged particles to speeds near the speed of light. There are two basic design geometries. The linear accelerator has charged particles that travel down a straight line. The particles can start at opposite ends of a long tunnel and collide into each other. The linear accelerator works best with electrons because they are a thousand times lighter than protons. A high percentage of the energy put into the accelerator goes into speeding up the charge (Schwartz, 1997). But electrons generate large amounts of synchrotron radiation. Protons generate less radiation but cannot achieve the same velocities. Synchrotron radiation is caused any time a charged particle accelerates. When a particle accelerates in a straight line it is called brehmsstrahlung radiation. The (simplied) formula for calculating the radiations 2 power is: P = 2ke3 2 a2 , where k is Coulombs constant, e is the elementary charges value, c is the speed 3c of light, = 1 (v/c)2 is a factor to account for relativistic speeds, and a is the acceleration. (When the speed is less than 10% the speed of light, 1). This equation applies, for example, for the power radiated by a (radio-) antenna. When a particle accelerates in a circle or curve it is called synchrotron radiation. 2 The same formula applies except the acceleration is found from: ac = r This means for circular motion: 2 4 P = 2ke3 2 2 Because the varies with speed, the -factor for an electron moving near the speed of light 3c r can be 1013 times greater than for a proton. This means that accelerating electrons is more diicult than the accelerating protons. In order to keep synchrotron radiation as small as possible protons are used and as the speed increases the radius must also increase. If the charges were placed in an energized ring, then they could continually be pumped up with energy to reach relativistic speeds. Because the proton generates less synchrotron radiation, it would make for a more viable candidate for acceleration in a circular collider.
Overview
When particles move at relativistic speeds, their energies are large enough to generate new particles when colliding with other particles. Huge amounts of energy can also overcome the strong nuclear force holding particles together. This may allow scientists to see whats inside the protons and neutrons. To achieve these high energies, a bigger collider needs to be built. CERN is the French acronym for European Nuclear Research Centre. This collider is located at the foot of the Jura mountains straddling the border between France and Switzerland (CERN, 2009). CERN built www.ck12.org
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its rst synchrotron accelerator in the late 1950s. The rst synchrotron gained notoriety in 1959. Since then several new colliders have been built on top of existing colliders at CERN. The new colliders either use the previously built colliders for pre-staging or the existing tunnels. The current LHC is no dierent. It uses the tunnels that were nished in 1989 for the LEP, Large Electron-Positron Collider. The LEP ceased running in November 2000 to make room for construction of the LHC (CERN Courier, 2001). The LHC is retrotting the LEPs tunnels with the most advanced superconducting magnets and updating its detectors to collect new data. There are currently six experiments requiring six dierent detectors at the LHC (CERN, 2009). When Einstein came up with his theory of general relativity he could not foresee the practical applications of this theory today. But a hundred years later, the theory of general relativity is used to calculate your position on the planet using a GPS-enabled device, (TED, Patricia Burchat: The Search for Dark Energy and Dark Matter, 2008). The LHC is doing science for the sake of education to answer some of the big questions such as: What causes mass? What is dark matter? Are there more than three spatial dimensions? The implications in science and technology of these answers is not yet known. But in a hundred years, it may have a profound eect on society (TED, Brian Cox: An Inside Tour of the Worlds Biggest Supercollider, 2008). ALICE: A Large Ion Collider Experiment Collisions in this section will be 100, 000 hotter than the sun. Looking for the particle responsible for mass. Investigating of quarks can be freed from protons and neutrons (CERNALICE Collaboration). Size: 26 m long, 16 m high, 16 m wide (CERN, 2008). Mass: 10, 000 tons (CERN, 2008). Look up ALICE on Google Earth to see its location.
ATLAS: A Toroidal LHC ApparatuS It is a general purpose detector. Looks at mass while searching for evidence of: the Higgs particle responsible for mass. dark matter. The ATLAS is the largest particle detector in the world (CERNATLAS Experiment 2008). Size: 46 m long, 25 m high, and 25 m wide (CERN, 2008). Mass: 7000 metric tons (CERN, 2008). Look up ATLAS on Google Earth to see its location.
CMS: Compact Muon Solenoid It is a general purpose detector. Looks at mass while searching for evidence of: the Higgs particle responsible for mass. dark matter.
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Unlike the ATLAS it will look for this evidence using dierent techniques (CERNCMS Outreach). It generates a magnetic eld 100,000 times stronger than the Earths. Size: 21 m long, 15 m wide, and 15 m high (CERN, 2008). Mass: 12, 500 metric tons (CERN, 2008). Look up CMS on Google Earth to see its location.
LHCb: Large Hadron Collider Beauty Looking to answer the question of why is there so little antimatter in our region of the universe (CERNLHCb Experiment, 2008). Size: 21 m long, 10 m high, and 13 m wide (CERN, 2008). Mass: 5600 metric tons (CERN, 2008). TOTEM: TOTal Elastic and Diractive Cross Section Measurement Looks at the size of the particles and the beams luminosity. This will complement the CMSs data and give some quality assurance. Size: 440 m long, 5 m high, and 5 m wide (CERN, 2008). Mass: 20 metric tons (CERN, 2008).
LHCf: Large Hadron Collider Forward Produces cosmic rays under laboratory conditions to look at how cosmic rays interfere with our atmosphere. Two detectors. Size: 30 cm long, 80 cm high, and 10 cm wide. Mass: 40 kg each.
Overview
Several scientists have called the LHC the largest scientic experiment in the world (Cox, 2008). To successfully accelerate the particles to relativistic speeds, the particles must be energized in stages. The circular geometry of the LHC and the fact that it is built using previous machines makes this possible. When launching a rocket to the moon, the rocket has multiple stages. Each stage pushes the rocket a little faster. The LHC does something similar to get the protons up to speed (www.YouTube.com What is CERN Large Hadron Collider LHC? End of the World? Search for God Particle and Micro Black Holes, 2008). www.ck12.org
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Figure 5.2: In the early stages, hydrogen gas is ejected into a chamber. Using electricity to generate a large electric eld, the electrons are stripped from the atom. Protons are then sent into the linear accelerator. This is stage one of ve for the process. This collection of charges contains protons. This collection is called a The device accelerating the bunch is called the lineac 2. By the time the proton bunch reaches the end of the tube it will be traveling at the speed of light. That is fast enough to go around the Earths equator two and a half times in one second. The charge injection process is repeated to create a collection bunches. This many bunches creates a beam of protons.
Figure 5.3: Upon leaving the lineac 2, the proton bunch enters stage 2. This booster stage consists of rings with a radius of meters. The packets are accelerated by electric elds. The electric elds are pulsed in such a way to speed up the packets and more tightly pack the protons together. Powerful magnets with a B-eld perpendicular to the direction of motion steer the packets in the circular rings. The packet leaves this stage at % the speed of light.
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Figure 5.4: Now in stage 3 of the acceleration, the packet is in the proton synchrotron. In this ring the bunch gets closer to the speed of light. Upon leaving this ring the protons will move as if they are 25 times heavier than when they were at rest. The proton will stay in this ring for seconds and reach a speed of % the speed of light before leaving the ring. Each proton will leave the ring with .
Figure 5.5: In stage 4 of the acceleration process the bunch enters a larger ring. This ring is called the super proton synchrotron. It has a radius of about . Energy added in this ring will increase the mass of the proton to 450 times its resting mass. At this point, each proton will leave the ring with an energy of . When the bunches leave this stage, half will enter the large ring traveling clockwise. The other half will leave the ring traveling counterclockwise (See Figure 6).
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Figure 5.6: The packet enters the large ring. The large ring has a radius of . The protons will travel around 11,000 times per second. This large ring contains two tunnels. The beams will travel in opposite directions until they are directed to a location for a head-on collision. Each proton will reach an energy level of while traveling at % the speed of light. This head-on energy generates a temperature of (www.YouTube.com, What is CERN Large Hadron Collider LHC? End of the World? Search for God Particle and Micro Black Holes, 2008; CERN, LHC Beams, 2008).
Overview
Inertia is one aspect of mass. The larger the mass of a resting object, the harder it is to move that object. But what causes mass? Is gravity related to particles the same way an atoms charge depends on the protons and electrons it holds? When you incorporate the standard model into the familiar formula for universal gravitational attraction you get a variable that keeps appearing in the mathematics. This is a small part of a formula that is handwritten on about 35 lines of notebook paper. And in this formula the H variable keeps appearing. The H variable represents a particle called the Higgs. Because the Higgs particle is responsible for a force, it is a boson. Somehow stu attracts Higgs particles. The more Higgs particles you attract, the more your motion is retarded. This is termed inertia and it can indicate the mass of an object. If the Higgs particle exists then it will lend more support for the standard model of subatomic particles. If the something dierent from the particle is found, then the fun really begins as new theories are developed and old ones are modied (Brian Cox: An Inside Tour of the Worlds Supercollider, 2008). The ATLAS experiment at the LHC is designed to search for this particle (Cox, 2008). Previous experiments have hinted toward this particles existence but were inconclusive. It has been determined that a more energetic collision is needed in a chamber with more sensitive detectors in an eort to nd more conclusive evidence (CERN, History, 1999).
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Overview
The standard model appears to be incomplete. While it does describe many phenomena and can predict many more, there are a few concepts it does not adequately describe.
Electron Size
According to the standard model, when examining the forces involved in the electron, it cannot be any smaller than 1017 m due to repulsion in the electron cloud.
Figure 5.7: According to the standard model, the electron cannot be smaller than in diameter. This is due to the internal forces pushing outwards. However, an electron is approximately in diameter. If you include superparticles, using the concepts of super-symmetry, then this smaller size is allowed by this modied standard model.
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Figure 5.8: Comparison of the Two Standard Models the LHC will experience temperatures of 10 million billion degrees Celsius, or 1 1016 C. This is 500 million times hotter than the Sun, (www.YouTube.com, LHC accelerator at CERN, 2008).
The Search
Evidence of super symmetry (SUSY) lies in nding tangible evidence of superpartner particles. Some evidence has already been found at experiments at Fermilabs Tevatron, KEKs KEKB e + e collider in Japan, and PEP II e + e storage ring at Stanford Linear Accelerator Center in the United States (U.C. Department of Science, Particle Physics as Discoverys Horizon, 2006). A superpartner is related to the particles in the standard model.
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In the standard model the particles can be divided into particles responsible for mass and particles responsible for force. The electron, e , muon, , the tau, , the three neutrinos, e , , , and the quarks are responsible for mass. The photon, , gluon, g, Zboson, Z, and the Wboson, W are responsible for force. In other words, the 12 quarks and leptons pictured on the left in the standard models table are called fermions and are responsible for mass. The four particles in the last column on the right are called bosons and are responsible for all the forces. If the Higgs particle is conrmed in collider experiments, the standard model table could change to look something like the table below (Cox, TED, 2008).
Figure 5.10: This chart shows how the standard model could change if evidence of the Higgs particle is substantiated. The supersymmetry model says that matter and force are not separate but somehow connected. Because of this connection, every fermion has a supersymmetric partner boson and for every boson there is a supersymmetric fermion. These supersymmetric particles are called the superpartners for the particles in the standard model. The superpartner particles are dierent from their counterparts by having half a quantum spin dierence. They also have specic names and symbols.
Figure 5.11: The chart shows how the standard model could change if the superpartners are found. www.ck12.org
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The symmetrical particles for the fermions are the superpartner bosons. The suix, ino, is added to the name. The symmetrical particles for the bosons are the superpartner fermions. The letter, s, is added in front of their name. Table 5.2: Particle and the Corresponding Symmetrical Particle Fermion quark electron neutrino muon tau boson photon gluon W Z Higgs Symmetrical Boson squark selectron sneutrino smuon stau symmetrical fermion photonino gluino Wino Zino Higgsino
Many of the superpartners are very heavy. This means they are shortlived during and after a collision and can only be created by converting a lot of kinetic energy to mass. The LHC could provide enough energy to create these superpartners. One theory has the sneutrino as being responsible for dark matter.
Overview
There is much that scientists dont know. When astronomers peer into space, they take pictures and make observations about the change in locations of the stars and galaxies above. From this data they propose theories and make sense of motions. One of the most exciting events is when the galaxies and stars dont behave as predicted. Scientists then begin to think how and why they are getting unusual results. Eventually a theory will arise that is supported more than others. It does not mean that it is correct, it may just be the most heavily tested at the time. Now is one of those times and dark matter and dark energy is one of those theories. When astronomers look at the speed of each planet in our solar system, they see that the farther away the planets are from the Sun the smaller the planets velocity. This can be calculated according to Newtons law of universal gravity and the concepts of circular motion. This concept extrapolates to the motion of galaxies as well as our solar system. But when astronomers look at the motion of other galaxies to examine the velocities of the stars in the systems, the results do not match the expectations. Instead, after a certain distance the speeds remain relatively constant.
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Astronomers measure the mass of a galaxy by looking at the average luminosity of the galaxy and the star density. This luminosity is then proportioned to our Suns luminosity to mass ratio. If Newtons law of universal gravity is used to verify the the motion of the galaxies, then it turns out that more mass must be in the galaxy than can be accounted for. About 50 percent or more of the needed mass is unaccounted for (Imamura, 2008). This is too much to be accounted for by the unseen planets in the galaxys solar systems. Not enough additional objects can be seen using frequencies above or below the visible light spectrum to account for the 50% missing mass. Because this mass is not giving o any form of energy in the electromagnetic spectrum, it is given the name dark matter. Dark matter is not detectable by looking in the electromagnetic spectrum. The collisions at the LHC may discover evidence of dark matter. It could nd a connection between the lightest super partner and dark matter, or it may nd evidence of multi-dimensions supporting string theory. A lot is to be determined (Green, 2008).
Figure 5.12: If there is a massive galaxy between the Earth observer and the distant galaxy, the light could be bent toward the Earth as pictured above.
Figure 5.13: The Earth observer will see the galaxy as if the galaxy cluster in the middle were not there. The observer will see the galaxy at the end of the dotted line. Because Earth exists in a threedimensional space, the Earth observer will see more than these galaxies. He will see an innite number of galaxies. All these galaxies will form a distorted ring in space. This distorted ring is called an Einstein ring. For these rings to appear in images, there must be something in between the Earth and the observer. It is theorized that that something is dark matterA substance that does not reect or emit any energy in the electromagnetic spectrum but does exert the forces of gravity on photons. www.ck12.org
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Figure 5.14: The distant galaxy will also emit other rays that will bend around the galaxy to reach the Earth.
Figure 5.15: This means that the Earth observer will see the distant galaxy in another position.
Figure 5.16: Using the Hubble telescope, astronomers have discovered many visual examples of an Einstein ring. This is an image of Galaxy Cluster Abell 2218. In this image you can see white circular streaks. These streaks form the image of Einsteins rings.
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Vocabulary
B-eld The abbreviation for magnetic eld. The use of the letter b is rumored to have come for the variable b that was used in a published paper by Michael Faraday. boson A subatomic particle, such as proton, that has no quantum spin. They follow the description given by Bose and Einstein. These particles are responsible for forces in the universe. bunch A collection of electrons or nucleons. For the LHC a bunch equals 2808 charges. CERN European Organization for Nuclear Research: The Abbreviation originates from the original title, Conseil Europen pour la Recherche Nuclaire. collider A machine in which two particles are guided into a headon collision. Coulomb The Systems Internationals standard unit of charge. Abbreviated with a capital C. Named after Charles Coulomb. dark matter A substance with mass that does not emit, absorb, or reect any type of electromagnetic energy. E-eld The abbreviation for electric eld. electric eld A force eld that moves objects with a charge that is positive or negative. Measured with the standard Systems International units of a Newton/Coulomb or the non-standard unit of a volt/meter. electron volt A small unit of energy directly proportional to the charge of an electron. eV: Abbreviated eV. fermion A subatomic particle, such as electrons, a quantum spin of a half. They follow the description given by Fermi and Dirac. These particles are responsible for mass. giga Prex standing for billions. Example: A 4 gigabyte hard drive stores four billion bytes of information. hadron A subatomic particle including baryons and mesons. Higgs A subatomic particle believed to be responsible for mass. Direct evidence of its existence has not been found as of February 2009. Joules The Systems Internationals standard unit of energy. Abbreviated with a capital J. Named after James Joules. kinetic energy The energy associated with moving objects. LHC Large Hadron Collider. www.ck12.org
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lineac Linear accelerator used to accelerate subatomic particles to high velocities. magnetic eld A force eld that aects moving charges. Natural sources are iron, nickel, cobalt, etc. The standard Systems International unit is the tesla. mega Prex standing for millions. Example: Six megavolts is six million volts. tera Prex standing for trillions.
Figure 5.17 4. A particle of negligible mass moves between two plates of a linear accelerator as shown in Figure 18. How much energy (in eVs) does the particles energy change by? 5. A particle of negligible mass moves between two plates of a linear accelerator as shown in Figure 19. How much energy (in eVs) does the particles energy change by? 6. What is the centripetal acceleration needed to turn a particle with a mass of exactly 100 protons traveling at 2.00 108 m/s around a ring the size of the LHC (circumference = 27 km)?
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Figure 5.18
Figure 5.19
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CERN Courier, Jan., 25, 2001, http://cerncourier.com/cws/article/cern/28361 (CERN History) CERN, 2008, http://public.web.cern.ch/public/en/LHC/LHC-en.html(CERN Experiments) CERNALICE Collaboration, 2008, http://aliceinfo.cern.ch/Public/Welcome.html CERNATLAS Experiment, 2008, http://atlas.ch/ CERN, 2008, http://public.web.cern.ch/public/en/LHC/TOTEM-en.html CMSOutreach, 2008, http://cms-project-cmsinfo.web.cern.ch/cms-project-cmsinfo/index. html CERNLHCb Experiment, 2008, http://lhcb-public.web.cern.ch/lhcb-public/ CERN, 2008, http://public.web.cern.ch/public/en/LHC/LHCf-en.html What is CERN Large Hadron Collider LHC? End of the World? Search for God Particle and Micro Black Holes.. CERNOutreach, 2008, http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach (Number data) . CERNOutreach, 2008, LHC Beams, http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach beam.htm CERN 1999, History, http://lhc.web.cern.ch/lhc/general/history.htm LHC accelerator at CERN . Oice of Science, U.S. Department of Energy, US/LHC Large Hadron Collider 2008, Particle Physics as Discoverys Horizon, http://www.uslhc.us/LHC_Science/Questions_for_the_Universe/Undiscovered_ Principles Imamura, Jim, 2008. Lecture notes, http://zebu.uoregon.edu/~imamura/123/lecture-2/mass. html Hooper, Dan, October 2007, www.YouTube, Supersymmetry and the Search for Dark Matter. Cox, Brian, TED Technology Engineering and Design, Brian Cox: An Inside Tour of the Worlds Biggest Supercollider, http://www.ted.com/index.php/talks/brian_cox_on_cern_s_supercollider. html Greene, Brian, 2008, TED Technology Engineering and Design, Brian Greene: The Universe on a String. http://www.ted.com/index.php/talks/brian_greene_on_string_theory.html
Image Sources
(1) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (2) Tony Wayne. . CC-BY-SA. (3) Tony Wayne. Size of the Electron. CC-BY-SA. (4) Tony Wayne. Galaxy Cluster. CC-BY-SA. (5) Tony Wayne. LHC Stage 1. CC-BY-SA. (6) Tony Wayne. Galaxy Cluster. CC-BY-SA. (7) Tony Wayne. LHC Stage 5. CC-BY-SA. (8) A. Fruchter and the FRO Team STSsci-STSci-PRC00-08. Galaxy Cluster Abell 2218. CC-BY-SA.
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(9) Tony Wayne. Galaxy Cluster. CC-BY-SA. (10) Tony Wayne. LHC Stage 4. CC-BY-SA. (11) Tony Wayne. . CC-BY-SA. (12) Tony Wayne. Visual representation of the standard model if superpartners are found.. CC-BY-SA. (13) Tony Wayne. Visual representation of the standard model if the Higgs particle is substantiatied.. CC-BY-SA. (14) Tony Wayne. LHC Stage 1. CC-BY-SA. (15) Tony Wayne. . CC-BY-SA. (16) Tony Wayne. LHC Stage 3. CC-BY-SA. (17) Tony Wayne. LHC Stage 2. CC-BY-SA. (18) Tony Wayne. Visual Representation of the Standard Model.. CC-BY-SA. (19) Tony Wayne. Comparison of the Two Standard Models. CC-BY-SA. (20) Tony Wayne. Galaxy Cluster. CC-BY-SA.
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6.1 Outline
1. What is modern physics?What is quantum mechanics and why did it develop? What part of physics was not complete? What is relativity and why did it develop? What part of physics was not complete? Question 1 How do you see? Question 2. Why cant we see atoms? Objects are made of atoms and light is reecting o of them, right? Why dont we see the little balls that make up the object? Question 3. So how do we know atoms exist? Question 4. How do we know the basic structure of an atom? Question 5. How do we know there are electrons? Is it the same experiment as for the nucleus? Question 6. Why are there neutrons in the nucleus with the protons? Question 7. What are quarks and how do they play a role inside the atom? Question 8. What are alpha particles and where do we get them? Question 9. What really is radioactivity? Why do some elements emit or put o streams of alpha particles? Do any elements emit particles other than alpha particles? Question 10. What is Quantum Mechanics and why did it develop? What part of physics was not complete? Question 11. What is the photoelectric eect? What does it mean to say that matter has wave-like properties? Question 12. What is Relativity and why did it develop? What part of physics was not complete? 2. What parts of modern physics are still being researched? Question 13 : What can be considered the big problem facing physicists today? 3. What are the implications of some of Modern Physics (including nanoscience, dark matter, black holes, parallel universes, and the graviton)? Question 14 : What are some of the implications of quantum mechanics and relativity? In the news there
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is mention of string theory, black holes, parallel universes, and other bizarre things. This chapter has been written as a series of questions in the eort to lead you through an understanding of how modern physics came about, some of its components, some of the still lingering problems in its theories, and some of its implications. This is by no means an exhaustive discussion and you are urged to read further and go deeper by asking experts. What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate schooland you think Im going to explain it to you so you can understand it? No, youre not going to be able to understand it. Why, then, am I going to bother you with all this? Why are you going to sit here all this time, when you wont be able to understand what I am going to say? It is my task to convince you not to turn away because you dont understand it. You see, my physics students dont understand it either. That is because I dont understand it. Nobody does. (FeynmanQED)
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on top of them taking big steps or would you march taking tiny steps? You would probably use small steps so that you wouldnt miss any of them because ants are small. Now, have you ever noticed dust oating in the air near a bright window? How come you cant see the dust oating anywhere except near the window? Well, the light coming through the window has lots of dierent wavelengths and some of the wavelengths are small enough that they bounce o the dust and are reected into your eye for your brain to interpret. All of the long wavelengths of that light just pass right over the dust, just like if you take a big step over a colony of ants, and just like long wavelengths of sound would pass right over the baby. All the short wavelengths of that light would hit the dust and bounce o. (Incidentally, shorter wavelengths [higher frequencies] are higher energy waves for light and longer wavelengths [lower frequencies] are lower energy waves for light.) This idea is used in (physical rather than geometric) optics: you need to much the wavelength with the size of the object in order to see it. How does this relate to seeing atoms? Atoms are very tiny. Scientists have found that if you line up ten carbon atoms, they will be about 1 nanometer long. A nanometer is very small. Hold up your hands to about the size of a meterstick (close to a yardstick). If you could divide that meterstick up into 1, 000, 000, 000 (or 109 , or a billion) little equal parts, one of those parts would be the size of 10 carbon atoms lined up in a row. Its hard to imagine a number like a billion because we dont usually think about it. The wavelength of violet light, the shortest wavelength of visible light, is 400 nanometers long, which is 400 times larger than the ten carbon atoms lined up. Metaphorically, thats like taking steps that are two meters long when trying to kill ants that are half centimeters long. You would not kill too many ants taking steps that big, and similarly you cant see tiny atoms shining violet light on them. The violet light is just too big and wont reect o the tiny atoms so you could see them. The violet light will just pass right over the atoms. And even when we do shine light on atoms with a wavelength that is small enough to bounce o the atom and into a detector, some complications occur. It turns out that light causes changes in the atom because light carries energy.
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Question 3: So how do we know atoms exist? Consider cutting a piece of paper in half. Now cut one of the resulting halves in half again. Continue doing this. How many times can you cut it before it can no longer be cut? Is there a limit? Around 400 BCE the Greek philosophers Democritus and his instructor Leucippus considered this and decided that there must be a tiniest part that cannot be divided and they called it an atom. They theorized that matter has whole building blocks. You can relate it to your skin cells. Your skin cells are your basic building block for your skin, and if you cut one in half, you no longer have a skin cell. The cell is the smallest whole of your skin. Democritus and Leucippus theorized that there must be a smallest whole of matter. This happened way before instruments were developed to detect atoms. There wasnt much in the development of the theory of atoms until much later, around the early 1800s. Summarized very briey, chemists were taking very careful measurements of masses and ratios of combining elements (such as hydrogen and oxygen to form water) and found that the ratios of elements in compounds is xed. This analysis is attributed to John Dalton, although he was building on Antoine Lavoisier, among other chemists. The signicance of this follows: Elements combine together in specic ratios, and this must mean that there are smallest wholes that can be added, but you cannot add a part of a whole. For example, you can have three atoms of hydrogen, or four atoms of hydrogen, but not three and a half. Dalton found that if you have 1 gram of hydrogen and combine it with 8 grams of oxygen, you get water. The ratio of oxygen to hydrogen is 8 : 1, and this is xed. You can also get water with 16 grams of oxygen and 2 grams of hydrogen, or any multiple of this specic ratio. Dalton considered this and decided that there must be atoms, or little whole chunks of matter, which give this ratio. Now physicists can see atoms by using as electron microscope, which uses electrons to magnify objects. Question 4: How do we know the basic structure of an atom? We can see atoms in other ways. We can see evidence of atoms. Close your eyes and feel the tabletop. You can tell its a table by how it feels, right? What if you were not allowed to feel it with your hands, but could touch and poke it with a stick? Would you be able to tell that its a table? It might take awhile, but you probably could gure out that its heavy by trying to push it with the stick. You could probably gure out that its hard by poking it. You could probably gure out how big it is by tracing along the tabletop with the stick. You could get a pretty good mental picture by just using a stick. (You should play a little game and try this.) Similar things have been done with atoms. In 1909, two British scientists, Hans Geiger and Ernest Marsden, took a sheet of very, very thin gold foil (like aluminum foil, except gold), and sent tiny particles toward it to detect what makes up the foil. Imagine taking a tennis ball shooter (like what tennis players use to practice their swing) and pointing it toward the chain fence that surrounds the tennis court. Instead of shooting tennis balls, shoot ping-pong balls instead. You would notice that some of the ping-pong balls would go right through the fence and some would bounce back, depending on what part of the fence they hit. Geiger and Marsden did the same thing, except on a smaller scale. They sent alpha particles (4 Henuclei) toward the foil and noticed that most of them went straight through the foil, but some of them bounced back. Please note that the momentum of the alpha particles they sent toward the gold foil was very high, and they completely expected them to pass right through. Ernest Rutherford, a scientist who used this experiment to develop his ideas on the structure of an atom describes it as shooting bullets at tissue paper. There was no reason for Geiger and Marsden to expect any reection of the alpha particles, but thats exactly what they observed. Because most of the alpha particles went right through, the scientists knew there must be empty space (where the electron cloud is, actually), and because some of the alpha particles were repelled back, the scientists knew there must be a dense core in the middle of the empty space. Thats the indirect evidence of the existence of the nucleus of the atom. Question 5: How do we know there are electrons? Is it the same experiment as for the www.ck12.org
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nucleus? Not quite. And actually, the evidence for the electron came before the evidence for the nucleus. It was J. J. Thomson in 1897 who was able to construct an experiment that helped scientists conclude that electrons are negatively charged. Thomson created a series of experiments applied to a stream of electrons (at the time they were called cathode rays) that were propagated through a vacuum tube (cathode tube). Thomson saw that the cathode rays had a negative charge, and thought he might try to separate the negative charge from the ray, but when he used a magnet, which bends negative charge, he saw that the whole ray bent, and he could not separate the charge from the cathode ray. From this and a couple of other experiments on the cathode rays he came up with the following hypotheses. His rst hypothesis was that the cathode rays themselves are charged particles, because he was unable to separate the charge from the ray. His second hypothesis was that these particles were part of the atom (a smaller particle that makes up the atom), because when he calculated the mass-to-charge ratio, he found that it was much smaller than the atom. At the time most scientists thought of the atom as indivisible, so to nd a smaller particle was unbelievable for many scientists. His third hypothesis was that these particles were the only building-blocks of the atom, which turns out to be incorrect, as we now know. Thomson proposed that because the atom was known to be neutral, perhaps these electrons swam around inside a cloud of massless, positive charge. His model was sometimes called the plum pudding model. Of course, we know this turned out to be incorrect because Geiger and Marsden with Rutherford were able to show us the nucleus. These days, scientists detect particles using particle accelerators. Here in Virginia, we have a particle accelerator at Jeerson Lab in Newport News (there are other particle accelerator labs in places such as Switzerland, Illinois, and California). Basically, scientists shoot particles at atoms and then watch where the particles go. Scientists can cause an electron to eject from an atom and watch its path, which helps them learn basic things about atoms and particles (particles leave tracks that scientists can detect). Question 6: How do we know that there are protons and neutrons in the nucleus? We already know that particles with opposite charges attract, like the proton and the electron. This attractive force is what keeps the electron in its orbit (or cloud) around the nucleus. Its similar to the way the Moon is attracted to the Earth and the Earth to the Moon. The gravitational pull (from the gravitational force) between the Earth and the Moon pulls the Moon inward. Likewise, the electric pull (from the electromagnetic force) between the electron and the proton pulls the electron inward. Particles with the same charge repel one another. For example, if you put two protons near each other, they push each other away. So, how can a nucleus full of protons stay together? Wouldnt the protons all repel each other like they repelled the alpha particle in the gold foil experiment? What glues them together in the nucleus? Well, its what physicists call the strong force, or the strong interaction. The strong force is whats responsible for the binding energy, which is the energy that glues together protons and neutrons in the nucleus. Without the neutrons, the protons would y apart because of the electromagnetic force (like charges repel). Therefore, the strong force has to be bigger than the electromagnetic force that causes the protons to repel each other. For example, suppose that you and your brother are pushing each other away. To keep you close together and prevent you from pushing each other apart, your parents would have to hold you together with a greater force than the force you and your brother are using to push each other apart. That force holding you together (your parents arms) is like the strong force that holds the protons together in the nucleus, even though they push each other apart. The neutrons provide part of this force, although protons themselves also contribute to the strong force. The strong force exists only in short range, meaning that the protons repel in general because of their charges, but if they are close enough (and they have to be very close), a dierent force (the strong force) attracts them together. If the nucleus just had protons, the short-range strong force would not be enough to hold the protons together, especially if its
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an atom with lots of protons (lots of repulsion force). Its the neutrons that add enough of the strong force to keep them together because they dont contribute to the repulsion (neutrons have zero charge and thus do not repel). The neutrons only contribute to the strong force, the force of attraction. The strong force is many more times greater than the electromagnetic force that causes the protons to repel. Its worth mentioning again, though, that the strong force only exists at very short ranges. That means if protons or neutrons are far apart, the strong force does not aect them. Only when they are close neighbors does the strong force create a large result (Weidner 415). Question 7: What are quarks and how do they play a role inside the atom? You have probably heard of the term quark and are wondering how it ts in the whole picture. We can explain the quark in terms of the strong force we just learned about. Recall how an atom is made up of smaller parts: electrons and nucleons (i.e., protons and neutrons). Electrons zip around the outside of the atom and protons and neutrons are inside the nucleus. Scientists presently think that electrons are fundamental particles, meaning that there is nothing that is smaller that composes an electron. However, neutrons and protons are not fundamental particles because there are particles that come together to create neutrons or protons. Think about it this way, just like a building is made of smaller components, such as bricks, in the same way protons and neutrons are composed of smaller quarks. The bricks are the quarks. The electron is like a brick, in that it is the smallest part. There does not seem to be anything smaller that builds an electron (so far). We call the electron a lepto (a dierent kind of brick than a quark). Feynman and others explained in the 1940s that the Coulomb force between electric charges is mediated by the exchange of (virtual) photons (or light particles, see below). This theory is called the Quantum Electrodynamics [QED] and remains one of the triumphs of theoretical physics. Likewise, quarks inside the protons and neutrons interact with each other by force carriers called gluons. You might think of these gluons as how the quarks let other quarks know they are there. The gluons carry the force that keeps the quarks together, the action that also keeps the nucleons together in the nucleus. The quarks inside the neutrons and the protons communicate their force by way of gluon and stick together (hence, gluon is like glue). This theory is called Quantum Chromodynamics [QCD]. There are six dierent kinds, or avors, of quarks, and physicists have thought of some creative names (maybe these scientists have spent too much time in their oices alone!): up, down, top, bottom, charm, and strange quarks. A proton is made of two up and one down quark. The neutron is made of one up and two down quarks. Just think of the dierent types of quarks as dierent types of bricks used to make dierent things. Quarks also have another property similar to the property of charge that we see with electrons and protons, and physicists call it color. Please note that the color of a quark has nothing to do with colors that we see, its just a way of categorizing (they can be red, blue, and green). Question 8: What are alpha particles and where do we get them? There are many types of particles, and, in fact, physicists often call all the particles together the particle zoo. You already know some of them: electrons, protons, and neutrons. There are lots of other types as well. Scientists predicted some of these particles before they saw evidence of them in experiments because they saw patterns. There are probably more particles that are as yet undiscovered. An alpha particle is identical to the nucleus of a helium atom. If you look at the Periodic Table of Elements, the Helium atom has two protons and two neutrons (its the second element). At the time of the gold foil experiment, scientists knew a little about the element radium. Im sure youve heard of radium. It probably makes you think about radioactivity, and then you probably think of the Earth science lesson you had involving half-life. Radium is an element that naturally emits alpha particles (or helium ions). So radium puts o or emits streams of helium atoms. Geiger and Marsden pointed the radium in the direction of the gold foil, much like you would point a gun, and waited for the radium to
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naturally emit the alpha particles. Question 9: What really is radioactivity? Why do some elements emit or put o streams of alpha particles? Do any elements emit particles other than alpha particles? Radioactivity occurs naturally, but can also be triggered. Radium, for example, naturally radiates alpha particles, which is why it was a good element to use for the gold foil experiment. Recall how the electromagnetic force causes protons to repel one another. Also recall that there are three quarks in each proton and neutron that exert forces on each other by way of the gluon, called the strong force. The more protons there are, the bigger the strong force has to be in order to cancel out the electromagnetic force repelling the protons away from each other. This occurs by way of the neutron, because it adds no extra repulsion force, but does contribute to the strong force holding the quarks of the protons and neutrons together. This means that the more protons there are, the more neutrons are needed in the nucleus in order to balance out the repulsion force between the protons. Moreover, when you group more than about 83 protons together, no matter how many neutrons are included, the nucleus becomes unstable. This is where we get nuclear decay, which causes radioactivity. Instability of the nucleus can also occur if the nucleus has too many neutrons. We call nuclei that have lots of protons and neutrons heavy nuclei, and heavy nuclei are not stable. The atom tries to gain stability through various means. The three most common means for an atom to gain stability are as follows. The rst way is by ejecting alpha particles. The second way is by converting a proton to a neutron or a neutron to a proton (whichever is needed) by ejecting a beta particle. A beta particle is another name for an electron or a positron. A positron is a positively charged particle that has the same mass as an electron, but is positively charged. We have not talked about it yet, but neutrons themselves can convert to protons by releasing an electron (and a tiny particle called an anti-neutrino). When we say that the neutron releases an electron, we dont mean that the electron is somewhere inside the neutron and the neutron lets it out. Rather, the electron and antineutrino are essentially essentially created out of nothing, as strange as this may sound. The third way a nucleus gains stability is by releasing energy via a gamma ray or gamma emission. A gamma ray is just a photon or a bit of light. Sometimes we call this electromagnetic radiation. Gamma rays are on the high-energy, and therefore high-frequency and short wavelength side of the electromagnetic spectrum. Question 10: What is quantum mechanics and why did it develop? What part of physics was not complete? The more success the quantum theory has, the sillier it looks. (Einstein, Zangger) Quantum mechanics is the study of subatomic particles (particles smaller than the atom), like electrons, protons, neutrons, and light (photons), and how they interact. Anything you see now in the news about nanoscience deals with quantum mechanics. It may help you to know that about 10 carbon atoms lined up gives you the size of one nanometer. Nanotechnology is just the manipulation of atoms on the nanoscale. You may have seen atoms pictured like solar systems. The nucleus is like the Sun and the electrons orbit around the nucleus like the planets orbit our Sun. This is not quite what happens and scientists who study subatomic particles have found some very interesting results in experimentation and philosophical thinking, using logic (Einstein called these logic experiments thought experiments, or gedanken experiments). So what led scientists to think that the atom was like a solar system? And now what leads them to think that the atom is not exactly like a solar system? Lets explore the rst question by studying Ernest Rutherford who was a scientist around the early 1900s (just after J. J. Thomson discovered clear evidence of the negatively charged electron in the cathode tube). Recall the Geiger-Marsden experiment from earlier. Geiger and Marsden sent alpha particles (created by the natural radioactivity of radium) toward gold foil and they found that a small percentage of the alpha
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particles bounced back. This caused Rutherford to believe that a dense mass is located in the center of an atom, albeit small. In 1911 Rutherford theoretically placed the electrons zipping around the nucleus for his model of the atom. It was known that the overall charge of the atom was zero and if the electrons were around the outside of the dense center, then the center had to be positively charged to keep the whole atom neutrally charged. Rutherford explored this scenario and did some calculations, which produced some confusion. According to classical physics, the electron should release electromagnetic radiation while it orbits. In accordance with classical physics, all accelerated charged particles produce radiation, or in other words, release waves of light. The key word here is accelerated. Note that here we are referencing the familiar centripetal (circular) motion. Recall that any object moving in a circle is constantly changing direction, and for an object to change direction, there must be a force acting on it causing it to change direction. According to Newtons second law, if there is a force, there is an acceleration (Fnet = ma). The laws of electricity and magnetism then show that the electron must be releasing radiation because it is constantly accelerating (orbiting). As it releases the radiation, it will lose energy, and therefore it should spiral inwards toward the nucleus. According to this theory, all matter is unstable, and the amount of time it would take the electrons to collapse into the nucleus is only 0.00000010 s! There has to be a better theory for the structure of an atom, as this one does not work for two reasons. The rst is that the electrons would collapse into the nucleus. The second is that scientists would be able to detect a continuous (smooth) spectrum of radiation emitted by the spiraling electron, and they do not. The reason that the radiation from the electron is continuous is that the radiation emitted by the orbiting electron depends on the radius at which the electron is orbiting, and if the radius of the electron continuously decreases (toward the nucleus), then the frequency of the radiation produced by the orbiting electron must also change continuously, and in fact would increase (Weidner 175). What scientists do detect is radiation of discrete frequencies, meaning that there are no in-between frequencies emitted. Think of it this way: Electrons may emit a frequency of a or a frequency of b, but no frequencies in between. Therefore, the electron cant be spiraling inward, as it would have to emit each frequency associated with each radial distance from the nucleus. (A thorough analysis of the mathematics that govern this logic can be studied in a modern physics course, usually the course taken right after a general physics class in college.) Niels Bohr came next with an improvement on the picture of the atom around 1913 (he was a student of Rutherford and Thomson). Bohr suggested that the atom had a nucleus of positive charge like before and that the electrons orbit around the nucleus at specic radii, like our solar system (like Rutherfords model), with some modication. It does not describe the atom quite as accurately as Louis de Broglie does in the 1920s, but it does make some good leaps forward. He involves Einsteins idea of the photon, which we will discuss later. What Bohr did was discard the very idea that the orbiting electron would spiral inward (as predicted by classical mechanics), and proceeded from there. He considered the idea that the electron orbited at discrete radii instead. What is meant by this is that the electron can be at a distance of r or 2r from the nucleus, but nowhere in between. This would mean that the electron would not spiral inward and would have a certain amount of energy, the amount associated with that radius of orbit. For an electron to increase or decrease its distance from the nucleus, it would have to obtain or release a discrete amount of energy that would place it at the next orbit. In other words, if you dont give the electron enough extra energy, it wont jump to the next orbit, and it wont orbit in between. Perhaps thinking of orbits like a ight of stairs would help. You may stand on the rst step or the second step, but you cant stand in between. You have to use the exact amount of energy needed to get to the next step. Using enough energy to get halfway to the next step will not result in you oating in-between the second step and the rst step; that is preposterous! Its the same, according to Bohr, for the electron. This solves the problems Rutherfords www.ck12.org
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model had. This means that no energy is lost by orbiting, and therefore the electron does not spiral to the nucleus, and does not emit a continuous spectrum of light as it spirals inward. Bohr doesnt explain how the light (photon) is created, rather he just makes the connection that as the electron makes a quantum jump to a lower orbit (closer to the nucleus), it emits a photon whose frequency corresponds to the amount of energy lost in moving closer to the nucleus. So what are the shortcomings of Bohrs model of the atom? It does not account for the wave-mechanical nature of matter and light (its okay if you dont understand that phrase), nor can it account for atoms with more than one electron, and also it doesnt really explain why certain radii are allowed. We need a new scientist to take us a little further into understanding the internal structure of an atom. The next person to make a conjecture for the structure of an atom is the French scientist Louise de Broglie. First, lets discuss light for a moment. Sometimes we think of light as little traveling packets, called photons. Sometimes we think of light as waves, with a frequency and a wavelength. It turns out that both seem to be good descriptions of light depending on the nature of the experiment we want to understand. Well discuss this in a moment using Einsteins Nobel-winning experiment called the photoelectric eect. Louise de Broglie initially just made the assumption that matter had wave-like properties, and then followed the logic to its end by using mathematics. The idea is strange, but the mathematics produces an accurate model for physics, and corroborates with experimentation results well. The more we peer into the internal structure of atoms, the stranger things seem to be, and because we cant see inside directly, we rely on mathematics and indirect methods of analyzing the particles. Sometimes physics is stranger than science ction! Please note, though, that physics is always logical, just not always intuitive. The natural language of physics is mathematics, and mathematics by its very nature follows logical reasoning. However, its solutions are not always what we expect! Question 11: What is the photoelectric eect? What does it mean to say that matter has wave-like properties? Einstein was awarded the Nobel Prize in physics for interpreting the results of this ingenious experiment, rst performed by Heinrich Hertz in the late 1800s. The photoelectric eect explores the energy of electrons and the energy carried by light. What Hertz did was shine ultraviolet light on zinc and he found that it became positively charged, which could not be explained at the time. The striking nding is that electrons are observed as soon the light is turned on, rather than the several minutes predicted from classical theory (electricity and magnetism). What Einstein gured out was that electrons can be knocked from the metal through the energy from the light shined on the metal, with a few important reservations. First, it depends on the frequency of light that is used. If you use a frequency that is not high enough, the electron will not be aected. The frequency necessary depends on the amount of energy binding the electron to the metal, called the work function. Of course, all light has energy. Prior to the results of this experiment, scientists believed that if you shone light on an object long enough, the energy possessed by the light would build up in that object. What Einstein showed is that energy does not build up in the electron. One has to use a frequency of light that has enough energy to provide one swift kick (the energy of the light is proportional to its frequency, as predicted by de Broglie) to cause the electron to jump from its orbit. What are the ramications of this experiment? For one, we learn that light can be described as a little chunk of energy. For example, suppose your friend is standing on the edge of the deep end of a pool and you want to push him in. Suppose you use a tiny push and he doesnt fall in, so you push him again with a tiny push. Will he fall in this time? No. It doesnt matter how many tiny pushes you give him in a row, if the push isnt large enough, it wont overcome the friction he has between his feet and the ground and therefore he wont fall in. It only takes one push that is just large enough to make him fall in. Its the same with an electron. You can push on an electron with a bit of light as long as you want, but it wont be knocked out of its orbit unless the push you give it is suiciently large. This led Einstein to see light as little particles instead of waves. If light were a wave, one would surmise that the energy would build up over time, but if you think of light as a little packet or ball of energy, you can then see that if
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the packet doesnt contain enough energy, it will never cause a change in the electron. You can also think of the light hitting the electron as a collision like you studied in your momentum chapter. The energy from the light-packet (photon) is given to the electron, and if its not enough, the electron will not have enough energy to escape its orbit and eject from the atom (any extra energy will go into kinetic energy of the electron). This was a breakthrough for physics. But physicists have suicient evidence of the properties of light to also see it as a wave (the way it interferes with other light), so we say that light has wave-particle duality. It is neither an ordinary classical particle, nor an ordinary wave; instead it has properties that are similar to both a particle and a wave at the same time. The same can actually be said for all matter. The diiculty in seeing evidence of the wavelength of matter results from how very tiny its wavelength is. Light has a long wavelength relative to the wavelength of matter. (Please note that light is not made of matter, rather its just a bit of wiggling energy made of electricity and magnetism. Its a strange concept.) So what evidence do we have that matter itself is also a wave? If we look at stu, we dont see it waving. Of course, earlier we said that de Broglie just made the assumption mathematically and the theory followed from there to produce accurate mathematical relationships. Experimentally we now have evidence as well (so we dont have to just rely on an assumption that de Broglie used), and the wavelength for matter is called its de Broglie wavelength. Now you might ask: How in the world could one test to see if matter, such as an electron, has a wavelength? Lets consider how we know that light has wave-like properties. We know that light waves interact, or interfere. We know that if two beams of light overlap, i.e., when two crests or two troughs overlap, we get constructive interference (the amplitudes add together) and when a crest and a trough overlap, we get destructive interference (they cancel out for that position). You should recall this from your lessons on light and sound. If we could cause two electrons to interfere like that, we would know that electrons, and therefore matter, behave like waves. Physicists have been able to do this. (If you would like to view some great pictures or diagrams of this, visit http://en.wikipedia.org/wiki/Double-slit_experiment. Picture two little slits parallel to one another (like two cuts in a thick sheet of paper). Now picture shining a beam of light through these slits. The light would pass through the two slits and form two beams on the other side, but they wouldnt just be two columns of light. Think about how light shines through a keyhole. On the other side of the keyhole the light spreads out. This result is called diraction. The light shining through the two slits will diract and form two beams of light that spread out. Because they spread out, they will overlap and interfere, and if you place a screen for the beams to shine on, you should see the pattern of interference. Where the two beams crests overlap, you have a bright place on the screen, and the same for two troughs. Where you have a crest interfering with a trough, the two beams will cancel out, and you will have a dark spot. (You can show this by using a simple, handheld laser pointer and a piece of hair. Simply tape a piece of paper on the wall where the beam will shine and hold a piece of hair in the path of the beam. You will see a series of bright and dark spots formed by the interfering beams. The piece of hair serves as an obstacle around which the laser has to diract on either side.) Scientists have done the same experiment with electrons, called the double-slit xperiment. They shot electrons through two slits and used a detecting screen to show the pattern they made. If the electrons behave like little balls (picture baseballs being thrown through two slits) one should see two bright spots across from the slits where the electrons hit. If the electrons behave like waves, one should see an interference pattern just like that of the beams of light. What scientists found is that an interference pattern emerged when releasing many electrons one particle at a time. Its as if each electron interfered with itself and formed the pattern, one hit at a time on the screen. It doesnt make intuitive sense, and wrapping your mind around such a foreign and abstract concept is diicult, however, the mathematics that predicted this behavior for the electron (and all matter) is now supported by this experimental evidence. C. Davisson and L. H. Germer were the rst scientists to conrm the wave-nature www.ck12.org
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of electrons in 1927, followed by scientist G. P. Thomson. Question 12: What is special relativity and why did it develop? What part of physics was not complete? If you were to ride on a beam of light, what would light look like? This is the question that Einstein asked himself while he was just a teenager. We already discussed that light is a bit of electricity and a bit of magnetism oscillating, or waving, so what would this look like if we could travel with it? Suppose you are driving on a highway right next to another car and you are both traveling at the same speed. What do you see if you look over at the other car? Does it look like it is moving? To you, the other car may seem like it is at rest and that you are at rest as well (and that the ground is moving behind you). This observation is because there is no dierence in your speeds. So what would you see if you could travel next to a beam of light and look over at it just as you did with the other car? And, better yet, consider this. We know that in order for us to see, light bounces o an object and into our eyes, and our brains interpret the light signal. What if you are traveling at the same speed as light and you hold up a mirror. Would you see your reection? When you are sitting still the light bounces o your face, then bounces o the mirror to your eyes. If you are traveling at the speed of light, would the light ever be able to go ahead of you, bounce o the mirror, and then travel back to your eyes, or would you see a blank reection because you are traveling with the light? These are questions that Einstein spent many years thinking about before he developed any answers, and he built his theories on those of many other physicists that came before him. By the end of the 19th century the speed of light had been tested to a pretty accurate 299, 792, 458 meters per second (thats about 670, 616, 629 miles per hour!). What Einstein postulated is that light always travels at this speed, which we call c, no matter how fast you are going. Our experience tells us that if we are going 25 miles per hour and another car passes us at 30 miles per hour, that other car seems to be going only 5 miles per hour, which is called the relative velocity. This approximation works for us, but when you begin thinking at extremes, just adding or subtracting velocities does not work, however strange it may sound. In our experience we know that if we are in a truck going 25 miles per hour and throw a ball in the direction we are traveling, the ball will have the velocity of the truck (25 miles per hour) plus the velocity we give it, say 30 miles per hour. In the absence of air resistance, an observer on the side of the road would see the ball go 55 miles per hour. Along the same thread, suppose you are on a ship going 75% of the speed of light (0.75c) and you launch a missile at half the speed of light. According to Newton (and our intuition), the missile would have a velocity of 1.25c, which cannot happen if light is the maximum speed. In reality, according to special relativity, it is incorrect to simply add the velocities. The most important thing to remember is that light travels at a constant speed, and it is the fastest anything can travel. The way to combine velocities is a bit more complicated than that, but results dierent than Newton would have predicted only become apparent at very fast speeds. This is why here on Earth at a tiny speed of 70 miles per hour we dont have to worry about relativity. Light-speed is the limit for speed. Therefore, if light were coming toward you, and you started to run, it would still approach you at a speed of c, no matter how fast you run. If you are in a car going almost the speed of light and you turn on your headlights, the light from your headlights would still appear to travel away from you at the speed of light. If you are watching somebody drive by at nearly the speed of light and they turn on their headlights, you would see the light still travel at the speed of light. The velocity of the car does not add to make the light go faster, as you might suspect. This seems ridiculous, its true, but there is experimentation to support this. Lets look at some of the ramications. If we set the speed of light as a constant in all reference frames, whether you are moving or not, and we know that speed is displacement over time, then what must be varying from one observer to another is displacement and time. The variable v (which is c) cannot change for light, so the displacement and time
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must change. The concept is often explained by considering a ashlight on a moving vehicle. Suppose you are on a vehicle that is traveling at a constant speed. On this vehicle you have a ashlight mounted to the oor and pointed toward the ceiling and you watch as a beam of light travels from the ashlight toward the ceiling where you have placed a mirror. You use a stopwatch (a very fast one) to time how long it takes for the beam of light to travel from the ashlight to the mirror and back again. Because you know that light travels at a constant speed of c, you can calculate the distance over which the light traveled (c = distance/time). Now you jump o the vehicle (its still moving at a constant speed) and repeat the experiment, but this time you measure from the side of the road. Notice that when you were on the vehicle, the light only had to travel up to the ceiling and then back to the ashlight. When you are on the side of the road, the light has to travel a bit further. Consider the diagrams above to help clarify.
Figure 6.2: Diagrams of Distance Travelled by Light. Notice in the diagrams that the light as viewed by the observer on the side of the road has to travel farther to reach the mirror and then return back to the ashlight. If the speed of light does not change, how do we reconcile these two observations? Is one of the observers wrong? The explanation given by Einstein and special relativity is that time slows down for observers who are traveling faster. The faster you go as the observer, the more time slows down for you. This is called time dilation. So the passenger on the vehicle taking the time for the light will measure a longer time than a person on the side of the road measuring the same light traveling at the same instance (or same event). The person on the side of the road measures a longer time on their stopwatch (the stopwatch ticks faster, so more time passes). The person on the vehicle measures a shorter time (the stopwatch ticks slower, so less time passes). That is, the stopwatches tick at dierent speeds. With this remedy we have reconciled the problem with the speed equation: speed = c = ( distance)/( time). For the passenger on the vehicle: c = ( shorter distance)/( shorter time) For the observer on the side of the road: c = ( longer distance)/( longer time) And so the proportions remain intact and the speed of light can remain a constant. Not only is it true that time depends on the observer, but if we apply the laws of physics with this constant speed limit for light, then an objects size and mass depend on the relative speeds of observers as well. www.ck12.org
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What Einstein was able to show is that the faster your speed, the slower your time ticks (time dilation), and the faster you go in a straight line, the shorter you become in that direction (Lorentz contraction). It seems the deeper we probe and question, the stranger the explanations become! But we have also seen this with real experimental evidence. One case in which this phenomena has been observed is with the muon in particle accelerators. A muon is just a particle, like an electron except much heavier. When just sitting, a muon will decay (kind of like radioactive decay) into other particles in about two millionths of a second (very, very fast). If accelerated at nearly the speed of light, the muon has been measured to last about ten times longer. Imagine if you lasted ten times longer than you normally would. For example, if you would normally live for 80 years, youd live for 800 years if you were accelerated at such a rate! A factor of ten is quite signicant. However, with the slowing down of your clock comes the slowing down of all your functions, and therefore you would not get more done. You would digest slower, think slower, etc. Everything would slow down. Essentially, from the slow persons perspective, he or she would be living the same amount of life, just slower. Its just relative. As it turns out, going super-fast to slow down your clock is not the fountain of youth (Greene 42). We have also seen evidence of special relativity on an airplane. Scientists placed an atomic clock (a clock that works by detecting the back-and-forth movement of electrons by detecting the emitted frequencies) on a plane while measuring the amount of time the plane was in the air according to an observer on the ground. When comparing the time shown on the stopwatch from land to the atomic clock on the plane, there was a denitive dierence. The atomic clock measured less time, which means its ticking must have slowed down, evidence of special relativity (time dilation). Another important application of special relativity is the global positioning system, or GPS. GPS uses satellites that are orbiting the Earth and traveling very fast, to locate positions, say, of cell phones. Because of their fast speeds, the clocks inside the satellites tick slower. Furthermore, there are the eects of general relativity. General relativity predicts that time ticks faster the further the clock is from a massive object, like Earth. Therefore, according to general relativity, the clocks on the satellites will tick faster. Combining the eects of special relativity (time dilation) and general relativity (distortions in the fabric of space-time due to massive objects), the satellites have a slightly fast clock (slowed by the speed and quickened by the distance from our planet). Because GPS is used in measuring position, and time is a very important ingredient in calculating position, scientists have to take relativity into account to achieve any decent accuracy. Without using calculations considering time dilation the GPS would not work accurately (Pogge GPS). This all may seem hard to swallow and if you are really engaging your brain, it should. If you continue with your physics studies in college and take a modern physics course, you will get a more rigorous treatment of these concepts, and get to use actual mathematics to aid your brain in processing these new, strange theories.
6.3 Section 2: What Parts of Modern Physics are Still Being Researched?
A successful unication of quantum theory and relativity would necessarily be a theory of the universe as a whole. It would tell us, as Aristotle and Newton did before, what things are made of, and what kind of laws those things obey. Such a theory will bring about a radical shifta revolutionin our understanding of what nature is. It must also have wide repercussions, and will likely bring about, or contribute to, a shift in our understanding of ourselves and our relationship to the rest of the universe. (Smolin 264) Question 13: What can be considered the big problem facing physicists today?
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We use general relativity for the physics of the very massive (planets, stars) and we use quantum mechanics for the physics of the very small (electrons, protons), so in most situations, they do not overlap. However, there are at least two situations that would be small and massive. The rst is in black hole theory (black holes are very dense), and the second is in analyzing theory for the whole universe at the moment of the big bang. In trying to combine the theories of general relativity and quantum mechanics, physicists currently get nonsense answers like innity for calculated probabilities. The two theories at present cannot co-exist. This is the big problem for physicists today: The reconciliation of quantum mechanics and general relativity into a unied theory. It does not sit well with physicists when they have to stick two theories together that do not t properly, like a piece-wise dened function.
6.4 Section 3: What are the Implications of Some of Modern Physics (Including String Theory, Nanoscience, Dark Matter, Black Holes, Parallel Universes, and The Graviton)?
Question 14: What are some of the implications of quantum mechanics and relativity? In the news there is mention of string theory, black holes, parallel universes, and other bizarre things. One of the theories that is being explored as a possible unication theory (a theory that is more general and works to bring together quantum theory and general relativity) is string theory. The idea is that instead of the universe being composed of small point-particles, it is composed of innitely-thin, rubber-band-like strings that vibrate. Recall how earlier we said that protons are made of three quarks, but the electron is an elementary particle, and it has no building blocks. In saying this about the electron we say that it exists only at a point, and does not have any radius (its not the sphere that you may be picturing). If an electron took up any space at all, there would be some sort of building-block material that is smaller than the electron. However, string theory says that these tiny, vibrating strings are the basic building blocks of all matter (including electrons), and whats more, the theory seems to smooth out the problems that exist between general relativity and quantum mechanics. String theorists are attempting to rectify inconsistencies that have been observed by nding a more general theory that encompasses all of the laws of physics. The length of a string in string theory would be about a Planck length (Planck was a scientist who made great leaps in quantum mechanics), which is about one hundred billion billion times smaller than the nucleus of an atom (thats way too many zeros after the decimal to type here). They are so tiny that scientists cannot even begin to nd experimental evidence of them. Presently string theory lies in the realm of mathematical theory. How would we detect them? Well, it would help to know how scientists currently detect the particles inside an atom. It may seem archaic, but essentially physicists shoot tiny particles at other particles and then measure what happens. Its kind of like closing your eyes and trying to nd the shape and size of an object by throwing marbles at it and watching what the marbles do after they bounce o the object. What happens if you use big marbles as opposed to small marbles? You might gather by intuition that the smaller the marble, the better and more rened your understanding of the mystery object is. Have you ever played with that toy thats made of hundreds of pins in a frame? If you push your hand into the pins and then take your hand away, you can see a picture or relief of your hand, like a mold. If the toy only had a few pins, the picture of your hand would not be very clear, but because there are so many pins that www.ck12.org
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map out the landscape of your hand, you can see a very clear mold of your hand. The same is true for the atom smashers (the fond name we give the machines that speed up particles and shoot them at other particles). The smaller the particles are that we shoot, the better picture we get of the thing at which we are shooting. The problems with general relativity and quantum mechanics occur at lengths a bit shorter than a Planck length, which is about the size of a string, or so it is theorized. This is extremely tiny. If physicists can shoot strings at particles, perhaps they can see inside the atom to its very tiniest of structures. However, we have a problem, assuming strings do exist. When you give a string a lot of energy (higher frequency), after a certain point, it starts to grow in size (again, theoretically), which does not help the cause of trying to peer into the very tiny world of the subatomic (Greene 155). This growing eect is not expected until you try to pump the string with enough energy to probe scales that are smaller than that of a Planck length, so anything larger than a Planck length should still be accessible. Scientists are at a bit of an impasse here, but string theory has a ways to go if its going to be supported by experimental evidence. We are not even close to experimenting at this energy level. There is some hope for string theorists at the new Large Hadron Collider at CERN in Europe. The energy that this atom smasher can give the accelerated particles (the bullets being shot) is much less than it needs to be to see strings (its not even in the ballpark). However, physicists might be able to see the eects of string theory. For example, you may not be able to see around a corner, but you may be able to detect that somebody is standing behind the corner by seeing his or her shadow. You are not directly seeing them, but you see the eects of their existence. One of the results of string theory is that gravity is not just a eld, as you may have learned earlier in the year in your physics class, and physicists may be able to detect this by using the collider at CERN. Lets take a moment to discuss general relativity. Einstein helped us view gravity in a new way that is described in general relativity. We discussed special relativity earlier when we were exploring time dilation and Lorentz length contraction (and the constant speed of light), and we mentioned general relativity, but did not go into a conceptual description. General relativity addresses time and space as a fabric, and Einstein helps us visualize by telling us to picture the space-time fabric as a giant rubber sheet (although in reality space-time is not at like a piece of paper). On this rubber sheet you should picture all the celestial bodies (Sun, planets, stars, etc.) resting. The larger the object, the more it presses down on the rubber sheet. (Please suspend the fact that there is no gravitational force to pull the planets and stars down on the rubber sheet. This isnt quite a perfect metaphor.) This view of space-time helps us to better picture how gravity is communicated from object to object and helps us answer the question of how the Moon knows or feels the presence of the Earth, and thereby causes it to have its present motion. The problem with Newtonian physics is that there is no mention of how planets feel gravity. In Newtonian physics, gravity just is. Newton was aware of this problem as you can see in the quote below. Tis unconceivable [sic] that inanimate brute matter should (without mediation of something else which is not material) operate on and aect other matter without mutual contact. That gravity should be innate, inherent and essential to matter so that one body may act upon another at a distance through a vacuum, without the mediation of anything else by and through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it. (Newton, Letter) Picture yourself and an elephant standing on a trampoline. Even with your eyes closed you could sense the presence of the elephant (although you may not know that its an elephant) by the way it causes you to slide and lean in a little toward it on the trampoline. And, the larger the elephant, the greater it would aect your position next to it. Einstein managed to help us resolve Newtons problem by helping us see that the celestial bodies aect one another through the distortion of the fabric of space-time in which they exist.
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String theory seems to suggest that this isnt quite the end of the story, rather just a blurry view of the real universe. String theory suggests that there exists a small particle that physicists call the graviton that communicates the force, just like the strong force has the gluon to communicate between quarks (called a force carrier). String theorists believe that gravity is not a very weak force, as is now the general thought, but that its strength is lessened because it is spread over more than just our dimension, and that parallel universes exist. These gravitons are thought to travel between these folds of parallel universes, and they are expected to travel at the speed of light and to be massless (only massless particles can travel at the speed of light, a consequence of relativity). As a side note, you may wonder why presently the force of gravity is considered a weak force. It governs the motion of the planets and stars, so at rst thought it seems like it should be a very strong force. But consider how a balloon that you rub on your hair is able to lift your hair against the gravitational force of the earth that is pulling down on your hair. When you rub a balloon on your head, some of the electrons are rubbed o on your hair and transferred to a localized region (balloons are insulators, so any charge you transfer sticks right where you put it) on the balloon, and its a relatively few number of electrons. Just a few electrons can attract the now positively-charged hair on your head (by rubbing electrons o your hair you have taken away negative charge, which leaves an unbalanced positive charge), and lift it very easily, despite the pull of Earths gravity. Gravity, therefore, must be a very weak force as compared to the strong force of electromagnetism. But perhaps string theorists are on to something if the gravitational forces force carriers, gravitons, are spread out over more than one universe (parallel universe); then it would appear weak.
Applications
Black Holes You have probably heard the term black hole and wondered exactly what it is. First, a black hole is not really a hole, a term rst coined by John Wheeler in 1969. He called it a hole because it appears as a black, featureless area in space. So then, what is a black hole? Physicists think that a black hole is formed when a large star (a few times larger than our Sun) runs out of the fuel that maintains it, and because its so large, its own gravitational force pulls it into a dense area of matter that is small, but very massive. Recall from our discussion of general relativity that celestial bodies, such as stars and planets, distort and stretch the fabric of space-time, like giant bowling balls on a rubber sheet. This distortion of space-time aects the path of light, whether its light that may be traveling by, or light emitted by the star actually causing the distortion. Note that the gravitational force only pulls on objects with mass, like planets and stars and particles. Light has no mass, so gravity does not pull on light. However, the actual path on which it is traveling is aected, so the gravitation force does aect the path of light, just not by directly pulling on it. Scientists believe that when a large star collapses its mass becomes distributed over a very small volume (its very dense).This collapse greatly distorts the fabric of space-time, so much so that light cannot escape its distortion. For example, for a space shuttle to escape orbiting our Earth, it has to go a certain speed. This speed is called the escape velocity. The larger the gravitational pull, the faster the object must go to escape its pull (or the distortion of space-time). Think about it this way: When you go around a bend in your car, if you go slowly enough, its easy to maintain circular motion. However, if you speed up, there is a certain speed that will cause you to break free from the frictional force keeping you circling and you will slide o tangentially. The dierence with orbits is that the force causing the circular motion (or centripetal motion) is not friction (like it is with your car). Instead its the gravitational force (or the warping of space-time by Earths large mass). A collapsed star is very massive and creates such a gravitational force (or distorts space-time so much) that the path of light turns right back in toward the center. Its path cant overcome the warping of space-time. Because light is the universes speed limit, nothing else can even come close www.ck12.org
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to escaping the space-time warping from the collapsed star. Thus, a collapsed star is called a black hole, as nothing can escape it, not even light (Whitlock, Gravity). (Recall that to see something you need to detect light bouncing o of it.) Dark Matter Physicists have taken pictures of distant interacting bodies (like a grouping of stars), and after some calculations, have surprisingly discovered that there isnt enough matter there to cause such an interaction. In this case interaction means gravitational pull or orbiting. How can these stars be grouped when the mathematics doesnt seem to add up? Scientists are now conjecturing that there is matter that exists that does not reect light, or perhaps reects just a very small amount of light, so that it cannot be detected. They have called it dark matter. The term dark matter does not infer that it is dangerous or bad, or that its like black holes. Instead the term dark matter means that very little light reects, if any, so it appears dark, or undetectable, or it would be if it were not for the fact that the gravitational forces are not adding up correctly (another example of learning about somethings existence without really being able to see it). Antimatter The term antimatter may sound mysterious, so lets shine a little light on it. Antimatter was predicted before it was experimentally discovered by Paul Dirac, a theoretical physicist who was developing quantum mechanics. In his theory he predicted the existence of a particle that is the same mass as an electron, but has an opposite charge (positive). Later, this particle was called the positron. You might wonder what the dierence is between the proton and the positron, because you already know that the proton has a positive charge. Protons are very large as compared to electrons. Positrons and electrons are the same mass, just opposite in charge. When charged particles move through a magnetic eld, they spiral, and the direction of their spiral depends on their charge. Physicists saw evidence of particles spiraling in two dierent directions, implying opposite charges. However, the particles were the mass of an electron, thus showing the rst evidence of antimatter. Now physicists have been able to produce antiparticles in particle accelerators (like at CERN in Switzerland), however, they never last very long as they are almost immediately annihilated by their corresponding particle (matter and antimatter annihilate each other). For example, if an anti electron (positron) is produced, it will be annihilated by an electron in very little time. Scientists at CERN have been able to produce antimatter (an atom made of antiparticles); they created an anti hydrogen atom by causing a positron to orbit an antiproton. Again, it was short-lived, as it was annihilated by the prevalent electrons and protons we have. It is predicted by quantum mechanics that the creation and annihilation of matter and antimatter happens frequently, but is so fast we cannot detect it, and because the particles annihilate each other, conservation laws are not violated (they end up canceling each other out). This is an application of the Heisenberg uncertainty principle, which we have not discussed. We could continue to discuss and delve deeper into what all this means, all this unintuitive physics, and the implications for how we view our universe, and you should continue thinking and reading about our universe, but for now lets leave with a summary quote from a notable thinker. I know that this dees the law of gravity, but, you see, I never studied law. (Bugs Bunny)
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Einstein, Albert. Letter to Heinrich Zangger. 20 May 1912. CPEA, Vol 5 Doc. 398. Feynman, Richard. QED: The Strange Theory of Light and Matter. Alix G. Mautner Memorial Lectures. Princeton University Press. 1986. Greene, Brian. The Elegant Universe. 2000. Newton, Isaac. Original Letter from Isaac Newton to Richard Bentley. Newton Project. 25 Jan 2009. http://www.newtonproject.sussex.ac.uk/texts/viewtext.php?id=THEM00258& mode=normalized Pogge, Richard W. Real-World Relativity: The GPS Navigation System. 15 Dec 2008. http: //www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html Smolin, Lee. The Trouble with Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next. Boston: Houghton Miin, 2006. Weidner and Sells. Elementary Modern Physics. Boston: Allyn and Bacon, 1980. Whitlock, Laura. Ask an Astrophysicist. How Gravity Eects Photons. NASA Goddard Space Flight Center. 15 Dec 2008. http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/961102. html
Image Sources
(1) Angela Cutshaw. Diagrams of distance traveled by light. CC-BY-SA. (2) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (3) Angela Cutshaw. Diagrams of waves with dierent frequencies and wavelengths. CC-BY-SA.
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Chapter 7 Nanoscience
Tapas Kar. Nanoscience, 21st Century Physics Flexbook.
7.1 Introduction
The little word, nano, has been rapidly insinuating itself into our consciousness because of its big potential. In the media, nano has captured headlines in television news channels and almost every technical and scientic journal. A number of instruments with nanometer-scale resolution made this possible. We are entering the era of nanoscience and nanotechnologymany remarkable mysteries lie ahead and several fascinating developments are forthcoming. The application of nanotechnology has enormous potential to greatly inuence the world in which we live. From consumer goods, electronics, computers, information and biotechnology, to aerospace, defense, energy, environment, and medicine, all sectors of the economy are to be profoundly impacted by nanotechnology. Properties (chemical, electrical, mechanical, and optical) of materials used in these sectors changes signicantly in nanoscale than their bulk form. Expected impact of nanotechnology on dierent sectors is illustrated in the following pie-chart, created by Lux Researchan independent research and advisory rm providing strategic advice and ongoing intelligence for emerging technologies.
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Figure 7.1: Lux Research Pie Chart is in its late emerging stage in a number of applications. The U.S. government sponsored a National Nanotechnology Initiative in 2000, which was aimed at supporting and encouraging early growth.
What is Nano?
To understand nanoscience and nanotechnology, we have to rst know what is nano? Nano means dwarf in Greek and it is a prex in the metric scale. Thus, a micrometer (m) is one-millionth (106 ) of a meter and a nanometer (nm) is one-billionth (109 ) of a meter. Larger scales are easier to conceptualize than smaller scales. The following are some examples that provide a sense of scale (small) for milli-, micro-, and nanometer objects. www.ck12.org
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Understanding Size
Although in the United States the standard unit of length is foot, the meter is the standard unit of length used in many other countries. Let us rst examine the relationship between a foot and a meter. 1 foot = 0.3048 meter or 1 meter = 3.2808 feet 1 yard = 0.9144 meter or 1 meter = 1.0936 yards 1 mile = 1.609 kilometer or 1 kilometer = 0.6216 miles Online conversion calculator: http://www.onlineconversion.com/length_common.htm
Figure 7.4: Dime A CD or DVD is thinner than a dime. The diameter and thickness of a CD or DVD are 120 mm and
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Figure 7.5: CD or DVD We can see objects as small as 0.05 millimeter (mm)that is the limitation of the human eye. For example, the typical width of a human hair is 0.05 mm.
How Small is the Smallest Thing You Can See Under a Microscope?
The smallest object that can be seen under a microscope is about: 0.2 0.5 m (micrometer) = 0.0002 0.0005 mm = 0.0000002 0.0000005 m (meter) 1 1 m (micrometer) = (meter) 1, 000, 000 1 106 of a meter = (millimeter) 1000 If you could split a human hair into 50 separate strands, each would be about one micrometer (m) wide. www.ck12.org
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1 109 of a meter = 1,000,000,000 m (meter) 1 One-billionth of a meter = 1,000,000 mm (millimeter) 1 One-millionth of a millimeter = 1,000 m (micrometer)
If you could split a human hair into 50, 000 separate strands, each would be a nanometer (nm) wide. In fact, human hairs grow by one nm every few seconds. To see nanometer scale objects, we need an electron microscope, in which electrons are used instead of light, to see nanometer scale objects. An electron microscope can resolve objects about 1000 times smaller than an optical microscope, enabling magnications of 1,000,000 times, without loss of detail.
Step-by-Step Magnication
Periodic Table and description of Elements: http://www.webelements.com/ Periodic Table and description of Elements: http://www.chemicool.com/ Periodic Table and description of Elements: http://www.lenntech.com/Periodic-chart.htm So at the nanometer scale we see molecules (a combination of dierent atoms connected by bonds). For
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example, any form of water (ice, snow, water vapor) is a combination of two hydrogen (H) atoms and one oxygen (O) atom, where the oxygen-hydrogen distance is about 0.1 nm.
Figure 7.9: Water molecule. Red and gray balls represents oxygen and hydrogen atoms, respectively.
Figure 7.10: Dierent Objects on the Nanoscale. More examples of step-by-step magnication: http://micro.magnet.fsu.edu/primer/java/scienceopticsu/ powersof10/
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Nobel Prize winner Dr. Horst Strmer said that the nanoscale is more interesting than the atomic scale (> 0.3 nm) because the nanoscale is the rst point where we can assemble somethingits not until we start putting atoms together that we can make anything useful. On the nanoscale, we can potentially assemble atoms together to make almost anything. For example, oxygen and hydrogen found in the human body is mostly as a component of water (H2 O) molecule. Carbon, hydrogen, and oxygen are integral components of all proteins, nucleic acids (DNA and RNA), carbohydrates, and fats. The combination of all of these molecules creates the living cells of the body.
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1974
The term nanotechnology was coined by Tokyo Science University Professor Norio Taniguchi to describe the precision manufacturing of materials with nanometer tolerances http://en.wikipedia.org/wiki/ Norio_Taniguchi Why did it take so long to implement nanotechnology? Because there was no tool to see and work on such a small scale.
1981
Gerd Binnig and Heinrich Rohrer invented the scanning tunneling microscope (STM), which can image atomic-sized objects. Electron microscopes help technology to move from micro-to nanoscale. www.ck12.org
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1985
C60 fullerene (also known a buckminsterfullerenes or bucky balls), a new form of carbon, was discovered by Robert F. Curl, Jr., Sir Harold W. Kroto, and Richard E. Smalley.
1986
K. Eric Drexler, in his 1986 book Engines of Creation: The Coming Era of Nanotechnology, proposed the idea of a nanoscale assembler, which would be able to build a copy of itself. For more information about K. Eric Drexler: http://en.wikipedia.org/wiki/K._Eric_Drexler
1991
Sumio Iijima, a researcher at NEC in Japan, discovered the carbon nanotube; he went on to produce an advanced, single-walled version in 1993.
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Figure 7.17: Fullerene, diameter . A soccer ball is a model of buckyball, but times larger.
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Figure 7.19: Dierent forms of single-wall carbon nanotubes. These are hollow tubes made from carbon atoms and their diameters vary from 0.5 to 3 nm. The longest tube synthesized so far is a few millimeters long. The discovery of fullerenes and nanotubes helped to expedite nanotechnology.
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switch), 7, 200 crystal diodes (blocks electricity at certain conditions and allows it to pass when those conditions change), 70, 000 resistors (limits the ow of electricity), 10, 000 capacitors (collects electricity and releases it all in one quick burst), and around 5 million hand-soldered joints.
Figure 7.20: First Digital Computer ENIAC The size of the vacuum tube, which is a key component of the computer and other electronic devices (such as the telephone, radio, and TV), is about 5 30 millimeter (mm).
Figure 7.21: Vacum tubes The vacuum tube (invented in 1941) was replaced by much smaller millimeter scale transistors in 1955. In 1971, Intel introduced the rst microprocessor, which contained about 2300 transistors for use in a calculator. In the following year, Intel doubled the number of transistors in an 8bit microprocessor designed to run computer terminals. The number of transistors in current processors, such as in the Pentium 4 is more than a few million, and the size ranges between 0.2 m to 0.06 m each. Presently, Intels Duo-core chips contain 191 million transistors in 143 square millimeter area, and the Quad-core Itanium chip (launched in Feb. 2008) packs more than 2 billion transistors in 65 nanometers is almost the
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same size as the chip. The size of the transistor is further decreased by Taiwanese Chipmaker TSMC to 40 nm, and recently IBM developed a 22.9 nm chip.
Figure 7.22: Transistors Over the last 40 years, the size of the transistor, which is a key component of almost all electronic gadgets used today, was reduced in size from a millimeter to a micrometer to a nanometer. The mid-80s to 2006 07 marked the period when technological development was based on micro (one-millionth of a meter) size components, and hence, termed microtechnology. Similarly, the current use of nanometer sized components (size less than 100 nm) deem calling it nanotechnology. In the future, we will use single molecule transistors of sizes less than 1 nm. View animation of single molecule transistor: http://stm.phys.ualberta.ca/wolkow/molecular/WebMidRezAudio. mov
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Figure 7.23: A microprocessor incorporates most or all of the functions of a central processing unit (CPU) on a single integrated circuit (IC) or chip.
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Figure 7.25: Hard disk drive and hard disk Microtechnology In 1980, Seagate Technology introduced the rst hard disk drive for personal computers. It was 5 1/4 drive and held 5 MB.
Figure 7.26: The large drive is a full-height drive. The smaller drive is a IDE drive. These drives also contained the disk. Currently, a drive is able to hold more than worth of data. Nanotechnology Atoms will be used in future drives and about 1 million GB worth of data may be stored in one square cm area. In summary, miniaturization of man-made devices signicantly improves eiciency, capacity, and functionality of all electronic gadgets, and at the same time saves lots of electrical energy.
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The human eye cannot see electron wavelengths; therefore, we need a television-type screen or special photographic lm to make electron microscope images visible to human eyes. Electrons have a much smaller wavelength than light (400 700 nm) and thus resolve much smaller objects. The wavelength of electrons used in electron microscopes is usually 5 to 0.05 nm. There are two types of electron microscopesthe Scanning Electron Microscope (SEM) and the Transmission Electron Microscope (TEM). The SEM is a type of electron microscope that images the sample surface by scanning it with a high-energy beam of electrons. The electrons interact with the atoms that make up the sample, producing signals that contain information about the samples surface topography, composition, and other properties such as electrical conductivity. The TEM beam of electrons is transmitted through an ultrathin specimen, interacting with the specimen as they pass through and then scatter providing a 2-D image of the specimen. The Scanning Transmission Electron Microscope (STEM) is a combination of SEM and TEM. The other kind of electron microscope uses a probe that scans the surface of objects providing 3-D images of atomic networks at the surface. Extremely sharp metal points that can be as narrow as a single atom at the tip is used in scanning probe microscopes. The Scanning Tunneling Microscope (STM) is an example of this type of microscope. Another type of scanning probe microscope is the Atomic Force Microscope (AFM). As the probe in an AFM moves along the surface of a sample, the electrons in the metal probe are repelled by the electron clouds of the atoms in the specimen. As the probe moves along the object, the AFM adjusts the height of the probe to keep the force on the probe constant. A sensor records the up-and-down movements of the probe, and feeds the data into a computer to construct a 3 D image of the surface of the sample. AFM and STM enable us to work on atoms and design molecules the way we want by placing atoms by atoms. An excellent example is placing 48 iron atoms (step-by-step) to form a quantum coral (see image at the bottom right-hand corner of Figure 11 and check out this Web site http://www.almaden.ibm.com/ vis/stm/corral.html. Here are some additional links to electron microscope images: http://www.mos.org/sln/sem/sem.html
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Figure 7.31: Block Diagram of Atomic Force Microscope (AFM) http://www5.pbrc.hawaii.edu/microangela/ http://www.denniskunkel.com/ http://www.ou.edu/research/electron/www-vl/image.shtml Applications of Atomic Force Microscope (AFM): http://www.pacificnanotech.com/application_part.html
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without defects in structure can be made. Also SEM tip can be used to design and create nanostructures by placing atom by atom. This process is tedious and time consuming and is not useful for industrial purposes.
Magic of Carbon
Carbon is one of the most abundant elements. It is not only the key element in all known life forms, but it is also present in several common materials that we use in our daily life. For example, coal, gasoline, pencil, pitch, and aromatic compounds are all carbon based. Carbon has a unique capacity to form bonds with itself and many other elements making possible to form millions of compounds.
Figure 7.32: Diamond (left) and graphite (right) are two allotropes of carbon: pure forms of the same element that dier in structure.
Fullerenes
Fullerenes (also known as buckyballs) and carbon nanotubes are new forms of carbons that were discovered in the late 1980s. The rst fullerene reported was a hollow ball that contained sixty carbon atoms. There
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are 12 pentagons and 20 hexagons in C60 and each pentagon is surrounded by 5 hexagons and each hexagon is surrounded by alternating hexagons and pentagons. At present, several other cage structured fullerenes containing 50 to 540 carbon atoms are available. Traces of fullerene are available in nature and several chemical methods are developed to synthesize pure (99.9%) fullerenes. Carbon nanotubes are synthesized in laboratories.
Figure 7.33: Dierent forms of Carbon (allotropes of carbon) : a) Diamond, b) Graphite, c) Lonsdaleite, d) C60 (Buckminsterfullerene or buckyball), e) C540, f) C70, g) Amorphous carbon, and h) single-walled carbon nanotube or buckytube. Because of their unique structure and properties (semiconducting and electron acceptor), fullerenes can be used in dierent technologically based areas, such as the solar cell, trapping active molecules inside the cage, drug delivery, and bio-sensors.
Carbon Nanotubes
Carbon nanotubes can have dierent forms depending on how a single hexagonal graphitic sheet is rolled to form the nanotube. Depending on their structures, carbon nanotubes can be either metallic or semiconductors. Figure 7.34 is an illustration of single-wall carbon nanotubes (SWCNT). Double-wall and multi-wall (MWCNT) nanotubes are also synthesized in the laboratory. However, synthesis results in a mixture of all kinds of nanotubes and it is hard to separate them. This has hindered some applications of individual carbon nanotubes, and current research is progressing to separate them. However, it should be noted that nanotubes are not synthesized by rolling graphite sheet(s), tubes simply resemble rolled up graphite sheets. The following image illustrates the possibility of dierent forms of SWCNTs that can be related to rolling patterns of hexagonal networks of graphite sheets. It can be seen from the following tables that the carbon nanotube is lighter than aluminum but stronger www.ck12.org
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Figure 7.34: Dierent Forms of Single-wall Carbon Nanotube. than steel. Table 7.1: Material Single-wall carbon nanotube Multi-wall carbon nanotubes Steel Aluminum Titanium Elastic (GPa) 1210 1260 207 69 103 Modulus Strain (%) 4 1.5 9 16 15 Yield (Gpa) 65.0 2.7 0.8 0.5 0.9 Strength Density (g/cm3 ) 1.3 1.8 7.8 2.7 4.5
Table 7.2: Comparison of Stability, Electrical and Thermal Properties of CNTs with Other Material Used Currently Properties Size (diameter) SWCNT: 0.61.8 nm MWCNT: 2050 nm Temperature stability Thermal conductivity Stable up to 2, 800 C in vacuum, 750 C in air 141 Predicted to be as high as 6, 000 W/m K at room temperature Nanotubes Current Materials Electron beam lithography can create lines 50 nm wide, a few nm thick Metal wires in microchips melt at 600 1, 000 C www.ck12.org Nearly pure diamond transmits heat at 3, 320 W/m K
Figure 7.36: This image is a nanometer carbon nanotube, lled with several cobalt nanoparticles.
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Hydrogen storage Single electron transistors Molecular quantum wires Thermal protection Magnetic nanotube
Figure 7.38: dimethylaminoazobenzenesulfonic Acid Sodium Salt Some metals in bulk form also possess color. For example, gold is a yellowish orange color when its dimensions are more than 100 nm. The color changes to green when the particle size is 25 nm and to red/ruby at 25 nm. Similarly, silver is yellow at 100 nm, but blue at 40 nm. These forms of tiny crystals of gold and silver are termed nanocrystals.
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The stained glass windows in churches are good examples of gold and silver nanoparticles or nanocrystals. Medieval artisans unknowingly became nanotechnologists when they made red stained glass by mixing gold chloride into molten glass. That created tiny gold spheres, which absorbed and reected sunlight in a way that produced a rich ruby color. While some of these stained glasses were made more than 1000 years ago, their color has maintained its brightness and saturation.
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Figure 7.40: Quantum Dots areas such as in solar cells, displays, light emitting devices (LEDs) and life sciences. QDs will replace present organic dyes used as biosensors and biomedical imaging. Further reading: http://www.evidenttech.com/quantum-dots-explained.html
Risks Factors?
Because materials at the nanoscale behave dierently than they do in their bulk form, there is a concern that some nanoparticles could be toxic. Nanoparticles are so small that they could easily enter living cells and cross the blood-brain barrier, a membrane that protects the brain from harmful chemicals in the bloodstream. More powerful weapons, both lethal and non-lethal, may be created using nanotechnology. Because of their light weight, a small quantity of useful or harmful nanomaterials could easily be smuggled into the wrong hands.
Image Sources
(1) CK-12 Foundation. Single-wall carbon nanotubes. CC-BY-SA. (2) CK-12 Foundation. CD or DVD. GNU Free Documentation.
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(3) CK-12 Foundation. Table 1. Metric Scale and Prexes. CC-BY-SA. (4) Gerd Binnig. Public Domain. (5) CK-12 Foundation. Richard P. Feynman. Public Domain. (6) CK-12 Foundation. Single molecule transistor. Public Domain. (7) CK-12 Foundation. Future hard drive. CC-BY-SA. (8) Quantum Dots. CC-BY-SA. (9) Atomic Force Microscope. Public Domain. (10) Lux Research. [www.luxresearchinc.com Lux Research Pie Chart]. CC-BY-SA. (11) Bob Hanvey. Hard Disk Drive and Hard Disk. CC-BY-SA. (12) CK-12 Foundation. Human hair. CC-BY-SA. (13) How Small is a Nano?. CC-BY-SA. (14) CK-12 Foundation. Water Molecule. CC-BY-SA. (15) Dierent forms of Carbon. GNU Free Documentation, Version 1.2. (16) First digital computer ENIAC. Public Domain. (17) Dierent Objects on the Nanoscale. CC-BY-SA. (18) CK-12 Foundation. Step-by-Step Magnication. CC-BY-SA. (19) Fullerene. CC 3.0. (20) CK-12 Foundation. Single wall carbon nanotubes. CC-BY-SA. (21) Tapas Kar. 4-dimethylaminoazobenzene -4- sulfonic acid sodium salt. CC-BY-SA. (22) CK-12 Foundation. Dierent Forms of Single-wall Carbon Nanotube. CC-BY-SA. (23) Optical microscope. Public Domain. (24) Stained glass window. Public Domain. (25) How the Scanning Tunneling Microscope works. CC-SA 2.0. (26) Methyl orange. Public Domain. (27) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (28) Block Diagram of Atomic Force Microscope (AFM). Public Domain. (29) This image is a 12 nanometer carbon nanotub. CC-BY-SA. (30) CK-12 Foundation. Nature and man-made things in dierent scales. CC-BY-SA. (31) Scanning Electron Microscope. Public Domain. (32) wiki/File:Transistorer_(croped).jpg Transistors. Public Domain. (33) CK-12 Foundation. Table 2. Technology at a Dierent Scale. Public Domain. www.ck12.org
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(34) CK-12 Foundation. Time frames of Development of Technology. CC-BY-SA. (35) A microprocessor. Public Domain. (36) Heinrich Rohrer. CC 3.0. (37) Comparison of large and hard drive.. CC-BY-SA. (38) CK-12 Foundation. Dime. Public Domain. (39) wiki/File:Rembrandt_Harmensz._van_Rijn_013.jpg Diamond (left) and graphite (right). GNU Free Documentation. (40) Sumio Iijima. CC 3.0. (41) Vacum tubes. Public Domain.
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8.1 Ultrasound
Benets of Ultrasonography
Noninvasive: the probe does not breach the tissue and reliable imaging can be recorded without surgery. Inexpensive and routinely available in North America. Provides a clearer picture of soft tissues that do not image well using ionizing radiation. Provides ow rates for blood using Doppler technique. Provides immediate images which can be used to guide other procedures or surgeries. Can be repeated reasonably often. Can be used on possibly pregnant women and on fetuses.
Ultrasound imaging provides a view of the human body that is not accessible by other means. While more energetic electromagnetic beams like Xrays penetrate the body including bones, ultrasound imaging or sonography uses sound waves to image soft tissues. Ultrasound imaging/sonography has been used to image fetuses in pregnant patients. This is especially important because the fetus is sensitive to many kinds of energetic probes which might otherwise be used. Xrays, for example, are known to cause changes in fetal DNA. The ultrasound image can be produced in real time so that the image can be used to guide surgical procedures. The equipment is routinely available in North America for a nominal cost. Advanced ultrasound machines can use the Doppler Eect to determine speed of blood ow in arteries and veins. By analyzing the blood velocity, physicians can locate aneurysms and blood clots. Risks and Shortcomings of Ultrasonography
No known risks, but major medical organizations such as the World Health Organization have discouraged the popular practice of imaging fetuses to determine sex or to take home movies. Very limited ability to image structures with bone or air. See the section on impedance. Very long period (>30 minutes) of ultrasound associated with damage in small rodents. www.ck12.org
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Ultrasound has been generally recognized as a safe procedure when used for medically signicant imaging. The recent popularity of making home movies of the developing fetus has been discouraged by major medical organizations. There is some danger of thermal heating of the tissue by long exposures at high power. A recent study found that prolonged exposures of over thirty minutes to developing rat fetuses produced some genetic damage. This is an area of ongoing research.
More Risks
One of the risks of any diagnostic device is that the energy beam is thermalized by the body; that is, the incoming energy heats the tissue. Because homeostasis (maintaining the same temperature) is the hallmark of mammals, changing the temperature of target tissue is a problem. As a worst case scenario, assume that the probe delivers 10 W of energy to a cylinder of tissue 10 cm deep and 5 cm in diameter for 15 minutes. Also assume that there is no blood ow to the aected area so that the heat stays in the cylinder and that the technician does not move the probe for the entire 15 minutes. The energy delivered to the tissue is 10 W 60 s/min 15 min = 9, 000 J. The change in temperature of the tissue can be compued by assuming that the energy of the ultrasound is converted into heat:
Q = mc T Q = V c T Q T = Vc T =
where Q is the change in the internal energy associated with a temperature rise T , m is the mass of the tissue, c is the heat capacity, and T is the temperature. Substituting for the mass m = V, where $$ is the density and V is the volume, yields the second equation. Solving for the change in temperature gives the third equation. The nal temperature change is about 11 K, which is appreciable. Clearly, there is the possibility of changing the temperature of the target tissue under extreme circumstances.
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Figure 8.1: Rayleigh criterion for dierent wavelengths When the minimum of one peak just overlaps the maximum of the next peak, the two peaks are resolved. If the peaks are closer, then they cannot be told apart.
where vmaterial is the speed of sound in the substance and is the wavelength of the sound wave. The speed of sound in a solid or a uid depends on the density of the material, , and the stiness of the material. For uids, this is the bulk modulus, B, while for solids this is usually Youngs modulus, Y. The speed of sound in human tissue varies by more than a factor of three. The speed of sound depends on the density of the tissue. While human bone is fairly dense, subcutaneous fat is much less dense than water. The overall density of a human is just about that of water. How do you know? Because the average human just barely oats in water. The more muscle and bone that a person has, the lower that person oats. The other factor for determining the speed of sound is the stiness of the material. Again bone is fairly sti while subcutaneous fat is not. The stiness is measured by the bulk modulus for uids or Youngs modulus for solids. B sound = Table 8.1: Material air human soft tissue www.ck12.org Velocity (m/s) 331 1540
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Table 8.1: (continued) Material human brain & amniotic uid liver kidney blood muscle skull-bone fat Velocity (m/s) 1541 1549 1561 1570 1585 4080 1450
(Taken from http://www.yale.edu/ynhti/curriculum/units/1983/7/83.07.05.x.html) Which references Christensen, E. E., Curry,T. S., Dowdey, J. E.: Introduction to the Physics of Diagnostic Radiology. Philadelphia: Lea & Febeger, 2nd Edition; 1978: Chapter 25. Most soft tissues where the ultrasound is most eective have a speed of sound that is about 1550 m/s, which is about ve times faster than the speed of sound in air. Everyone is familiar with watching a distant event like a lightning strike where the light arrives almost immediately, but the sound of the thunderclap arrives some time later (about 5 seconds for every mile away).
or about 154 kHz. As the size of the target decreases, the frequency increases. To image a target that is 1.0 mm, the imaging beam must have a frequency of 1.54 MHz. The normal range of diagnostic ultrasound is 7 9 MHz. Now try some problems. 1. What is the frequency required to image an object that is 1 mm in diameter?
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2. What is the frequency required to image an object that is 0.2 mm in diameter? 3. A typical medical ultrasound is 9 MHz. What is the smallest object that can be imaged with this wave in human tissue?
Echolocation
The process of imaging is the same as the echo-locating sonar of a submarine or a bat. The observer sends out a brief pulse of ultrasound and waits for an echo. The pulse travels out, reects o the target and returns. The ultrasound machine uses pulses because the same device acts as both transmitter and receiver. If it continually sent out sounds, then the receiver would not hear the much softer echo over the louder transmission. The duty cycle of the ultrasound imager is the amount of time spent transmitting compared to the total time of transmitting and listening.
Figure 8.2: Reected Wave The pulse travels out and returns to the transducer where it is converted to electrical signal. But the same device is both sender and receiver. Duty cycle: emit pulse, wait, and listen. Same procedure as SONAR.
Wait Time
In order to be as eicient as possible the machine should send out the next pulse just after the target pulse arrives. To calculate the wait time: t= 2d 2 0.05 m = = 65 s material 1540 m/s
Notice that there is a very short return time for the echo. Some bats do this naturally and even change the duty cycle as they close in on their prey. Essentially, as soon as the bat hears an echo, it sends out a new chirp for additional information. The duty cycle is the amount of time that the probe is producing a pulse compared to the time that it is listening. www.ck12.org
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The size of the object that can be imaged with the transducer is a function of wavelength, therefore, the user should move to a transducer with the highest frequency and smallest wavelength. But, sound waves that travel are subject to attenuation, i.e., gradual loss of intensity.
Attenuation
All of this suggests that in order to image very small targets, the frequency of the ultrasound should be increased to something like 150 MHz, which is certainly technically feasible. But this is where the other aspect of sound travel through a medium comes into play. Waves can lose their energy by scattering or by absorption. Together these two processes are called beam attenuation. Scattering results from parts of the beam deecting from the straight path of travel. The most familiar example is the scattering of sunlight by our atmosphere. While the Sun is very bright, the rest of the sky is lit by the scattered light of the atmosphere. Scattering is often most important in mixtures and materials that are heterogeneous. The scattering of sunlight is aided by solid dust particles in the air. The most famous example of dust scattering sunlight was the spectacular sunsets that resulted from the famous Krakatoa volcano eruption in 1883. Scattering is not as important in ultrasound as absorption. The ultrasound beam is thermalized as the sonic energy is converted into tissue heating. This problem of absorption of the sound has a large impact on the amount of echo that returns to the probe. Scattering Where a portion of the wave deects from the straight line path and is lost. Typically this will happen more often in materials that are heterogeneous (mixed) materials (not generally a problem here). Absorption Where the wave energy is converted into thermal heating of the material. Absorption is a major issue in ultrasound imaging.
Absorption Coeicient
The absorption process follows rst order kinetics, which is familiar from radioactive decay. The intensity is measured as a function of the distance x travelled in the tissue, i.e., I = I(x). The change in intensity I = I(x + x) I(x) is proportional to the intensity and the distance x, i.e., I = I(x) x. Here is the absorption coeicient in units of inverse meters. It follows that the intensity decays exponentially with distance, I(x) = I0 ex where I0 is the initial intensity at x = 0. The absorption coeicient governs how rapidly the sound is absorbed. The absorption coeicient is the product of two factors, the medium and the frequency of the ultrasonic beam. The frequency matters because the transfer of energy from the ultrasound beam to the tissue is more eicient if the frequency of the ultrasound matches the frequency of a (microscopic) process in the tissue. This is called resonance and is familiar from a child on a swing set. The parent exerts a small force each time the child is closest to him/her. If these forces are synchronized with the oscillations of the child, theses pushes add up, resulting in a large amplitude [and thus energy] of the child.
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The competition between the need for the highest possible frequency to provide a good target image and the need for the lowest possible frequency to return a good echo is usually made around 7 2 MHz, which provides reliable imaging of objects that are 0.20 mm in size. The absorption coeicient is the product of a coeicient that depends on the medium and the frequency f of the ultrasonic beam,
= 2 f I(x) = I0 e2
f x
This means that as the frequency is increased (wavelength decreases), more of the signal is absorbed by the medium. Some Sample Amplitude Absorption Coeicients Table 8.2: Tissue blood abdomen fat soft tissue muscle bone lung y sound (s/m) 2.1 106 5.9 106 7.0 106 8.3 106 2.3 105 1.6 104 4.7 104
(Taken from Irving, H. B., Physics of the Human Body. Berlin: Springer Verlag; 2007:562.)
Sample Problems
1. Calculate the attenuation for a 15 MHz ultrasound that penetrates average soft tissue for a distance of 5 cm and returns to the transponder. Consider the initial beam to be 100%. 2. Repeat the calculation for a beam with twice the wavelength (7.5 MHz). 3. Repeat the calculation for a beam with half the distance (2.5 cm) and twice the wavelength (7.5 MHz).
Equipment
The probe for the ultrasound is a transducer. This is a crystal of piezoelectric material (piezo is Greek for pressure or squeezing). The most common example of piezoelectricity is the high school demonstration with a crystal mounted under a lever. As the lever is rocked the crystal is squeezed and electric sparks shoot across a gap beside the crystal. A somewhat similar, but not exactly the same eect, is the triboluminescence of crushing Wint-O-Green Lifesavers. Try It! Take some fresh Wint-O-Green Lifesavers into a darkened room with a mirror. Put one in your mouth and crush it. You should be rewarded with a visible blue spark as the electrons are moved out of the sugars www.ck12.org
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by crushing. The Wint-O-green avor, methyl salicylate, (oil of wintergreen) acts to convert the normally ultraviolet light of this transition into visible light. When an electric eld is applied to the crystal it contracts, as the eld is reversed, the crystal expands. The contraction and expansion produce a pressure sound wave. The same process works in reverse so that when the echo comes back the pressure wave produces an electrical signal. The compression high pressure will cause the crystal to produce one electric eld and the low-pressure rarefaction will produce the opposite electric eld.
Figure 8.3: Transducer Transducers consist of a piezoelectric material. A varying electrical signal will cause the material to contract and expand, which produces a pressure sound wave. The same process can work in reverse: A sound wave hitting the piezoelectric material will give rise to a varying electrical signal.
Another Problem
The other transmission problem occurs when the ultrasound wave encounters a new medium with a dierent speed of sound. As the incident intensity of the beam encounters the new material it is either reected or transmitted (refracted). The amount transmitted plus the amount reected must be equal to the incident amount in order to conserve energy. The amount that is reected compared to the amount that is transmitted depends on a property of the two materials called the impedance. The short answer is that if the impedances of the two materials match then the ultrasound is transmitted. If they dont, the wave is reected. The ultrasound wave represents a beam energy that must move into the new material if transmission is to occur. But, because the materials are dierent, the materials have dierent speeds of sound. An analogy is an airplane delivering packages to an airport that then transfers those packages to trucks for home delivery. If the air freight plane delivers packages (energy) faster than the trucks can haul them away, packages pile
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up. In that case, the packages are sent back (re-elected). Impedance, or complex resistance, can also be found in electricity, specically in an LRC circuit. The inductor (L), resistor (R), and capacitor (C) have a natural resonance where the electric current is the highest. Two circuits with matching impedances will resonate together. In the case of ultrasound, two processes can happen when the beam changes two media: Reection and refraction. As the ultrasound pressure wave hits the boundary between the media, there can be no net pressure so exactly how much is reected and how much is transmitted (refracted) depends on the impedances of tthe media. If the impedances match then ALL of the incident intensity if transmitted. For example, if the material is the same on both sides, then the beam is transmitted and not reected.
More Impedance
The intensity, I, of some pressure sound wave is dened as the power P per unit area A. But the power is the kinetic energy per unit time. Expressing the mass of the kinetic energy term as the product of the density, , and the volume, V, yields the fourth equation. But the volume is simply the unit area A times the distance that the wave travels at the speed of sound c in time t, i.e., V = A ct. I= = = = = = Ek P = A At 0.5 mag 2 At A ctavg 2 2At cag 2 2 1 (c)ag 2 2 1 Zag 2 2
Cancellation of the A and t terms yields an equation with the impedance, Z, dened as the speed of sound in the material times the density of the material, Z = c . The impedance, Z, of human tissue is not that dierent from water, but is markedly dierent for air and bone. This is natural because of the dierences in speeds and densities of the materials compared to water. This means an ultrasound beam will travel quite readily from water to tissue to muscle and back but will reect o of air (lungs) or bones. The amount of reectance can be quite remarkable as the table shows. The amount that is transmitted plummets from 99.8% for water and soft tissue to 0.10% for air and soft tissue. Table 8.3: Substance air water fat muscle www.ck12.org Density ( kg/m3 ) 1.29 1, 000 940 1, 058 Speed (m/s) 340 1, 496 1, 476 1, 568 Impedance (Pa s/m) 439 1, 490, 000 1, 390, 000 1, 600, 000
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Table 8.3: (continued) Substance bone Density ( kg/m3 ) 1, 785 Speed (m/s) 3, 360 Impedance (Pa s/m) 6, 000, 000
Transmission
The actual intensity that is transmitted can be calculated by taking the ratio of the impedances times the incident intensity. The key point to see here is that the ultrasound wants some of the energy to be reected in order to have an echo for imaging. But when all of the energy is reected, nothing beyond that material can be directly imaged. The most common example of this impedance mismatch is the way that sounds travel very far across water surfaces like lakes. It is not uncommon to hear conversations that are occurring a mile away as if they were in the same room. Another example of this impedance mismatch between water and air is the way that sounds are mued when underwater. Next time you are in a pool, have someone yell at you while you are underwater. The sound reects but does not transmit into the water. At the same time, if someone clangs on a pool ladder in the water, the sound travels quite well to your underwater ears. Itransmitted = Ztransmitted Iincident Zincident
The fraction transmitted is dependent on impedance matching. For water/air, Z/Z = 3, 416. Almost all of the sound wave is reected whether from air to water or water to air. Table 8.4: Interface water/soft tissue fat/muscle bone/muscle soft tissue/bone bone/fat soft tissue/lung air/muscle air/water air/soft tissue Reect (%) .23 1.08 41.23 43.50 48.91 63.64 98.01 99.89 99.90 Transmit (%) 99.77 98.92 58.77 56.50 51.09 36.36 1.99 .11 .10
Notice that just getting the ultrasound beam into the body is a problem as most of the energy is reected at the air interface. A special gel is used to make good acoustic contact (match impedances) between the transducer and the body.
Doppler Eect
The Doppler eect is the change in the frequency as heard by a listener compared to the frequency emitted by the source. As the listener moves closer to the source, the listener encounters more waves in the same
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time. The listeners frequency is higher than the source. The reverse is true for the listener who moves away from the source. An analogous change occurs when the listener is stationary and the source moves. The summary of the Doppler eect is that when the distance between the source and listener decreases, the listener hears a higher frequency. When the distance between the source and the listener increases, the listener hears a lower frequency. (c ) L fL = fS c Moving Listener A similar argument for a listener moving away results in the same equation but with a minus sign. fL is the frequency that the listener hears. f s is the frequency of the source. vL is the velocity of the listener. ( f L = fS Moving Source fL is the frequency that the listener hears. f s is the frequency of the source. v s is the velocity of the source. c is the speed of sound. ( f L = fS c L c S ) c c S )
Moving Source and Listener Notice that if the source and the listener are moving in the same direction at the same velocity, the result is that the frequency is unchanged. As the listener and source close in on each other, the frequency will increase. As the listener and the source move away from each other, the frequency will decrease. Use relative motion to simplify problems by stopping the slower object.
Doppler Demo
Now Try This! 1. Either use a tuning fork on a string or a constant frequency speaker (a piezoelectric buzzer with a 9-volt battery). 2. Start the buzzer or strike the tuning fork. 3. Twirl around your head and listen for the Doppler shifted sounds. 4. This is more eective as the speed of the sound source increases. Did you hear it? That was the Doppler eect! What does this have to do with Ultrasound? www.ck12.org
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Blood Flow
Ultrasound can be used to diagnose the speed of blood ow by using the Doppler eect. The procedure is non-invasive because it doesnt require inserting a probe into the blood vessel. If the blood is owing at 2 cm/s in the blood vessel then the Doppler eect calculation shows that the change in the Doppler frequency is about 0.0025%, which would be incredibly diicult to measure. But the imaging system doesnt measure the frequency directly; instead, it mixes the echo with the known original signal to produce a beat frequency (which is just the dierence between the two frequencies). In this case the beat frequency is 178.3 Hz , which is very easy to accurately measure. This is a common practice in physics to measure an unknown signal precisely by mixing it with a known frequency to produce a beat frequency. ( c )
L
f_L = f_S
( ) 1570 m/s+0.02 m/s fL = 7.000000 MHz 1570 m/s0.02 m/s fL = 7.000178346 MHz
cS
Image Sources
(1) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (2) GSU. Wavelength vs. Frequency. CC-BY-SA. (3) CK-12 Foundation. Reected Wave. CC-BY-SA. (4) Transducer. Public Domain.
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vibrational motion (such as the oscillations of a buttery wing), or any combinations of linear, rotational, and vibrational motion are also possible. In this chapter, however, we are interested only in linear motion because it is the simplest type of motion and it provides a framework upon which more complicated types of motion of bodies can be characterized.
Vocabulary
distance, average speed, position, velocity, acceleration Distance Distance is the amount of length that a body has moved from one instant of time to another instant of time. It is always a positive number because it is just an amount, and says nothing about a direction. For example: 30 m, 35.6 ft, 25.138 cm. Average Speed Speed, in general, is a measure of how fast a body is moving without regard to the direction of motion. Average speed (symbol, save ) is dened as the total distance a body travels per unit time interval. Because distance and time, t, are positive quantities, speed is always a positive quantity. We can express save mathematically in the following way: save = Total distance traveled Total time elapsed
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Example 4 On average, a bug runs 25.276 centimeters in 2 minutes: save = 12.6 cm min
Instantaneous speed (symbol s) is a bodys speed at a particular point in time. An oil trucks speedometer displays the trucks instantaneous speed. Position Position refers to the location of a body at one instant of time with respect to some reference position. It is a vectormeaning it is both a magnitude and a direction. Vector quantities usually have symbols that are written in boldface type, which we will use here. A Cartesian coordinate system provides a convenient reference frame for you to use to locate a position. In this case, position can have either a positive or a negative value. Example 5 y = 40 mi directly north of home Example 6 x = 6.0 cm along the xaxis in a Cartesian coordinate system Example 7 A = 39.7 m at an angle of 25 east of the +y axis in a Cartesian coordinate system Velocity Velocity is a term used to specify not only the speed of a body, but also its direction. Like position, velocity is also a vector. The Cartesian coordinate system provides a convenient reference frame for its direction. In this case, velocity can have either a positive or a negative value. There are two types of velocity, average velocity (symbol vave ) and instantaneous velocity (symbol v). Average velocity is dened as the change in the position (called a displacement) of a body during a particular time interval. Because position is a vector, average velocity can be positive or negative in a Cartesian coordinate system. The average velocity, written in terms of the change (symbol ) in the initial position, xi , and the nal position, x f , is: vave = Example 8 x f xi x = t t f ti mi directly north h mm s
vave = 40 Example 9
Instantaneous velocity (boldface symbol v) refers to the velocity of a body at one particular instant of time. If the direction is not specied (as in the oil trucks speedometer), then Instantaneous Velocity and instantaneous speed are equivalent. Example 11 v = 30 Example 12 m northward s mi h
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Acceleration When there is a change in the instantaneous velocity of a body during a particular time interval, the body possesses an average acceleration (symbol aave ). Because velocity is a vector, average acceleration can be positive or negative in a Cartesian coordinate system. The average acceleration, written in terms of the change (symbol ) in the initial velocity, vi , and the nal velocity, v f , is: aave = v f vi v = t t f ti
The instantaneous acceleration (symbol a) is the acceleration of a body a particular instant of time. In this lesson we will only consider accelerations that are constant in time. For that reason, a = aave . Example 14 a = 30 Example 15 m northward s2 mi h2
a = 20 Example 16 a = 6.0
m in the +y-direction s2
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Figure 9.1: A motion diagram of the oil truck dripping oil at a constant time interval. In (a) the distance between adjacent dots increases successively by a factor of as time increases from left to right. This indicates that the oil truck is increasing its speed (accelerating) at each successive interval. In (b) the distance between adjacent dots are equal for the rst of motion, indicating that the oil truck is moving with constant speed. For the last of motion the distance between adjacent dots are also equal but larger in length, indicating a greater constant speed. the smaller mouth of the bottle, and then drill a hole into the cork, just large enough to insert a medicine dropper. You can then ll the empty portion of the bottle with the uid and the medicine dropper will be your dripper. The advantages of using motion diagrams are that you get a quick, visual idea of the type of motion involved. You can determine average speeds or average velocities, but not instantaneous speeds or instantaneous velocities. Also, for very long periods of motion, motion diagrams become impractical because of the quantity of dots involved and the time needed to analyze the dots.
2. Graphing Motion
Unlike motion diagrams, graphs provide more accurate information by providing a continuous visual description of motion. Graphing motion usually involves making a two-dimensional plot of an instantaneous variable (distance, position, velocity, or acceleration) as a function of time. Average values of these variables can also be determined from these graphs. Let us now return to the oil truck, which started from rest and was eventually traveling at 3 m/s in 6 s. Three possible ways in which this motion could be interpreted are as follows: 1. Starting from rest, the truck immediately traveled at a constant speed of 3 m/s for the next 6 s (virtually impossible to do). 2. Starting from rest, the truck steadily increased its speed, reaching a speed of 3 m/s in 6 s. 3. The trucks instantaneous speed is 3 m/s at t = 6 s as indicated by the trucks speedometer (or by police radar). The trucks speed could have been any value before 6 s. Which interpretation is the correct one? We can answer this question if we have a graphical description of the motion for each of these three possible interpretations. Interpretation 1 Lets start by looking at a distance versus time (and/or position versus time) graph for the oil truck based on interpretation 1. Figure 2(a) shows the oil trucks distance increasing at a constant rate as a function of time, starting from rest. The rate of speed is determined by the slope of the red line, which is positive. Notice that we can determine the exact distance, D, that the oil truck has moved at each instant of time in a continuous manner. With the motion diagrams, we could only determine the distances the oil truck www.ck12.org
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moves at discrete instances of time. The oil truck could also be moving in the opposite direction. In this case, we could plot a position versus time graph that would show the oil truck moving in the negative x direction, another possible motion based on interpretation 1. Figure 2(b) shows the oil truck moving in the negative x direction at a constant rate of speed. It is also moving with a constant negative velocity based on the slope of the red line.
Figure 9.2: A Distance Versus Time Graph Interpretation 2 Now let us turn our attention to interpretation 2 of the oil trucks motion in the 6 s period. Figure 3(a) shows a distance versus time graph (in red) in which the trucks distance increases at a greater rate as time increases from t = 0 to t = 6 s. Therefore, the speed of the truck also increases, but at a constant rate, as shown by the increasing slope of the tangent lines (small black lines) to the (red) curve. The instantaneous speed (3 m/s) at t = 6 s would be the slope of the tangent line to the (red) curve right at t = 6 s. Figure 3(b) shows the same distance versus time graph of the motion for the oil truck as in (a). The average speed, vave , between any two points on a distance versus time curve can be obtained by determining the slope of the line connecting those two points (in black).
Figure 9.3: Graphs of distance, , in meters versus time, , in seconds for the oil truck. In (a) the trucks distance increases at a greater rate as time increases from to . The speed of the truck also increases, but at a constant rate, as shown by the increasing slope of the tangent lines (the small black lines) to the red curve. In (b) the between times and , determined by the slope of the black line, is . Interpretation 3 In this interpretation, a distance versus time graph (or a position versus time graph) could show any shape as long as the slope of the tangent line to the curve at t = 6 s gives a value of 3 m/s. Rather than determining tangents to the curve in these graphs at various points in the motion, it would be better to plot the speed (or velocity) versus time of the oil truck. Figure 4 (a) shows a velocity versus time graph (in red) of the motion of the oil truck. In this graph, the oil truck rst accelerates at a constant rate aave = 0.67 m/s2 , then accelerates at a constant rate of aave = 0.33 m/s2 to a nal velocity of 3 m/s.
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Figure 4 (b) is an acceleration versus time graph of the motion of the oil truck based on the information in Figure 4(a). From these few examples, we can see now how graphing can be used to give us a more complete description of the motion of a body.
Figure 9.4: Graphs of the velocity of the oil truck versus time . In , the trucks velocity is zero at time . As time increases, the trucks velocity increases at a constant rate until a velocity is reached at . After that time, the trucks velocity increases at a lower rate until it reaches a velocity at . Graph (b) shows that the trucks average acceleration from to , and an , from to .
Experimental Setup The experimental setup used to graph your motion is shown in Figure 5. A motion sensor is connected to an interface box which in turn is connected to a computer. The interface box translates the signals from the motion sensor into the computer. The computer displays these signals, either as a position, a velocity, or acceleration as a function of time. How the Motion Sensor Works When describing the motion of an object, knowing where the object is relative to a reference point, how fast and in what direction it is moving, and how it is accelerating (changing its rate of motion) is crucial. The motion sensor is a sonar ranging device using high-frequency pulses of sound that reect from an object to determine the position of the object. The ultrasound pulses travel at a constant speed ( 343 m/s in air at room temperature). As the object moves, the change in its position is measured many times each second as the pulse travels back and forth from object to sensor. Positioning the Motion Sensor and Computer Mount the motion sensor on a table or support rod so that it is aimed at your midsection when you are standing in front of the sensor. Clear the area for at least 3 meters (about 9 feet) in front of the motion sensor. Position the computer monitor so you or your lab partner can see the screen while you move in front of the motion sensor. www.ck12.org
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Figure 9.5: Experimental setup for motion sensing. A motion sensor, interfaced to a computer, is directed at the midsection of a student. The student moves toward or away from the motion sensor and the sensor monitors the students movement. A computer gives a graphical display of the motion. General Procedure In this activity, the motion sensor will measure your position, velocity, or acceleration as you move. The computer plots your position, x, on a graph as a function of time, t. Moving away from the motion sensor could be considered motion in the positive xdirection, and moving toward the sensor considered motion in the negative xdirection. Tips for Better Data Acquisition Always stay in line directly in front of the motion sensor when at rest or when in motion. Try to avoid unnecessary movements that might be sensed. Be sure that the area around you is clear of all obstacles that may interfere with the motion sensor and cause a false reading. Never stand closer than 0.5 m or farther than 4.0 m from the motion sensor. Otherwise, your position will not be correctly determined by the motion sensor. Starting at 0.5 m in front of the motion sensor (your x = 0 position) use masking tape to mark the oor at 0.5 m intervals going away from the motion sensor for a total of 3.0 m. Once you have marked positions on the oor and you want the detector to produce readings that agree, stand at the 2.0 m mark on the number line and have someone reposition the motion sensor until the reading on the computer shows a position x = 2.0 m. Complete your drawings on the graphs in an idealized form rather than showing many small wiggles. Note: It is very diicult to obtain accurate acceleration versus time graphs with the current motion sensors available due to the nature of the sensor. Procedural Steps 1. Figure 6 shows six columns: (a) through ( f ). Each column is headed by a Description of motion of your motion or a set of empty lines. 2. Below each description of motion are three graphs: a position versus time graph, a velocity versus time graph, and an acceleration versus time graph. They represent your motion in front of the motion sensor. Some graphs are complete, others are to be completed. 3. The challenge of these tutorial exercises is to predict the descriptions of the motion, to complete the remaining graphs based on the information given, and to write a description of the motion in the
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empty lines at the head of particular columns. Complete each column with your predictions one at a time, instead of checking several problems at once. Use the motion sensor to check your answers. Figure 7 shows the correct answers.
Vocabulary
energy, kinetic energy, work, gravitational potential energy Energy Energy is a term that we hear over and over again. It is what a body possesses that allows it to do work. The more energy a body has, the more work it can do. Energy comes in many forms: chemical, electrical, nuclear, and mechanical. In this chapter we are interested only in the mechanical energy of a body. We can divide mechanical energy into two types: kinetic energy (symbol KE) and potential energy (symbol PE). The units we will use for KE and PE in this chapter will be joules (symbol J). Kinetic Energy Kinetic energy, KE, is the energy possessed by a body (of mass, m) that is moving with instantaneous velocity, v. It is expressed mathematically as follows: 1 KE = mv2 2 Notice that kinetic energy is always positive because the square of the velocity is also positive. If a body is not moving, then KE = 0. www.ck12.org
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Example 1 A boy with a mass of 60.0 kg runs with a velocity v = 0.500 m/s. His kinetic energy is ( )2 1 KE = 2 (60.0 kg) 0.500 m = 7.50 J s Gravitational Potential Energy Potential energy is always associated with interactions between two or more bodies; in the case of gravitational potential energy, we consider a body with mass m in the gravitational pull of the Earth. GPE is the potential energy a body possesses based on its position relative to a reference level (usually the Earths surface). We can express GPE mathematically as: GPE = mgh where:
m = mass of crate g = acceleration due to gravity = 9.80 m/s2 h = height above a reference level
Notice that the higher a body is from the Earths surface ground, the greater its GPE. Example 2 Lift a 2.00 kg box to a height of 1.50 m above the Earths surface. The gravitational potential ( ) m energy of the box (relative to the Earth) is GPE = (2.00 kg) 9.80 s2 (1.5 m) = 29.4 J Example 3 A 0.500 m tall stool sits next to you on the Earths surface. When you lift a 2.00 kg box to a height of 1.50 m above the Earths surface, the gravitational potential energy of the box (relative to the top of the stool) is now: ( m) GPE = (2.00 kg) 9.80 2 (0.500 m) = 4.90 J s Work Work is a term associated with our daily activities involving physical and mental stress, goals to accomplish, deadlines to meet, etc. Of course, the work we refer to as labor is dierent than the true denition of the word work (symbol W) we shall study in this chapter. When bodies need an applied force to move them, work is being done on the body by that applied force. The work that is done by the applied force causes the energy of the body to change. In this chapter we will study the work done by one or more forces on one or more bodies, determine the types of energy involved, and draw connections between the work done on the bodies and the energies changes in the bodies. First, however, we need to identify what we call work.
Identifying Work
Work, W, is dened as the product W = Fapp d Where Fapp is the force applied to a body (either a push or a pull) and d is the displacement of the body.
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Using the units of Fapp are newtons, N, and the units of d are in meters, m. So, work, W, is in units of Joules, where 1 Nm = 1 J. Note that Nm is never used as a unit of work (or energy); rather it is reserved as the unit for torque. The Free Body Diagram To determine all the work being done on a single body, we need order to clarify all the forces acting on the body. To this end, a free-body diagram of that body is usually created. The purpose of the free-body diagram is twofold: 1. to treat the single body as a point mass having the same amount of mass as the original body, but with a volume concentrated at one point. The reason for doing this is to circumvent any rotation that may actually occur when one or more forces are applied to the body (you cant rotate a point). We are only interested here in the work that causes the body to move in one dimension. 2. to show all the forces on the body of interest, not the forces the body may impose on other bodies. In creating free-body diagrams, forces acting on the point particle are always drawn as pull forces. That is, the head of the vector arrow representing each force always points away from the point mass. Think about it. Pushing or pulling on a body in the same direction, with the same amount of force, creates the same motion of the body.
W = T d_ = (50.0 N)(0.50 m) = 25 J
This is the maximum amount of work done on the crate by the rope.
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Figure 9.12: A person pulling on a wooden crate that has a mass of using rope. We will consider the ropes mass to be so small that it can be neglected. The person pulls the crate with a force of magnitude , for a distance . There is between the box and the oor. The net eect of all these forces is the movement of the box to the left. (b) A free-body diagram of the crate. The wooden crate in Figure 8 is acted upon by two forces in the vertical direction, gravity (pulling downward) and the normal force (pushing upward) from the ground. The total work, WT , done by these two forces on the wooden crate is WT = Wgravity + WNormal = 0 + 0 = 0 Because the two forces are balanced, the box does not move in the vertical direction. Hence, WT = 0. Note that the work done by the normal force is always zero.
) m 2 s
) m 2 s
Solving for Fapp , we get Fapp = 0.12 N. Work of Two Applied Forces on an Object
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Figure 9.13: A person pushing a crate , initially at rest , with a constant applied force, , on ground. The direction of is indicated by the red arrow. The crate is pushed a distance, d, causing work, W, to be done on it by the . The work increases the velocity of the crate from to . Consider the situation shown below in Figure 10. Person 1 and dog 2 are each pulling on a crate that has a mass of 2.0 kg. Both person and the dog are pulling on a rope attached to the crate. We will consider the rope so light that we can neglect its mass. There is friction between the crate and the oor.
Figure 9.14: Person 1 and dog 2 are each pulling on a crate using ropes (with negligible mass) attached to the crate. The crate has a mass of . Person pulls with a force of magnitude . The dog pulls with a force of magnitude . The magnitude of the kinetic frictional force between the crate and the oor is . In what follows, answer the questions pertaining to Figure 10. Procedural Steps 1. In the box to the right of the dog in Figure 10, draw a free-body diagram for the crate, showing all the forces on it. Label each force and its magnitude. 2. The crate in Figure 10 moves a distance x = 5.00 m to the right. Calculate the work done by each applied force on the crate and the total work, WT , done on the crate, W1 , W2 , and W f riction 3. Determine the kinetic energy change, KE, of the crate. 4. Determine the nal velocity of the crate.
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Figure 9.15: The brown crate has two paths, and , available to it to descend from height, , to ground level . Path is a ramp with negligible friction inclined at to the horizontal (ground). Path is a series of steps. The work done by gravity, , to the crate from height to ground level is the same for each path. is independent of the path taken and only depends on the dierence in height, . Free-body diagrams of crate for paths and are shown in the small boxes above the crates.
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Figure 9.16: The mass-pulley system is made up of two identical masses (crate and crate ) connected by a string that runs over a pulley with negligible friction. Crate is initially held at rest on table top. The friction between crate and the table top can also be considered negligible. Both masses are . When crate is released, both crates move .
Figure 9.17: This illustrates a crate placed on a ramp that is inclined to the horizontal. The coeicients of static friction and kinetic friction are and , respectively, between the crate and the incline. The straight-line distance from the crate to the end of the incline is . Procedural Steps 1. In the boxes to the right of the mass-pulley system in Figure 13, draw a free-body diagram of the crate, showing all the forces on it. Label each force and its magnitude. 2. Determine the work done by each force on the crate and the total work, WT , done on the crate. www.ck12.org
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3. Using the work-energy theorem, determine the change in kinetic energy, KE, of the crate as a result of the 2.0 m movement. 4. Determine the change in gravitational potential energy, GPE, of the crate as a result of the 2.0 m movement down the incline. Is gravity a conservative force? (i.e., work done is independent of the path taken)
Image Sources
(1) John Ochab. A man pushing a crate. CC-BY-SA. (2) John Ochab. A Distance versus Time Graph. CC-BY-SA. (3) John Ochab. Description of Motion: graphs a and b. CC-BY-SA. (4) John Ochab. The mass-pulley system. CC-BY-SA. (5) John Ochab. A motion diagram. CC-BY-SA. (6) John Ochab. Description of Motion: graphs e and f. CC-BY-SA. (7) John Ochab. Experimental setup for motion sensing. CC-BY-SA. (8) John Ochab. Answers: graphs a and b. CC-BY-SA. (9) John Ochab. A Distance Versus Time Graph. CC-BY-SA. (10) John Ochab. Description of Motion: graphs c and d. CC-BY-SA. (11) John Ochab. Crate on ramp. CC-BY-SA. (12) John Ochab. Person pulling a wood crate. CC-BY-SA. (13) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (14) John Ochab. Answers: graphs c and d. CC-BY-SA. (15) John Ochab. A Velocity versus Time Graph. CC-BY-SA. (16) John Ochab. A brown crate with two paths. CC-BY-SA. (17) John Ochab. Answers: graphs e and f. CC-BY-SA. (18) John Ochab. A person and a dog pulling on a crate. CC-BY-SA.
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10.1 Purpose
The purpose of this chapter is to provide several examples of physics experiments that utilize 21 st century technology. The technology highlighted in this chapter is the PASCO GLX Handheld Interface. The Xplorer GLX captures, analyzes, stores, and prints data quickly without the use of a computer. It can also be connected to a computer to make use of the datastudio graphing software. Most of the included labs are intended to be used by teachers that are new to this technology. As a teacher becomes comfortable with the technology, more advanced and inquiry based labs are easily done as extensions. The following labs are written specically for the GLX. However, they can be easily modied to use the PASCO PasPort Interface for collecting data with a computer. Older analog Interfaces from PASCO, such as the 500 and 750 Science Workshop Interface can use PASPORT digital and analog adapters that allow you to use PASCOs latest technology to be used without having to replace sensors. Although the PASCO technology is documented in this chapter, other companies such as Data Harvest Educational Inc., Fourier Systems Inc., Texas Instruments Inc., and Vernier Software & Technology Inc., also oer probeware technology. It is hoped that this chapter, which is only a beginning, will spark interest with physics teachers to begin using 21 st century technology. If teachers are already using probeware technology, then this may serve as an additional resource. The idea of the physics FlexBook is an evolving supplemental physics resource. Additions to this chapter might include generic laboratory experiments that can be followed with any brand of probeware.
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Figure 10.2: Velocity of a Motorized Cart Lab Description This activity uses a motion sensor to measure the motion of a motorized cart as it moves at dierent speeds. Although constant velocity is straightforward, the graphical representation of constant velocity involves many fundamental concepts of kinematics. The slope of a plot of position versus time is the speed
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of the object. Students will describe the relationship between the slope for each plot of data and the physical quantities represented by the slope. Download Lab: http://www.pasco.com/file_downloads/ experiments/pdf-files/glx/physics/03-Vel-of-cart-SV.pdf.
Figure 10.3: Acceleration Due to Gravity Lab Description This activity uses the motion sensor to measure the motion of a ball as it falls and bounces. The motion of the ball is recorded and displayed, allowing students to analyze the position and velocity of the ball. A velocity versus time graph can be used to nd the acceleration of the ball. Students will compare the experimental value of acceleration (slope of velocity versus time) to the accepted value for the acceleration due to gravity.Download Lab: http://www.dentonisd.org/512719301176/lib/512719301176/_files/ 05_Free_fall_SV.pdf
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Figure 10.4: Acceleration on an Inclined Track This activity uses a motion sensor to measure the motion of a cart as it moves up and down an inclined plane. The Xplorer GLX is used to record and display the motion. From the collected data, students can determine whether the acceleration up and down the inclined plane is constant. Download Lab: http: //www.aug.edu/hbusch/Phsc1011%20Files/Lab%202%20Accel%20on%20an%20inclined%20track.pdf.
Figure 10.5: Newtons Second LawConstant Force Lab Description The purpose of this activity is to determine what happens to an objects acceleration when the net force applied to the object stays constant but the mass of the system is changed. A motion sensor is used to measure the motion of a cart that is accelerated by a net force. The Xplorer GLX is used to record the
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motion and display and analyze the velocity of the cart. Download Lab: http://www.rblanski.com/ files/Lab_P08_Newton_s_Second_Law_Constant_Force.pdf.
Figure 10.6: Static and Kinetic Friction Lab Description The purpose of this activity is to investigate static friction and kinetic (sliding) friction. A force sensor is used to measure the force on an object as it is pulled across dierent surfaces. The Xplorer GLX is used to record and display the force versus time. The data can be used to determine the static and kinetic friction and then nd the coeicients of static and kinetic friction. Download Lab: http://www.aug.edu/hbusch/ Phys%202211%20Files/Lab%207%20Friction.pdf.
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Figure 10.7: Momentum in Collisions This activity uses two motion sensors to measure the motion of two carts before and after they collide. The momentum of each cart before and after the collision can be compared. Download Lab:http://www. pasco.com/file_downloads/experiments/pdf-files/glx/physics/14-Momentum-SV.pdf.
Figure 10.8: Conservation of Energy Lab Description This activity uses a motion sensor to measure the motion of a ball as it falls from a given height. The
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Xplorer GLX is used to record and display the motion. Students can use the data to determine the balls gravitational potential energy and kinetic energy. A change in potential energy can then be compared to the nal kinetic energy. Download Lab: http://authors.ck12.org/wiki/images/2/26/FLX_ VA_LaboratoryActivities_cons_energy_11.doc.
Figure 10.9: Work and Energy Lab Description This activity uses a force sensor to measure the force applied to a cart by a string attached to a descending mass. A motion sensor is used to measure the motion of the cart as it is pulled by the string. The Xplorer GLX is used to record and display the force and the motion. Students can use the data to determine the work done on the system and the nal kinetic energy of the system. They can then compare the work done to the nal kinetic energy. Download Lab: http://www.pasco.com/file_downloads/experiments/ pdf-files/glx/physics/19-Work-energy-SV.pdf.
Figure 10.10: Sound Wave Properties Lab Description This activity uses the built-in GLX Sound Sensor to measure the sound waves from various sources.The Xplorer is used to record and display the data. Students can analyze the sound waves to determine www.ck12.org
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the period, frequency, and wavelength of each sound. Download Lab: http://www.pasco.com/file_ downloads/experiments/pdf-files/glx/physics/27-Sound-waves-SV.pdf.
Figure 10.11: InferenceBeat Frequency Lab Description This activity uses the built-in Xplorer GLX Sound Generator to create two sound waves with slightly dierent frequencies. The GLX Stopwatch is used to record the amount of time for several beats to occur. Students can also determine the period of the beats and calculate the beat frequency. They can also compare the beat frequency to the dierence in frequency. Download Lab: http://www.pasco.com/file_ downloads/experiments_of_month/glx/beat_frequency_with_glx/Beat-Frequency-with-GLX.zip.
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Lab Description This activity uses two fast-response temperature probes to measure the change in temperature of equal quantities of hot water in two dierently colored aluminum cans. The temperature is recored for each container for a period of 15 minutes as the temperature cools. Students can then determine which aluminum can transfers thermal energy fastest. Download Lab: http://www.pasco.com/file_downloads/ experiments/pdf-files/glx/physics/30-Transfer-energy-SV.pdf.
Image Sources
(1) Newtons Second LawConstant Force. CC-BY-SA. (2) Acceleration Due to Gravity. CC-BY-SA. (3) Work and Energy. CC-BY-SA. (4) Conservation of Energy. CC-BY-SA. (5) Static and Kinetic Friction. CC-BY-SA. (6) Velocity of a Motorized Cart. CC-BY-SA. (7) Position-Match Graph Lab. CC-BY-SA. (8) Transfer of Energy. CC-BY-SA. (9) InferenceBeat Frequency. CC-BY-SA. (10) Momentum in Collisions. CC-BY-SA. (11) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf (12) Sound Wave Properties. CC-BY-SA. (13) Acceleration on an Inclined Track. CC-BY-SA.
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11.1 Theory
Two Sides of the Same Coin
Since this chapter is about statistical physics, its good to start by considering how this eld diers from traditional statistics. In statistics, scientists use empirical data to estimate some unknown number or set of numbers (called parameters). Lets say we have a coin and we dont know what the probability of it landing on heads after during a ip is, but we do know that this probability remains constant between ips. A question that a statistician might ask is, How can we best estimate, based on repeated trials, this probability our parameter for this particular situation? Imagine we ipped this coin 1,000 times and found that it landed on heads on 715 of those. In this case, our best estimate for the probability lets call it P that the coin lands heads on any single toss would 715 be the ratio 1000 , or about 72 per cent. Theres no guarantee that P is actually equal to .715; this is just a best guess. Broadly, the problem facing statisticians can be summarized as, How can we translate collected data (like the results of the 1000 ips above) into an estimate of the unknown parameter (P above) that is eicient and accurate? For the coin example, it seemed reasonable to guess that the percent of ips that result in heads in a long series of trials is a good estimate for the probability that the coin lands on heads after a single trial but is it the best possible estimate? Using statistical methods, it is possible to show it actually is, but we wont prove that result here.
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As simple as the coin example is, our treatment of it shows two major purposes of statistics: to nd ways of estimating parameters from some data and to study the eiciency and accuracy of such methods. Statistical physics (part of it, at least), on the other hand, uses the coin as the basis of a model the random walk. We are no longer interested in estimating parameters, but in trying to model some more complicated situation by reducing it to a series of coin ips. Like the example above for statistics, the random walk model described below gives a simple glimpse into the methods and results of statistical physics.
Three-Step Case
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possible outcomes of a three step random walk where P = 1 (its a fair coin, the walker is equally likely to 2 step left or right at every step). The various possibilities for this case are illustrated below:
The rst case is equivalent to ipping three heads in a row and therefore taking three right step, the second to ipping the sequence heads, tails, heads, and so on. Since in this case the coin is fair, the eight outcomes 3 (step combinations) shown above are equally likely to occur: they each have a probability of 1 = 1 . 2 8 These outcomes, however, do not all result in dierent end locations (the four low arrows) for the walker: this is determined by the dierence between the number of steps taken to the right and the number taken to the left. So while only one outcome corresponds to an end location of three steps to the right or three to the left, three outcomes correspond to an end location of one step to the right or one to the left. So the eight equally likely outcomes result in four possible end locations that are clearly not equally likely. Since the outcomes are mutually exclusive (you cant have more than one of them occur at the same time), the probability that a particular end location (such as one step to the right of the starting point) occurs will equal to the sum of the probabilities of the outcomes that lead to it. Since all eight outcomes are equally likely, this will also equal the number of relevant outcomes multiplied by the probability of a single one. Therefore, the probability of ending one step right is:
1 8+ 1 + 1 =3 1 = 3
8 8 8 8
This reasoning allows us to nd the probabilities of the other possible end locations as well, noted below the arrows on the graph above. This grouping of possible end locations and their respective likelihoods is an example of a probability mass function, which in general is a list of events with associated probabilities. Therefore, the upper list of outcomes that happen with probability 1 is a probability mass function as 8 well, although obviously not as relevant to us. In more mathematical language, we can represent of a set of events as (x1 , x2 , x3 , . . . , xi ). The associated probabilities, meanwhile, are written as (P(x1 ), P(x2 ), P(x3 ), . . . , P(xi )). For instance, the end locations of the three step random walk described above will can be written as (3R, 1R, 1L, 3L) and their probabilities 1 as ( 8 , 3 , 3 , 1 ). We can plot this distribution on a graph where nal displacement (alternatively, number 8 8 8 of steps the two quantities will be equal if we set the step lengths to 1, which we can with no loss of generality) from the origin is on the x-axis, while its probability is on the y-axis:
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Above, we have answered the question posed at the beginning of this section for the three-step case. That is, we have completely determined what the likelihood of the walker being in any possible location is at the end of this walk.
General Case
Now let us try to generalize these results to to a random walk with P = 1 and N steps. The intuition 2 we obtained from considering the simple case above can be summarized as follows: to nd the probability mass function of the end location of a random walker, one should rst consider all the possible outcomes, nd the ones that lead to the same end locations, and multiply their number by the probability of a single outcome. The diagram below is analogous to the one for three steps, but now with N steps. We can divide the possibilities into 2N equally likely outcomes, this time each with probability 21 . The question is, How N many outcomes lead to a given end location, say, D steps to the right (as posed above)?
There is still only one outcome that leads to each of the two extreme locations, when all steps are taken either right or left. Therefore, their probabilities are 2N but what about the other locations? To nd their respective probabilities, we rst use the fact that end locations depend on the dierence between the number of steps taken to the left and right (and not their order) to pose the problem in a slightly dierent way. Let L be the number of steps taken to the left, and R to the right. Since the total number of steps is N,
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N = L + R Total steps
If the walkers winds up D to the right of the origin, she must have taken D more steps to the right than to the left:
We have solved for the necessary number of steps left and right in terms of known quantities, N and D. At this point all that remains is nding how many ways there are to take 1 (D + N) steps to the right out of a 2 total of N steps: this will give us the number of outcomes that lead to end location of D steps to the right. In the three step case, for instance, ending one space right of the origin required taking two steps right and one step left; there are three discrete ways to take two achieve this (the left step can be rst, second, or third), and so three outcomes that lead to that location. For the case of N total steps and 1 (D + N) steps to the right, the correct result will be given by the ways 2 of choosing formula from combinatorics: the number of ways to choose 1 (D + N) positions for the right 2 steps out of a total of N positions. This is written as
Since each outcome has probability 2N , the probability of nding the walker a distance D steps to the right of the origin is given by the following formula: ( ) Probability of being D steps away after N steps
P(D) = 2N
1 2 (N+D)
We have now answered our original question (nding the probabilities of various end locations) for all unbiased (P = 1 )) random walks with constant step lengths. Again, we can plot this distribution for 2 several dierent cases:
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When the number of steps becomes large, the distribution begins to look like a bell curve; here is the plot for N = 100:
Problems
1. What is the dierence, in terms of end probability distributions, between random walks with even and odd numbers of steps) 2. Why are we justied in setting the step lengths equal to 1 in modeling all constant step length random walks? 3. Solve for the probability mass function of end locations for a four-step random walk analogously to the three-step example above (illustrating it also). Then, graph this probability mass function. 4. Our proof for the general case can be called right-biased in two ways. This question settles both: (a) We found the probability of being D to the right of the origin, but the probability distributions were graphed as symmetrical. First, explain why this must be true in terms of possible outcomes and end locations. Then, show that the formula for P(D) can be used to nd probabilities to the left also, that is, prove that P(D) = P(D) using the formula above and the denition of factorials. www.ck12.org
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(b) We also found the number of outcomes that lead to an end displacement of D in terms of steps taken to the right. Use the result from the previous part to show that using 1 (N D) the 2 number of steps to the left corresponding to a nal distance of D steps to the right in the derivation of the general result would not have changed it. 5. Derive the probability mass function for a biased random walk (that is, steps in one direction are more likely than in the other, or the coin has a higher probability of landing heads than tails). Hint: the outcomes will no longer be equally likely, but what about outcomes that lead to specic end locations? (a) Graph a few of these distributions.
11.2 Simulations
Although random walks can be challenging to understand using math alone, they can be easily simulated on computers. Random walk simulations can be written in practically any programing language. In this chapter, I show how to write all the programs I mention in Python, a free, open source programming language used by scientists throughout the world. It can be downloaded at http://www.python.org/download/ (this chapter is written for the 2.xx version of Python, which should be supported into the mid 2010s). I also use the Pylab package, which adds extra functionality to Python (and is also freely available); it us used by scientists throughout the world for mathematical computing. Its available at http://www.scipy.org/PyLab. Words like programing, computationally, and simulations make this subject seem much more diicult than it is. In reality, it requires very little programing knowledge. The following section goes over the basics we need. All you need to start is a computer with the above software installed.
Programing Basics
You are probably familiar with computer programs, like browsers, mp3 players, and games. On a very fundamental level, these can be described as tools that let you access the processing power of a computer in specic ways. Computer programing languages allow one to create her own programs and have more direct control over how the processing power is used. Writing complex programs like mp3 players is a pretty involved task, but we only need basics for random walk simulations. Good (very brief) outlines for beginning Python can be found at http://www.ibm.com/developerworks/library/lcheatsheet3.html and at http://docs.python.org/tutorial/introduction.html. More about Pylab, which adds a lot of computational functionality to Python, can be found at http://www.scipy.org/Cookbook. What follows is not a general introduction to programing, but an outline even briefer than those mentioned above of the concepts and methods needed to run simulations that illustrate the results from the rst part of the chapter in the Python programing language. Because of Pythons interactive nature, the simulations we run should be relatively familiar to anyone who has used a graphing calculator for a math class, though that is certainly not a requirement. I hope to provide enough of an introduction that a motivated student can gure the rest out through interactive exploration. That said, interested readers can resolve their problems, nd more information on Python, and learn more about computational programing in general by following the links above.
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data) and functions (ways to create and modify data). In everyday use, the word objects can be used like interchangeably with stu, or things in reference to anything that exists, has a name, takes up space, or contains information. This isnt far from the way programing languages interpret objects as well. In python objects can be things like numbers, lists of numbers, pictures, les, and other entities that store data. Going along with the real world analogy, lets think of something that is typically referred to as an object, like a table. When we speak of a particular table, we imply that it is an object that belongs to the category of things that can be called tables. In the same way, objects in programing languages are concrete instances of broader categories called types. In Python, things that belong to the type integers, for instance, are numbers like 2, -4, 5, etc. We will need several types of data structures to store the results of our simulations later. A concrete object (instance of a type) can have a name, or reference; no two objects in use at the same time can have the same name (otherwise the computer would not be able to tell them apart). We can create new references to objects using variables more on this later. If objects main purpose is to store data, functions allow us to create and manipulate this data. Functions generally take one or more objects or parameters as input, perform some procedure on or with them, and return one or more objects as output. For instance, in Python, the square root function can take a number as an input, take its square root, and return this number as an output. We will need several dierent functions for our random walk simulation. Python has some built in types of objects and some built in functions, such as those mentioned above. Additionally, like all object oriented language, it provides the framework for creating new, user dened, types and functions. These can, in turn, use the pre-existing ones. There are also many libraries of functions and types made by people throughout the world useful for specic purposes freely available on the internet. Heres a summary of the important terms:
Objects: Entities (of many dierent types) that contains data. Variables: Strings or characters that reference, or name, objects. Functions: Algorithms for manipulating data.
Using Python
The paragraphs above were pretty abstract; now we translate that theory into actual programing. Before we can run any simulations, we need to understand how to interact with the computer. Luckily, this is easy in Python due to its interactive interface. One of the nice features about Python is that it can be run interactively using a command line: in other words, you can see results in real time rather than having to type all your programs in a separate text le that needs to be translated for the computer to run it. Most computer languages require this translation, called compiling. In that situation, running a simulation is diicult: you have to write a complete set of instructions before starting, since once the program is compiled it generally cant be altered. These sets of instructions are called computer programs. Due to Pythons interactive compiler, we can just perform the steps one by one ourselves. At this point, lets start the Python interface and explore some of the functionality that we will need. Note: in the screenshots, the lines starting with In [#]: are input (typed in), while those with Out [#] are output returned by the computer. www.ck12.org
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The equal sign (=) assigns variables values in Python. When I type a = 5, I am creating a number object which stores the value 5 and referencing it with the letter a. We assign variables to objects we are using in order to keep track of them and access them more easily (each object has a unique identity but can have many variables refer to it, like a,b, and c do below). If I input a variable name (a quick warning: variables are case sensitive, and can include more than one letter ant number), Python prints a description of what it references (not all are just one number, some are much more complicated sets of data):
A useful data type that we will need for our simulations is the list. Like their name suggests, list objects can store multiple entries of data. Lists can be created using the [a,b,c,. . .] construction. To access the nth element of list b, we us the construction b[n]:
When accessing lists above, what we usually call the rst element is indexed with a 0, and this is important to keep in mind. Finally, a note on referencing: certain characters and strings, such as all numbers, are related to pre-dened functions or types and cannot be used as variables (see error message below).
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Many objects in Python can be altered after they are created. For instance, one can change the elements of a list object one by one:
A slightly obscure point about the last example is that certain objects cant be altered after they are created, such as numbers. But what does this mean practically? Simply that if you use the same variable for two dierent numbers, two dierent number objects are created in succession instead of one object being changed; this distinction wont have many implications for us.
Functions
Functions, broadly speaking, oer various ways to process or modify data. The general method of calling functions is FunctionName(Inputs). For instance, the square root sqrt() function takes a number and returns its square root.
The zeros() function and the range() functions both take a number as an argument and return lists. The rst returns an array (type of list) of zeros of a given length, while the second returns a list of numbers that starts with zero and ends one below the given number. We will use these functions to help us create structures that will store the data in our simulation.
The choice() function takes a list as an input and returns an element, chosen at random. We will use this to pick steps randomly from the list [-1,1].
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A quick note on Python arithmetic: Python has several types devoted to numbers. When arithmetical symbols and functions are used, results are often determined by the types of the arguments. For instance, when one integer is divided by another, an integer is returned: not necessarily the actual result. Converting integers to oating point numbers using the oat() operation solves this issue, but this problem does not aect the simulations below anyway.
Finally, we will need a construction called a for loop. It is a structure that allows us to perform some action for every element of a list. So, to use it, we rst need a list to iterate over. The loop is then dened in terms of a variable name. Below, we use x. This variable is only dened for the duration of the loop, and takes value equal to the elements of the list. The loop basically does the following: 1. 2. 3. 4. Sets the variable equal to the rst element of the list. Performs some action Sets the variable equal to the second element of the list Performs some action again
For instance:
The print function, as you may have guessed, nds the value of an expression and then writes it on the screen.
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At any step, we can use the choice([1,-1]) function to pick a step at random. By setting the step lengths equal to one (as we did in the rst part of the chapter), we allow -1 to represent a step left and 1 a step right. A list object, initially all zeros (created by the zeros() function), can be used to store data about positions. At any step, we will modify the corresponding element of the list to hold the location at that step. In other words, a list for a three-step walk could be (0,1,2,1) which would correspond to a step right, another step right, and then a step left. If we plan on modeling an N step random walk, this list should be of length N+1, since the rst location is 0 by default (starting location). According to the outline above, our simulation for a random walk of N steps has to start by dening the two lists described above. For this example, N = 10:
Lets consider what this loop does over its rst two iterations: 1. Set x equal to the rst element of the Steps list, specically, x = 0. 2. Sets Positions[0+1] the second (why?) element of that list equal to ((either 1 or -1) + Positions[0]). Positions[0] is equal to 0, though, because that is the starting location). 3. Sets x equal to the second element of the Steps list, x = 1. 4. Sets Positions[1+1] the third element of that list by adding 1 or -1 (picked randomly) to the previous location, Positions[1]. 5. So on. . . Accordingly, after running the loop, the kth element of the Positions list should correspond to the location of the walker after k steps. Indeed:
This case corresponds to steps right, right, left, right, left, left, right, right, right, right. Of course, others could have been possible, as we can see by running the loop again (doing this resets the simulation why?).
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remember: it is one shorter than the positions list (there is a starting position, but no starting step). So, if there are N+1 positions, we have to use range(N+1) for a set of x-values. The following graph is produced when we input plot(range(N+1), Positions):
This image corresponds to the Positions list given above (why?). By default, when Python plots a two sets of points, it will connect them with lines. To overcome this (if we want) we can use this construction: plot(range(N+1), Positions, o). This will use dots (os) instead of lines:
We can add more lines to a plot once it is created. Python will do this automatically (and using dierent colors). Below, I run the random walk simulation twice more and plot them one by one:
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(a) Run the random walk simulation, but only save the last location. (b) Store it as an element of the list. (c) Repeat many times.
For instance, lets say we have a list with 1000 elements called data which contains 1000 end locations from random walks with 10 steps. We can create a histogram of the values using the function hist(data,20). The 20 refers to the number of bins that the computer will use. Using twice the number of steps for the number of bins makes sense because it scales the x-axis in the same way our probability distributions did in the rst part. Heres the aforementioned graph (compare to the 10-step case as it was derived earlier): www.ck12.org
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Problems
1. What sequences of steps lead to the following Positions lists: (a) [0, 1, 0, -1, -2, -3, -2, -1, 0, -1, 0] (b) [0, -1, -2, -1, -2, -1, -2, -3, -2, -1, 0] (c) [0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 2]
2. Look at the wikipedia page on random walks. Now, look at the gure included: http://en.wikipedia.org/wiki/File: Walk_example.svg. This gure was generated in Python. Explain how the program that creates it works, and run it yourself. 3. Find a way to obtain lists of end locations needed to make histograms. You will have to use nested for loops in one of three ways (in each case, make sure to store the end location at the end of each run in some list): (a) Alter the program used on the wikipedia page. (b) Repeat the for loop we used to generate Positions lists several times. (c) Finally, the quickest way is probably to nd a way of simulating a random walk that only results in the end location, then run that simulation several times. 4. One way to model biased random walks is to increase the number of arguments in the choice() function we use. Think about how this relates to altering the parameter P in random walks from the rst part of the chapter. Is there a one to one correspondence, or is one method more comprehensive? (a) Find equivalent ways to represent several biased random walks using the two methods above. Then, use the method from the rst part of the chapter to graph the probability mass function, and the method from the second half to create a histogram. Do they look alike? How do histograms relate to probability mass functions?
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12.1 Introduction
To the Teacher Science teachers have always been concerned about engaging students in the content of the subject that they teach. It is very easy for students to see science as a complex, fact-driven eld of study in which all of the basics have already been determined. Especially with the great advances in personal computers and technology in general over the past few years, it is easy for students to view traditional science courses as not relevant to their everyday lives. Teachers are always on the lookout for approaches that make instruction both relevant and enjoyable while at the same time maintaining academic rigor. One approach that is starting to gain traction in education is in the use of models and simulations (MODSIM) in the classroom. MODSIM is a promising approach for several reasons. The rst is that it allows instruction to be studentdriven. Students can explore a model or simulation at their own pace and in a manner that makes sense to them. Another is that models and simulations today use state-of-the-art technology. It seems more real to the students because the models and simulations use the same types of technology that they use in other areas of their lifecomputers, Web-based programs, and gaming systems. Finally, using a MODSIM approach to study a phenomenon mirrors what happens in the world of science and engineering outside the classroom. Students can think and act like scientists and engineers as they explore for themselves how varying conditions aect a process or system without having to have the physical system available to them. This chapter will examine some sample models and simulations using two dierent programs, Squeak and STELLA. Each program represents a dierent approach to using MODSIM for instruction. Squeak is an animation-based program while STELLA is a mathematics modeling based software package. For each program there is background information for the instructor, activities for students, and an answer key. The student activities are written as pull-out sections from the other two sections.
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12.2 Squeak
What is Squeak?
Squeak is a free, open-source, object-oriented, multimedia authoring environment that runs on many platforms and can be used to construct active learning environments for all ages. Programs can be written in the Squeak environment by novices using graphical programming tiles or by experts using Smalltalk. Developers around the world are continually adding functionality to the open-source Squeak image. In fact, Squeak is written in Squeak. Everything in the Squeak world is an object. Each object has properties and can send messages to other objects. The objects are like actors on a stage. Each object can be imbued with actions that create interactive experiences for learners and authoring is always on. Squeak is currently being rewritten from the ground up and is the basis for many new collaborative programming environments and exciting developments. Two activities have been developed using Squeak. The rst activity introduces students to the use of lasers to measure aerosol and cloud thickness. The second activity introduces students to Squeak programming through the study of motion in one dimension. These programs can be opened by following this link, http://www.pcs.cnu.edu/~rcaton/flexbook/flexbook.html.
determining the relationship between laser intensity and aerosol/cloud thickness determining the thickness of an aerosol and/or cloud developing the algebraic equation that relates laser intensity to thickness developing the exponential equation that relates laser intensity to thickness writing a Squeak script to make a fth test cell.
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Determine whether the relationship between intensity and thickness is directly proportional, inversely proportional, always decreases by the same factor, or none of these. Show work to support you answer. 3. Challenge 2 In the calibration activity, each cell was 1 cm thick. In real life, clouds and aerosols can be meters or even kilometers thick. Scale the cell thicknesses to meters using the following information: For aerosols, 1 cm = 3500 m For clouds, 1 cm = 70 m In the Squeak program, click on the aerosol challenge and cloud challenge buttons to measure the laser intensity as it passes through aerosols/clouds of various thicknesses. Record these intensities and determine the corresponding thickness in meters. Explain the method you used to determine the thickness in meters. 4. Challenge 3 Based on your graphs, write an algebraic equation for the relationship between laser intensity and thickness for both aerosols and clouds. Show work to support your answer. 5. Challenge 4 Plot the natural log (ln) of intensity vs. thickness for both the aerosol and cloud calibration data. Write an exponential equation for each relationship. 6. Challenge 5 Construct a fth test cell using Squeak. Save the edited program and submit it to your teacher.
Table 12.2: Cloud Calibration Data Table Number of Test Cells 1 2 3 4 Laser Intensity .750 .563 .422 .316
Challenge 1 The intensity decreases by the same factor every time the thickness increases by the same amount. Challenge 2 Below is a sample calculation for aerosol and cloud thickness. Students answers will vary depending on the thickness of the aerosol or cloud they measure. They may also determine the thickness www.ck12.org
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using the graph they created. Aerosol Laser intensity reading = .281 From the data table above, a 1 cm thickness gives a laser intensity reading of .500, so .500 .281 = 1 x x = .562 cm From the information provided in the simulation, 1 cm in the simulation is 3500 m, so
.3500 1=
x .562 x=19678
Cloud Laser intensity reading = .553 From the data table above, a 1 cm thickness gives a laser intensity reading of .750, so
cm
70 1=
x .737 x=51.6
Challenge 3 Aerosol equation: y = .50 x Cloud equation: y = .75 x Challenge 4 Table 12.3: Transformed Aerosol Calibration Data Table Number of Test Cells 1 2 3 4 ln (Laser Intensity) .693 1.38 2.07 2.76
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Equation: ln(y) = .144x + .594 or y = 1.81e.144x Table 12.4: Cloud Calibration Data Table Number of Test Cells 1 2 3 4 ln (Laser Intensity) .288 .574 .863 1.15
Any value that can be changed is next to a blue box with a description of the value. Once you have changed the values, click and hold the ! in the yellow circle to run the simulation. Click the yellow reset button to return to the original conditions. The scripts that are running the simulation are in the green boxes.
The challenge you are given is: Use Squeak to create a simulated microworld that shows how a body moves under the action of a force law of your choice. Illustrate the motion with graphs. Write an instruction manual for your project and be sure to include an explanation of how your project works. To be sure your instructions are clear, test your manual on others not familiar with your project to see if they can follow your instructions. As an additional challenge, explore the relationship between Newtons second law (F = ma) and energy in your microworld. www.ck12.org
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12.3 STELLA
What is STELLA?
STELLA is a software package used by thousands of educators and researchers for model building and simulation. It can be used to study everything from physics to economics, literature to calculus, chemistry to public policy. Based on Systems Thinking and System Dynamics, STELLA is a powerful tool for creating environments that allow students at all levels to learn by doing. STELLA models provide endless opportunities to explore by asking what if and watching what happens, inspiring the exciting ah-ha! moments of learning. Developed by isee systems, STELLA is available in both Windows and Macintosh versions. For more information, visit www.iseesystems.com www.iseesystems.com. There are four basic building blocks to a STELLA model: stocks, ow, converters, and connectors. Each is dened as follows: A stock is a quantity that is accumulated or depleted. The value increases or decreases over time. A stock is represented by a rectangle in the model. A ow represents those actions or activities that cause the stock value to increase or decrease over time. A ow is represented by a large arrow with a valve in the middle. If the arrow points toward the stock then it causes the stocks value to increase over time. If the arrow points away from the stock then the value of the stock will decrease over time. It is also possible to have a biow, which means the value of the stock can increase and decrease over time. A converter is used to represent additional logic important to the model. Typically, a converter modies a ow. Converters are represented by circles. A connector connects related items together. A connector can be an action (causes something to change) or informational (shows a qualitative relationship). Connectors are represented by wire arrows. In order to use the models below, teachers should download the free isee Player at http://www.iseesystems. com/community/PhysicsFlexBook.aspx. The program can be saved and loaded on as many student computers as are needed. Students may also load this software on their home computers.
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nished experimenting when you can answer each of the questions below. How does the magnitude of the displacement aect the period, frequency, and amplitude of the pendulums motion? What happens when the displacement is a negative value? What is the signicance of this in the physical world, i.e., what dierence would you observe if you were actually swinging the pendulum? How does the strings length aect the period, frequency, and amplitude of the pendulums motion? Grandfather clocks use a pendulum to keep time. If a grandfather clock was running slow, would you make the pendulum shorter or longer? Why? How does the mass of the bob aect the period, frequency, and amplitude of the pendulums motion? To answer the following questions you should look at page two of the graph, which displays velocity vs. displacement. To see page 2, click on the dog ear at the bottom left corner of the graph. What is the displacement when velocity is at its maximum? If you were watching a pendulum, where would the bob be when maximum velocity is achieved? What is the velocity when displacement is at its maximum? Where would the bob be at this point? Why are velocity and displacement sometimes negative?
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What is the displacement when velocity is at its maximum? If you were watching a pendulum, where would the bob be when maximum velocity is achieved? The displacement is zero when velocity is at a maximum. At this point the pendulum is in the middle (rest) position. What is the velocity when displacement is at its maximum? Where would the bob be at this point? Velocity is zero when displacement is at a maximum. The bob would be as far right or left as it was going to travel. Why are velocity and displacement sometimes negative? The negative sign indicates the direction of travel.
Directions and Questions for Coee with the President and Prime Minister
To complete this activity, go to http://www.iseesystems.com/community/PhysicsFlexBook.aspx and download the Coee with the President and Prime Minister model. Open the model with the isee Player and click on Background and Context to read about the problem you will be investigating. Return to the home screen. Before clicking on Conduct Experiments, answer the following question: Whose coee do you think will be hotter? Why do you think so? Click on Conduct Experiments and follow the directions. Continue to the next screen and record the coee temperatures below. Presidents coee temperature: Prime Ministers coee temperature: Read the Understanding Why pages. After you have examined the graph, answer the following question: What assumptions are being made about the temperature of the cream added to the Presidents and Prime Ministers coee? Click on the Part 2 Experiments link. On this page you can manipulate the times that cream is added as well as the insulating power of the cups. By experimenting with these inputs you will be able to answer the following questions: What happens to the temperature dierence at the end of each run as the time dierence between when each person adds their cream increases? Why does this happen? What happens to the temperature dierence as the insulating power increases and decreases? Why?
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Answer Key for Coee with the President and Prime Minister
Whose coee do you think will be hotter? Why do you think so? Answers will vary. Presidents coee temperature: 123o F Prime Ministers coee temperature: 107o F What assumptions are being made about the temperature of the cream added to the President and Prime Ministers coee? The temperature of the cream added by the President and the Prime Minister is the same. The change in temperature caused by the addition of the cream is independent of the temperature of the coee when the cream is added. What happens to the temperature dierence at the end of each run as the time dierence between when each person adds their cream increases? Why does this happen? As the time dierence between when the cream is added increases, the temperature dierence at the end of the 20 minute run increases. After the cream is added to one cup of coee, both cups cool and the temperature dierence between the two decreases. When the cream is added to the second cup of coee, the temperature dierence is again immediately increased. What happens to the temperature dierence at the end of each run as the insulating power increases and decreases? Why? The better the insulation, the less temperature change there is over time for the individual cups of coee. This means that once the cream has been added to both, the two cups of coee are closer to being at the same temperature. This is observed because increasing the insulating power reduces the amount of heat exchange between the coee and the surroundings.
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On this page, three graphs are displayed: position vs. time, position vs. velocity, and restoring force vs. position. Click on each graph to read a description of the graph. Now press the Run button and watch the graphs plot as the experiment proceeds and answer the following questions. Note that you may run the simulation multiple times without exiting this page if you need to see a replay of the simulation. When is the velocity of the bungee jumper zero? What is happening to the bungee jumper when the velocity is zero? When is the velocity of the bungee jumper at a maximum? Where is the bungee jumper at this point? Does the restoring force increase or decrease when the bungee jumper rst jumps? When is the restoring force at a maximum and a minimum? Go back to the Experiment screen and run several dierent trials with dierent masses and numbers of bungee cords. After each run, go to the Review Results page and look at the graphs. What eect does changing the mass seem to have on the total displacement (amplitude), velocity, and restoring force? What happens to the number of bounces (period) as the mass changes? What eect does changing the number of bungee cords seem to have on the total displacement, velocity, and restoring force? What happens to the number of bounces (period) as the number of bungee cords changes? The bungee jumper represents a mass-spring system, with the jumper acting as the mass and the bungee cords acting as the spring. Do more bungee cords correspond to a stier spring or a looser spring? Explain. Return to the home page and click on Extended Experiments. You will now be able to control the platform height (initial displacement) and force of gravity as well as the mass and number of bungee cords. Experiment to determine how gravity aects the total displacement, velocity, restoring force, and period of the system. Write a paragraph to describe these eects.
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As the mass increases, the range of values of restoring force increases. What happens to the number of bounces (period) as the mass changes? The higher the mass, the fewer the bounces and longer the period. What eect does changing the number of bungee cords seem to have on the total displacement and restoring force? As the number of cords increases, the amplitude decreases. As the number of cords increases, the slope of the line for restoring force vs. position becomes steeper. What happens to the number of bounces (period) as the number of bungee cords changes? The number of bounces increase and the period decreases. The bungee jumper represents a mass-spring system, with the jumper acting as the mass and the bungee cords acting as the spring. Do more bungee cords correspond to a stier spring or a looser spring? Explain. More bungee cords are the same as a stier spring. The stier the spring, the less displacement there is. When the number of bungee cords is at a minimum, the jumper never bounces back. What is the eect of gravity on the total displacement, velocity, and period of the system? Write a paragraph to describe the eect. Gravity increases the displacement and velocity of the jumper, but has no eect on the period.
Image Sources
(1) http://www.doe.virginia.gov/VDOE/Instruction/Science/ScienceCF-PH.pdf
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Figure 1: The launch abort system for the Pad Abort-1 (PA-1) ight test is positioned on the launch pad in preparation for the test at the U.S. Armys White Sands Missile Range in New Mexico. The uncrewed, integrated ight test will evaluate the ability of a launch abort system to pull an astronaut crew to safety in the event of an emergency on a launch pad. (Attribution: NASA) The LAS has been tested and according to the NASA web site [2], NASAs 97-second ight test of Pad Abort 1 (PA-1) was launched at 7 a.m. MT on May 6, 2010, at the U.S. Armys White Sands Missile Range, New Mexico. PA-1 is the rst fully integrated ight test of the launch abort system being developed for the Orion Crew exploration vehicle. Refer to NASAs myexploration web site for a video of the test ight and more details about the LAS and the Pad Abort 1 ight test. The information gathered from the test will help rene design and analysis for future launch abort systems, resulting in safer and more reliable crew escape capability during rocket launch emergencies [3]. From the description of the LAS, it is clear that systems can be very complex and you can have systems within systems. The LAS is a subsystem of the CEV system, just as Earths atmosphere is a subsystem of the Earth system and the atmosphere plays a very important role in the LAS. It is clear that systems and subsystems can interact to further complicate the modeling.
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larger full-scale objects. Richard Feynman (1918-1988), who won the Nobel Prize for fundamental work in quantum electrodynamics, helped determine the cause of the failure in the Challenger disaster of 1986. In the processes of doing physics or engineering, practitioners use modeling and simulation to help understand theories/laws and to design/understand real-world products.
Figure 2: Modeling and simulation using a computer can play a very important role in the above process. (Attribution: Randall Caton, CC-BY-NC-SA.)
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in understanding motion in three-dimensions by breaking the problem into separate one-dimensional problems. From centuries of study, scientists have determined that position, velocity, and acceleration are the important and necessary quantities to describe motion. See the Kinematics chapter in this book for more discussion of one-dimensional motion: motion diagrams, observing motion with motion sensors, graphing of motion and understanding graphs of motion. Also see Laboratory Activities in this book for experiments on motion.
Velocity
Velocity is how fast you move through space. It is the rate of change of position with time. Average velocity is dened as the displacement divided by the time elapsed. For large elapsed times, average velocity gives us a very rough idea of how rapidly we moved through space and sometimes not even that. For the round trip described above the average velocity is zero even though we may have been moving at a reasonable average rate during the whole trip. The concept of average velocity becomes most useful when we consider its limit as the elapsed time interval approaches zero. Then we get a measure of the rate of motion at the instant in question. We call the limit of the average velocity as the elapsed time approaches zero the instantaneous velocity. When we speak of motion in the y direction we call this vy . If you know calculus, it is the derivative of position with time dy/dt. Graphically, instantaneous velocity is the slope of the tangent line to the y vs. t curve at the time in question (see the Kinematics chapter for more detail). The concept of instantaneous velocity is essential to a further understanding of motion.
Acceleration
Velocities arent always steady - they often change with time. Sometimes the magnitude (or absolute value) of the velocity increases, like when you drop a ball, and sometimes velocities decrease or stay steady. It is valuable to realize that we use the same process in dening rates of change of velocity as we did above when dening rates of change of position. Acceleration is the term we use for rate of change of velocity. Average acceleration is the change in velocity divided by the elapsed time. Again it is instantaneous acceleration, or the limit of the average acceleration as the elapsed time approaches zero, that tells the best story about motion. If you know calculus, it is the derivative of velocity with time dvy /dt. Graphically, instantaneous acceleration is the slope of the tangent line to the vy vs. t curve at the time in question. On Earth, freely falling objects accelerate downward at roughly 10 m/s2 . Once the rocket engines cut o, the only force on the rocket is gravity (if we neglect air resistance) and we say the rocket is in freefall accelerating downward at a constant rate even if its motion continues upward for some time. www.ck12.org
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Force
Force is a push or a pull. When an elevator accelerates upward, the motor exerts a force on the cable attached to the elevator to pull it up. Even when the elevator is going down, the motor must exert a force on the cable to keep the elevator from falling too fast. When you stand on the oor, the oor exerts an upward force on you to keep you from falling through the oor. When you stand on a scale, the oor exerts an upward force on the scale and the scale in turn exerts an upward force on you, which is read on the scale. Complicated - isnt it?
Gravity
Now that you understand the basics of motion, we can discuss an important force in many cases involving motion. All masses attract each other. We call that gravity. The Earth is a very large mass equal to about a trillion billion elephants. The Earth attracts the rocket towards its center and this often causes the rocket to return back to Earth. The acceleration of a freely falling body near the surface of the Earth is given the symbol g. How would things change on the Moon or other planets? You can explore that later.
Weight
When you stand at rest on the oor, the oor pushes up on you to keep you from falling through. The value of this force is your mass times the acceleration of gravity and we call the value of this force your weight W = mg. This is what you actually feel because of Earths gravity.
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Approach to Learning
Students can learn a lot by looking at examples of how others solved a particular problem and then modifying the solution to adapt the problem to less stringent assumptions. It is in the modication process that the students understanding of the solution is tested and where valuable learning takes place. We will follow that process in this chapter. Using Etoys, a simulation of the rocket, with simplifying assumptions will be available for the learner to modify to a more realistic solution. As more complexity is included along the way, the learner will be rewarded with a sense of accomplishment and a deeper understanding of modeling, simulation and motion.
General Assumptions
We start by making several assumptions to simplify the problem of the LAS and relax several of them as we approach a more realistic model. 1. The acceleration of gravity is constant at 9.8 m/s2 . This is a reasonable assumption near the surface of the earth, but at the International Space Station, that acceleration has already dropped to around 0.9 the value that it has at the Earths surface. We can relax this assumption without too much eort. 2. We will neglect air resistance. This isnt a reasonable assumption and it is one of the rst we will attack with Eulers method. 3. The rocket engine will be modeled as providing a constant upward acceleration. This isnt realistic and we will later model the rocket engine with a constant gas exhaust velocity relative to the rocket and a constant rate of mass decrease of the rocket fuel. However, we wont be able to model the actual engine used by NASA, as that is proprietary information proprietary information is information www.ck12.org
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4.
5. 6. 7.
or intellectual property owned by a company and protected from unauthorized distribution for the purpose of allowing the company to make money. We will neglect the fact that Earth is rotating. Newtons laws apply in a coordinate system that isnt accelerating. Because the Earth rotates about its axis, a coordinate system attached to the Earth from which we observe the motion, is accelerating. Anything moving in a circle accelerates because the direction of the velocity is always changing. Velocity is a vector and has both magnitude and direction. If either the direction or magnitude changes, the velocity changes and the object is said to accelerate. The acceleration of a point on the equator is around 0.034 m/s2 . Although this is a small fraction of g, the eect is very noticeable for the range of motion of a typical rocket. In addition there is a smaller acceleration because the Earth revolves around the Sun 0.006 m/s2 and an even smaller one because the Sun revolves about the center of our Milky Way galaxy 2.4 x 10 -10 m/s2 , and so on... It isnt easy to do physics calculations in a rotating coordinate system and we will leave that as an independent investigation for the learner. Read about the Coriolis eect (Gaspard-Gustave Coriolis 1792-1843) if this topic interests you. We will neglect forces from winds. We will neglect departures of the Earth from a uniform sphere. We will neglect the buoyant force of the Earths atmosphere on the rocket.
Description
You can access this model after downloading and installing Etoys at squeakland.org. The URL for the model is http://www.pcs.cnu.edu/rcaton/exbook/exbook.html. If you cant nd a computer where you can install Etoys, it is possible to run the simulation o a memory stick. You can download a copy of EtoysTo-Go at squeakland.org and the simulation at http://www.pcs.cnu.edu/rcaton/exbook/exbook.html. The simulation is an Etoys project with an extension .pr. If the downloading process puts a .txt extension on the end, remove the .txt extension. You need to unzip the downloaded le and put Etoys-To-Go and the simulation on the same memory stick. The simulation .pr le should go in the Etoys directory.
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against the time on the horizontal axis. Work in groups and brainstorm how you should best collect and record the data. Have each member choose dierent burn times and accelerations for the rocket. Discuss your results in the group and compare with others data and plots. 1. How does the burn time aect your graphs shape? 2. How does the acceleration aect your graphs shape? Exploration 2: Record data in your notebook on the velocity of the rocket approximately every second until 10 seconds after the rockets burn time is over. Plot the velocity on the vertical axis against the time on the horizontal axis. Work in groups and brainstorm how you should best collect and record the data. Have each member choose dierent burn times and accelerations for the rocket. Discuss your results in the group and compare with others data and plots. 1. What is special about the shape of your graph? Remember, there is always error in real data, so your graph may not be perfect. Try to visualize what the ideal graph would look like. 2. How does the burn time aect your graphs shape? 3. How does the acceleration aect your graphs shape? Exploration 3: What would happen if you launched the rocket on the Moon or a planet other than Earth? Work in groups and have each member nd a value for the acceleration of gravity at the surface of the Moon or planet. Use books and the Internet. Be sure the units are meters per second per second so you can compare with the value given for Earth. If the values are in dierent units, look up how to convert the values to the needed units. Enter the new gravity values in the second from bottom box in the Control and Data Center. Take data on the new motions and plot it. Discuss your results in the group and compare with others data and plots.
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Model 4: Accounting for the Change in Gravity as the Distance from the Earth Changes
Earths force of gravity reduces as the rockets distance from Earth increases. To model this, assume an Earth with spherically symmetrically distributed matter and then you can replace the Earth with a point mass at its center. Newton held up publication his Principia until he could prove this was true. For the LAS at the launch pad, the variation of the force of gravity isnt a big eect because the CEV doesnt get that far from the Earths surface, but it is valuable for your education to model this behavior.
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A simplied 2D model of the LAS implemented in Etoys can be downloaded at http://www.pcs.cnu.edu/rcaton/exboo Explore this model to learn how it works. A simulation with changing mass would be a much better representation. The LAS + CM loses about a 7th of its mass during the abort motor burn. We used a simplied thrust curve for the abort motor because the actual thrust curve is NASA sensitive.
Extensions
Adapt the 2D LAS model to employ a rocket with constant exhaust velocity and constant rate of mass loss. Model a two-stage 1D rocket.
13.6 References
1. Orion Crew Exploration Vehicle Launch Abort System Guidance and Control Analysis Overview, John B. Davidson, Sungwan Kim, David L. Raney, Vanessa V. Aubuchon, Dean W. Sparks, www.ck12.org
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and Ronald C. Busan, Ryan W. Proud and Deborah S. Merritt 2. http://www.nasa.gov/externalash/myexploration/index2.html 3. Private communication, Christopher E. Giersch
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