General Physics 1: Quarter 2 - Module 3: Harmonic Motion
General Physics 1: Quarter 2 - Module 3: Harmonic Motion
General Physics 1: Quarter 2 - Module 3: Harmonic Motion
General
Physics 1
Quarter 2 – Module 3:
Harmonic Motion
Subject Area – 12
Self-Learning Module (SLM)
Quarter 2 – Module 3: Harmonic Motion
First Edition, 2020
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General Physics 1
Quarter 2 – Module 3:
Harmonic Motion
Introductory Message
This module was collaboratively designed, developed and reviewed by educators both
from public and private institutions to assist you, the teacher or facilitator in helping
the learners meet the standards set by the K to 12 Curriculum while overcoming
their personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help
learners acquire the needed 21st century skills while taking into consideration their
needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to manage
their own learning. Furthermore, you are expected to encourage and assist the
learners as they do the tasks included in the module.
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For the learner:
This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at your own pace and time. You will be enabled to
process the contents of the learning resource while being an active learner.
What I Need to Know This will give you an idea of the skills or
competencies you are expected to learn in the
module.
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What I Can Do This section provides an activity which will
help you transfer your new knowledge or skill
into real life situations or concerns.
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
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What I Need to Know
This module was designed and written with you in mind. It is here to help you master
the principles of Simple Harmonic Motion. The scope of this module permits it to be
used in many different learning situations. The language used recognizes the diverse
vocabulary level of students. The lessons are arranged to follow the standard
sequence of the course. But the order in which you read them can be changed to
correspond with the textbook you are now using.
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What I Know
PRE-TEST. Choose the letter of the correct answer. Encircle the letter of you
chosen answer on the answer sheet provided.
1. Which of the following is a mass on a spring undergoes Simple Harmonic Motion
and the maximum displacement from the equilibrium?
A. Period B. Frequency C. Amplitude D. Wavelength
2. Which of the following in a periodic process, the number of cycles per unit of
time?
4. What is the instantaneous velocity, when the mass reaches point x = +A?
A. Maximum and positive C. Maximum and Negative
B. Zero D. Less than maximum and positive
7. What is the instantaneous acceleration, when the mass reaches point x = +A?
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8. An object with a mass M is suspended from an elastic spring with a spring
constant k. The object oscillates with maximum amplitude A. If the amplitude of
oscillations is doubled, how will it change the period of oscillations?
A. 2T B. 4T C. ½ T D. ¼ T
12. The length of a simple pendulum oscillating with a period T is quartered, what is
the new period of oscillations in terms of T?
A. 2T B. 4T C. ½ T D. ¼ T
13. A simple pendulum has a period of 1 s. What is the length of the string?
A. 1 m B. 2 m C. 4 m D. ½ m E. ¼ m
14. A simple pendulum has a period of 2 s. What is the length of the string?
A. 1 m B. 2 m C. 4 m D. ½ m E. ¼ m
15. A simple pendulum has a period of 4 s. What is the length of the string?
A. 1 m B. 2 m C. 4 m D. ½ m E. ¼ m
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Lesson Amplitude, Frequency, Angular Frequency,
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Period, Displacement, Velocity and
Acceleration of Oscillating Systems
Learning Objective:
What’ s In
In the previous lesson we learned all about Kepler’s Laws of planetary motion
and circulation motion. Now let’s try to relate the concepts of circulation motion and
try to think what velocity and acceleration does the revolving body experience.
1st Law: The planets move about the sun in ELLIPTICAL orbits, with the sun at
one focus of the ellipse.
2nd Law: The straight line joining the sun and given planet sweeps_______________
_____________
Can be remembered as ________________.
3rd Law: The square of the period of revolution of a planet ABOUT THE SUN is
proportional of the cube of its mean distance from the sun.
A planet is in the orbit as shown below. Where are the two possible locations
for a Sun?
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What’s New
Material:
Rubber Ball
1. 2. 3.
Ball Drop at your knee Ball Drop at your waist Ball Drop at your chest
level level level
Hold the Ball at the level of Hold the Ball at the level of Hold the Ball at the level of
your knee and drop it into your waist and drop it into your chest and drop it into
a flat surface, and let the a flat surface, and let the a flat surface and let the
ball bounce on its on until ball bounce on its on until ball bounce on its on until
it will stop. it will stop. it will stop.
Questions:
1. After the First Bounce, did you notice that the ball return to its initial height?
2. Which of the following levels allows the ball to bounce the longest?
3. What word or term can you relate to the activity you performed?
What is It
Periodic Motion
Imagine attaching a spring to a ball while
hanging them vertically and you pull the ball at a
certain displacement from the equilibrium point.
It can be observed that the ball at high speed
seems to vibrate or oscillate, this motion is called
a Periodic Motion. Specifically, Periodic motion
is a motion that repeats itself in a definite cycle,
It occurs whenever a body has a stable
equilibrium position and a restoring force acts
when it is displayed from equilibrium.
It can be said at point A that the ball is at equilibrium, and no force is acting
on it. This point, where the spring isn’t stretched or compressed, is called the
equilibrium point. At point B, the ball pushes against the spring, and the spring
retaliates with force F opposing that pushing. At point C, the spring releases, and the
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ball springs to an equal distance on the other side of the equilibrium point. At this
point, the ball isn’t moving, but a force acts on it, which is force F, so it starts going
back the other direction. If friction and air resistance is neglected it can be said that
the periodic motion is directly proportional to the displacement (y), this motion is
called Simple Harmonic Motion.
Amplitude (A)
When the example moves in a Simple
Harmonic Motion, the distance made by the
object from its equilibrium point is called the
Amplitude. Specifically, amplitude is simply the
maximum extent of the oscillation or the size of
the oscillation.
Period (T)
Imagine shining a light in front of the ball as
it cast its shadow. Film the movement of the shadow
as shown in the image. Add a reference circle and
try to trace the wave formed from the film and relate
it to the circle. Each time an object moves around a
full circle, it completes a cycle. The time the object
takes to complete the cycle is called the Period.
When an object moves in a full circle,
completing a cycle, the object goes 2π radians. It
travels that many radians in T seconds, so its
angular speed, ω is:
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Displacement, Velocity & Acceleration
By observing the reference circle, we
can relate time with the movement of the
ball along the y-axis. Let amplitude(A) be
the hypotenuses, distance y be the
opposite and the angle be θ which is equal
to ωt.
From the said given we can get the
relationship of the amplitude to
frequency(f):
Convert angular frequency (ω) to frequency (f):
sin(θ) = sin(ωt) = y/A;
y = A*sin(2πft); where ω = 2πf
y = A*sin(ωt)
(Note: since the motion of the ball is a vertical motion, going up or down,
displacement y is the one computed.)
By using the equation above, and differentiating it with respect to time(t) we can
get the relationship of velocity(v) to frequency(f):
y = A*sin(ωt) Convert angular frequency (ω) to frequency (f):
𝑑𝑦
v= = 𝐴 ∗ ωcos(ωt) v = A*(2πf)cos(2πft)
𝑑𝑡
By using the equation above, and differentiating it with respect to time(t) we can
get the relationship of acceleration(a) to frequency(f):
𝑑2 𝑦
= −𝐴 ∗ 𝜔2 sin(ωt)a = Convert angular frequency (ω) to frequency (f):
𝑑𝑡 2
a = -𝜔2 𝑦 a = -(2πf)2 y
Since
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When amplitude (A) is 16 cm and the ball bounces
Example: back and forth in a span of 4 seconds. Find (a)
frequency and angular frequency, (b) acceleration
and velocity at x = 16 cm, and (c) acceleration and
velocity at x = 10 cm.
(a) Frequency (f) Angular Frequency (𝜔)
What’s More
Materials:
1 Paper Clip
6 Rubber Bonds
2 Push Pins
1 Flat Wood
1 Ruler
Procedures:
1. Place the push pin in 1 foot apart.
2. Connect the 3 rubbers bonds. To create 2 sets.
3. Connect the 2 sets of rubber bonds together with a paper clip
4. Place the both ends of the rubber to the push pin.
5. Stretch the paper clip 10 cm to the right.
6. Record the time when the paper clip reaches 10 cm to the left.
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Complete the Table
Amplitude Frequency Angular Period Displacem Velocity
(𝒎) (𝑯𝒛) Frequency (𝒔) ent (𝒎/𝒔)
(𝒓𝒂𝒅/𝒔) (𝒎)
Paper
Clip
Paper
Clip
with
Coin
Question:
1. What causes the paper clip to move back and forth?
2. Given the mass of the paper clip ____ g and the coefficient of the elasticity of the
rubber bond k=________. Solve for the Acceleration of the motion?
3. Given the mass of the paper clip and the coin ____ g and the coefficient of the
elasticity of the rubber bond k=________. Solve for the Acceleration of the motion?
4. What changes you observe between the two scenarios?
What I Can Do
Instruction: Solve and illustrate the problem given. And show your solution
and diagram legibly.
A 100-g body is attached at the end of a hanging spring with a spring constant
of 2,000 dynes/cm. It is displaced 10 cm from its equilibrium position and then
released.
(a) Calculate the period T.
(b) Find the maximum acceleration of the body.
(c) Find the acceleration of the body when it is 5.0 cm from the equilibrium position.
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Lesson
Learning Objective
1. Identify necessary conditions for an object to undergo Simple Harmonic
Motion
2. Analyze and solve problems involving Simple Harmonic Motion.
What’s In
_______1. Frequency a. The number of cycles that are completed per second
_______2. Period b. The time the object takes to complete the cycle.
equilibrium point.
an equilibrium.
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What’s New
Questions:
1. What factor allows your body to go back to its initial position?
2. What word or term can you relate to the following?
What is It
To calculate the force exerted by the spring to the body at each of point, we
need to use the Hooke’s Law for ideal springs. The equation is:
Fx = -kx (restoring force exerted by an ideal spring)
Where k = force constant
x = displacement from equilibrium
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Fx = F
-kx = ma
-kx = m(-𝜔2 𝑥)
(substitute acceleration [a] with the acceleration obtained
from the previous lesson, but instead of using the
amplitude y we use amplitude x since the motion is on
-kx = m(-𝜔2 𝑥)
the horizontal axis. [a =-𝜔2 𝑥])
-k = m(-𝜔2 )
−𝑘 𝑚(−𝜔2 )
=
𝑚 𝑚 (cancel out the common terms from both sides)
𝑘
= 𝜔2
𝑚
(equate the equation by angular frequency [𝜔])
2𝜋
𝜔 = 2𝜋𝑓 =
𝑇
2𝜋
𝑇 = , 𝜔 = 2𝜋𝑓
𝜔
𝑚
𝑇 = 2𝜋√ 𝑘
(Period [𝑇] of SHM)
From the previous derivations it can be seen that Simple Harmonic Motion (SHM)
is the combination periodic motion and a restoring force whose magnitude depends
on the displacement from the equilibrium position.
Example:
A 100-g body is attached is attached at the end of a hanging spring with a spring
constant of 2,000 dynes/cm. It is displaced 10 cm from its equilibrium position and
then released, Calculate the period T.
Given: m = 100 g k = 2000 dynes/cm
A = 10 cm x = 5.0 cm
𝑑𝑦𝑛𝑒𝑠
T = 2π√𝑚/𝑘 = 2π√100𝑔/(2,000 ) = 1.404 s.
𝑐𝑚
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What’s More
Materials:
2 feet yarn
2 feet connected rubber bands
1pc. 330 ml mineral bottle filled with water
1 meter wooden or bamboo stick
2 chair
1 tape
Procedures:
1. Using the 1-foot rubber bonds, tie the mineral bottle that is filled with water.
2. Put the wooden or bamboo stick between a distant chair. Using the tape assure
the stick will not move.
3. Tie the material prepared in procedure No. 1 at the middle of the wooden or
bamboo stick suspending it at a length of about an elbow’s length.
4. Place the bottled water on the to top of the stick where you tie it.
5. Release and top the bottle.
6. Observe its movement.
7. Repeat the procedure using the yarn.
Questions:
1. What do you observe when the bottled released?
Using the yarn? Rubber band?
2. What do you observe with the movement of the bottle with two
materials?
3. Will the frequency matters with the materials used? How about its
velocity?
4. In the activity, Is there a restoring force that allows the object to
restores its initial position?
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What I Have Learned
Instruction: Fill the blanks with the correct answer. Choose your answer on
the given set of words in the box.
The greater the spring constant (k), the 1)______the spring; hence a greater
force is required to 2)________or compress the spring. When force is greater,
acceleration is 3) __________, and the amount of time required for a single cycle should
4)________________(assuming that the amplitude remains constant). Thus, for a given
amplitude, a stiffer spring will take 5)_________ time to complete one cycle of motion
than one that is less stiff.
What I Can Do
Instruction: Solve and analyze the problem carefully. Show your solution.
The body of a 1275 kg car is supported on a frame by four springs. Two people
riding in the car have a combined mass of 153 kg. When driven over a pothole in the
road, the frame vibrates with a period of 0.840 s. For the first few seconds, the
vibration approximates simple harmonic motion. Find the spring constant of a single
spring.
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Lesson Application of Simple Harmonic
3 Motion
Learning Objective:
What’ s In
In the previous activity we have derived the equation for a mass-spring system
in harmonic motion now let’s try to expound the concept by applying the previous
derivation to some real-life situation where harmonic motion is seen.
a. At maximum stretch
b. At equilibrium
c. At maximum compress
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What’s New
Procedures:
1. Hold the Ball at the level of your chest and drop it into a flat surface, and
let the ball bounce on its on until it will stop.
2. Observe the movement of every ball.
Questions:
1. Which of the three bounce faster?
2. Why is it the three balls bounce differ to each other?
What is It
One of the most relatable phenomena that shows the concept of Simple Harmonics
Motion (SHM) is a pendulum. Now let us define what is it.
Simple Pendulum
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F = -mgsinθ
𝑥
And using sinθ =
𝐿
𝑥
F = -mg
𝐿
Since from the second law of motion by Newton, F = ma, we can equate the
two equations for the force. We can have then,
𝑥
ma = -mg
𝐿
𝑥 −𝐿
=
𝑎 𝑔
𝐿 −𝑥
Since T = 2π√−𝑥/𝑎, we substitute to .
𝑔 𝑎
Therefore, the period of a simple pendulum is T = 2π√𝑳/𝒈.
*Where L is the length of the string and g is gravitational acceleration (g = 9.81 m/s 2).
Physical Pendulum
√𝐿 = (T√𝑔)/2π;
L = T2g/4π2 = (6 s)2(9.81 m/s2)/4π2 = 8.95 m
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What’s More
Instruction: Perform the activity and observe it properly. And fill in the table to
answer the following questions below.
Materials:
1-meter yarn
1- Foot of connected rubber bonds
1pc. 330 ml mineral bottle filled with water
1 meter wooden or bamboo stick
2 chair
1 tape
Procedures:
1. Using the 1-meter yarn, tie the mineral bottle that is filled with water with a length
of around 50 cm (approximately an arm’s length).
2. Put the wooden or bamboo stick between a distant chair. Using the tape assure
the stick will not move.
3. Tie the homemade pendulum prepared in procedure No. 1 at the middle of the
wooden or bamboo stick suspending it at a length of about an elbow’s length.
4. Hold the suspended improvised pendulum at a height equal to the height of the
stick making sure that the string does not sag or bend.
5. Release the improvised pendulum.
6. Observe its movement.
Questions:
1. Which length of string had the greater period?
2. Which length of string had the greater frequency? Angular frequency?
3. What can be concluded between the length of the string to the value of period,
frequency, and angular frequency?
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What I Have Learned
Instruction: Underline the correct answer that will satisfy the statements
below.
What I Can Do
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Assessment
POST-TEST. Choose the letter of the correct answer. Encircle the letter of you
chosen answer on the answer sheet provided.
1. The length of a simple pendulum oscillating with a period T is quadrupled, what
is the new period of oscillations in terms of T?
A. 2T B. 4T C. ½ T D. ¼ T
2. The length of a simple pendulum oscillating with a period T is quartered, what is
the new period of oscillations in terms of T?
A. 2T B. 4T C. ½ T D. ¼ T
3. A simple pendulum has a period of 1 s. What is the length of the string?
A. 1 m B. 2 m C. 4 m D. ½ m E. ¼ m
10. In a periodic process, the number of cycles per unit of time is called?
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11. Which of the following in a periodic process, which the time required to complete
one cycle?
A. Period B. Frequency C. Amplitude D. Wavelength
12. What is the instantaneous acceleration, when the mass reaches point x = +A?
A. Maximum and can be positive or negative
B. Constant and doesn’t depend on the location
C. Zero
D. Slightly less than maximum and positive
13. An object with a mass M is suspended from an elastic spring with a spring
constant k. The object oscillates with maximum amplitude A. If the amplitude of
oscillations is doubled, how it will change the period of oscillations?
A. The period is increased by factor two
B. The period is increased by factor four
C. The period is decreased by factor two
D. The period is decreased by factor four
14. An object with a mass M is suspended from an elastic spring with a spring
constant k. If the mass of oscillations is quadrupled, how it will change the period of
oscillations?
Additional Activities
Aside from a pendulum as an example for simple harmonic motion, give three
(3) examples of phenomenon or motion that shows simple harmonic motion and draw
and label what acts as its restoring force and point of equilibrium.
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References
Holzner, S., 2011. Physics I For Dummies. 2nd ed. Indianapolis, Indiana: Wiley
Publishing Inc., pp.251-267.
Serway, R. and Faughn, J., 2006. Holt Physics. 10801 N. MoPac Expressway,
Building 3, Austin, Texas 78759: Holt, Rineheart and Winston, pp.368-403.
Young, H. and Freedman, R., 2008. University Physics With Modern Physics. 12th
ed. 1301 Sansom.e St.. San Francisco, CA 94111: Pearson, pp.419-439.
Young, H. and Freedman, R., 2021. University Physics With Modern Physics. 13th
ed. 1301 Sansome Street, San Francisco, CA, 94111: Pearson Education Inc.,
pp.437-456.
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EDITOR’S NOTE