Circular Motion
Circular Motion
Circular Motion
Please remember to photocopy 4 pages onto one sheet by going A3A4 and using back to back on the photocopier.
We know that angles can be measured in degrees. They can also be measured in something called Radians*, where
Angular Velocity
Angular Velocity is the rate of change of angle with respect to time.
Angular Velocity is measured in radians per second, (rad/s). =
The symbol for angular velocity is (pronounced omega).
To derive v = r
Remember we defined (in radians) as s/r: =
Divide both sides by t:
=
But = /t and v = s/t: =
Cross-multiply to get: =
Centripetal Force*
The force - acting in towards the centre - required to keep an object moving in a circle is called a centripetal force.
Centripetal Acceleration
If a body is moving in a circle the acceleration it has towards the centre is called Centripetal Acceleration.
For both of the definitions above you must refer the object moving in a circle and that the direction is in towards the
centre.
= a = r2
but because v = r we also have
These formulae are all on page 51 of the log tables, and you should have this page open in front of you when doing
these questions in order to familiarise yourself with them.
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Circular Satellite Orbits
4 2 R 3
Relationship between Periodic Time and Radius for a Satellite in Orbit* T2
GM
Derivation of formula:
We compare two formulae which we have for Force:
Gm1m2
The first is the Universal Gravitational Force formula: Fg
d2
mv2
The second is the Centripetal Force formula: Fc
r
Equate both forces (because both equations apply to satellite motion),
Cancel one m from both sides
Replace the d2 in the first formula with r2 (because in this scenario the distance between the satellite and the planet
also corresponds to the radius of the circle that the satellite is tracing out.
Cancel one r both sides
You now have GM
v2
R Equation (1)
{You must be familiar with using this equation as it comes up a lot and is not in the log tables}
2R 4 2 R 2
v v2 Equation (2)
T T2
4 2 R 3
Equating Equations (1) and (2) we get T2
GM
Note that there are a total of three formulae here; all three are either on the gravity page or on the circular motion page
of the log tables.
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Extra Credit
Centripetal Motion
centripetal means centre seeking.
Look at the image of the stone moving in a circular path above a persons
head.
If an object is to go round in a circle it must have a force pulling it inwards
towards the centre of the circle e.g. the pull of a string on an object being
swirled round. Without a force towards the centre of the circle, the object
will carry on in a straight line!
The force towards the centre is called the centripetal force but
and heres the important bit what causes this centripetal force
will change from situation to situation.
Xkcd.com/123
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*Relationship between Periodic Time and Radius for a Satellite in Orbit
Congratulations
You have just arrived at an equation which bookmarks a seminal moment in the history of science.
Around this time (late16th century) an astronomer called Johannes Kepler discovered empirically (i.e. by analyzing
data on the motion of planets) that the square of the periodic time of these planets (time for one complete orbit around
the sun) is proportional to the cube of their distance from the sun.
Kepler actually stole the necessary data from a colleague, Tycho Brahe, but thats nothing new in the world of
Science. We will conveniently ignore that for now.
Later on Newton came along and was able to demonstrate this relationship mathematically, by combining a well-
mv 2 Gm1m2
known equation for circular motion on Earth Fc with his own universal law of gravitation. Fg
r d2
We are about to follow in his footsteps and see exactly what he did and how he did it. Do not under-estimate the
importance of this exercise (yes you have to know it for exam purposes, but thats not why I consider it important).
This event had two very important consequences.
1. It showed that Newtons Law of Gravitation must be valid in its own right, which was very important in securing
Newtons reputation as a giant of science, both at the time and for posterity.
2. Even more importantly, it demonstrated that the heavens followed the same rules of science as those which
operated here on Earth.
This meant that they were a legitimate area of study, and so Astronomy (which in turn led to Cosmology) was
given an added respectability. Just to give a sense of what people believed at the time, Kepler had to spend much
of his time during this period defending his mother of charges of being a witch.
I can think of no modern discovery which compares with this. Even if we discovered life on Mars it really wouldnt be
that big a deal. For up to this point the heavens were considered off-limits the realm of God or the gods or whatever
youre into yourself. But now they could be shown to be just another series of objects which followed set rules, much
like cogs in a complicated clock. So God was being pushed into the wings. You could see why neither Martin Luther
or the Vatican Church would have been keen fans.
Kepler was following on the work of Nicolas Copernicus (known to science students down the ages as copper
knickers), who in turned showed that the Earth revolved around the Sun, not the other way around.
Galileos run in with the Church was because he supported Copernicus view, so Galileo never actually made that
discovery, but he was happy to use it to make fun of the church authority figures of the time. And we all know how
that worked out for him.
So this was really the dawn of science, and progress was hindered by medieval views of the astronomers themselves.
It took Kepler decades to realise that the orbit of the planets was elliptical in nature, not circular. He had assumed
initially that the motion had to be circular because a circle was a perfect shape (harping back to the teachings of
Pythagoras and Aristotle, among others) and therefore would have been more pleasing to God who obviously had
created the planets in the first place.
Similarly Newton, despite being heralded as one of our greatest ever scientists, spend up to 90% of his time trying to
date the creation of Earth by tracing who gave birth to who in the bible.
But then Newton had another problem. He realised that Kepler was correct in stating that the planets traced out
elliptical orbits, but even Newtons equations didnt fully match the path of the heavenly bodies; according to
Newtons equations the planets should slowly but exonerably drift from their current pathways. He couldnt figure out
why this didnt happen after all, his equations seemed to be perfect in every other way. And Newton believed that he
was getting his ideas directly from God. Which doesnt leave much room for admitting you made a mistake.
We now know that while Newtons equation are very accurate, we actually need Einsteins Theory of General
Relativity to explain why they dont precisely describe the motion of the planets.
Its interesting to note that Newtons explanation was that God must step in every so often to gently nudge the planets
back into their preferred orbits. Now as you now know, invoking a deity to explain discrepancies in scientific
observations is the antithesis of Science. So perhaps Newton wasnt actually so mighty after all. This is partly why he
is sometimes referred to as the last sorcerer rather than the first scientist.
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So now were up to Einstein. His general theory of relativity suggested that the universe was expanding, but just like
all of his predecessors he was a man of his time, and this coloured how he saw the world. It was believed at the time
that the universe has always been the way it is now (this is referred to as the Steady State theory). Einstein figured
that there must be some mistake in his paper so he introduced what he called a cosmological constant which
basically amounted to a fudge factor which altered the implications of his calculations and prevented the universe
from expanding.
Which was all very well until Hubble (he of the Hubble telescope) showed that the universe was actually expanding
after all.
Doh!
So there you have it. This has been my attempt to put some context on the derivation that we are about to carry out. It
is our chance to repeat one of the greatest moments in the history of science.
So you have two options; you can consider this exercise to be a pain in the ass or you consider it an incredible
privilege to be in a position where you can follow in the footsteps of giants.
I think we know which option I go with.
And dont be afraid to tell your parents this tonight; they may well throw their eyes up to heaven but if they do thats a
slight on them not on you.
The amount of formulae that need to be learned in Physics can be rather intimidating, so any help you can get to
alleviate your suffering should be availed of. This chapter contains more formulae than most others, and possibly as a
result of this students often try to avoid questions on circular motion when they can.
I suggest you get a sheet of A4 paper and put all these equations on it, showing how they are interconnected.
Taking a big picture approach like this should help to reduce the intimidation factor.
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Exam Questions
2. [2006]
Derive the relationship between the velocity of a particle travelling in uniform circular motion and its angular
velocity.
3. [2002]
A particle travels at a constant speed of 10 m s-1 in a circle of radius 2 m. What is its angular velocity?
4. [2006]
(i) A student swings a ball in a circle of radius 70 cm in the vertical plane as shown. The angular velocity of the ball
is 10 rad s1. What is the velocity of the ball?
(ii) How long does the ball take to complete one revolution?
5. [2005]
Define centripetal force.
6. [2004]
Give an expression for centripetal force.
7. [2004]
(i) Centripetal force is required to keep the earth moving around the sun.
What provides this centripetal force?
(ii) In what direction does this centripetal force act?
8. [2009]
A skateboarder of mass 70 kg has a speed of 10.5 m s1 as he enters a circular ramp of radius 10 m.
What is the centripetal force acting on him?
9. [2004]
The earth has a speed of 3.0 104 m s1 as it orbits the sun.
The distance between the earth and the sun is 1.5 1011 m. Calculate the mass of the sun.
10. [2009]
The moon orbits the earth. What is the relationship between the period of the moon and the radius of its orbit?
11. [2008][2005]
Derive the relationship between the period of the ISS, the radius of its orbit and the mass of the earth.
12. [2005]
(i) A satellite is in a circular orbit around the planet Saturn.
The period of the satellite is 380 hours. Calculate the radius of the satellites orbit around Saturn.
13. [2008]
The international space station (ISS) moves in a circular orbit around the equator at a height of 400 km.
(i) Calculate the period of an orbit of the ISS.
(ii) After an orbit, the ISS will be above a different point on the earths surface. Explain why.
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Exam Solutions
1. Angular velocity is the rate of change of angle with respect to time.
2. = s /r
/t = s/rt
= v /r
v=r
4.
(i) v = r = (10)(0.70) = 7.0 m s-1
(ii) T= 2r/v = 2(0.70)/v = 0.63 s
5. The force - acting in towards the centre - required to keep an object moving in a circle is called Centripetal Force.
mv2
6. Fc
r
7.
(i) Gravitational pull of the sun.
(ii) Towards the centre.
mv 2
8. Fc = 70 (10.5)2/10 = 771.75 N
r
Gm1m2 mv2 GM
9. Fg and Fc Equating gives v2 Ms = v2R/G
d2 r R
Ms = (3.0 104)2 (1.5 1011)/ 6.7 1011 = 2.0 1030 kg.
Note
Strictly speaking the distance (or radius) is from centre to centre. In this case the question doesnt specify
whether the quoted distance is from surface to surface or from centre to centre, but it doesnt really matter
because the diameters of both spheres are insignificant relative to the overall distance. For an artificial satellite in
orbit around a planet however the diameter of the planet would have to be taken into account.
11. See notes Circular Motion chapter for a more detailed derivation.
13.
(i) R = (400 103 + 6.4 x 106 ) = 6.8 106 m
(ii) The ISS has a different period to that of the earths rotation (it is not in geostationary orbit).