Circular Motion
7. Force and Circular Motion*
This unit is about the forces that cause central acceleration in uniform circular motion. Recall that uniform
means constant speed, and circular means a constant distance from some point, which is the center of
the circle of motion.
Learning Objectives:
1. To understand the difference between change in speed and change in velocity.
2. To realize that an object in circular motion is not in equilibrium.
3. To figure out how to design a graph of experimental data to be a straight line with a
meaningful slope.
nd
4. To utilize Newton’s 2 law and the formula for central acceleration during uniform
circular motion in solving problems in context.
Before coming to lab, you need to read the sections in your book on uniform circular motion and on
Newton’s second law. Review also projectile motion
Read the following sections. (Section numbers may be slightly different depending on the edition of your
textbook: Check the section titles.)Study the following sections of your textbook before coming to lab.
There will be a check to see that you have done this.
Knight, Jones & Field (161): 6.1: Uniform Circular Motion, 6.2: Speed, Velocity, and Acceleration in
Uniform Circular Motion, 6.3: Dynamics of a Uniform Circular Motion, 6.4: Apparent Forces in Circular
Motion, 6.5: Circular Orbits and Weightlessness
Serway and Vuille (211): 7.4 Centripetal Acceleration
Serway and Jewett (251) 4.4: Uniform Circular Motion
In this scenario, your group belongs to the Big-Top
Amusement Company, makers of carnival rides and
games. You’ll be working on the chair swing, on which
the riders sit in chair-like seats that swing around in a
circle. To model this in the laboratory, you can use two
different masses, one M hanging from a string that passes
through a glass tube, and another m that swings around
in a circle, as shown at the right. The company wants to
know about what rate of rotation will be reasonably
comfortable, and what will happen if a rider drops some
object while the swing is turning. Of course, in the
process, you want to learn about the forces that act and
how they permit uniform circular motion.

m
Paper clip
M
.
Pre-Lab exercises: Before coming to lab, answer the following questions. The TA will check them off
when you get to lab to be sure you have them done, otherwise you cannot enter lab. The answers are to
be handed in with your report.
These questions lead you through a derivation of a formula that you will need in the lab. Normally, we
would expect you to think of these steps yourself. So, be aware that this is a type of problem you should
be able to do. One of the toughest parts is always figuring out what to do. Thus you are not facing the
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*© William A Schwalm 2012
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Circular Motion
choices you would have to face for a problem in the wild. Later, you will need to get into thinking of what
to do for yourselves. Imagine this as a sort of scaffolding you can use for now, and take away later.
We’re going to assume you have done your reading.
PRE-LAB QUESTIONS (Do these problems before you come to lab)
1. Write down from your reading the formula for acceleration
circular motion in terms of speed and radius of the circle.
a toward the center of the circle in uniform
2. In the figure on the previous page the mass m moves in a circle. Indicate where the center of the
circle is using a penciled-in x on the figure.
3. In uniform circular motion, an object is accelerating even though its speed is constant. Explain how
this can happen. How can an object accelerate without changing its speed?
4. If an object moves at constant speed v around a circle of radius r, what is the magnitude
acceleration, and toward what place does the acceleration vector
a  a of its
a always point?
5. In the laboratory, you will not want to have to measure  . Explain why this would be a pain in the
neck. Why do you not want to have to measure this angle directly?
(Parts 6 through 12 are for Phys 211 and 251, but not 161)
6. Now we are going to draw two similar right triangles and label the parts. Right triangle I has the
length
as its hypotenuse, and the radius r of the circle as its base. You may call the height h.
Triangle II is a force triangle. It has the tension F =
components of the tension force
F in the string as its hypotenuse. The
F acting on the moving mass form the base Fx and the height Fy .
Draw these two triangles here and label each side in each triangle with the appropriate quantity.

r
h
I have drawn
triangle I at
the left, now
you draw
triangle II.
7. How do you know the two triangles are similar? (Hint: consider the angle
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
.)
Circular Motion
8. Since the triangles are similar, and so corresponding sides are proportional, we have that
Fx What?

r
9. Since the speed is constant, derive a formula expressing the speed v in terms of the
circumference 2 r of the circle and the time T it takes to go around once.
10. Applying Newton’s law to the hanging stationary mass M, one can see that the tension F in the string
above the mass must equal its weight Mg. Explain in a sentence and include a Free-Body Diagram
(FBD).
11. Neglecting the mass of the string itself, and the mass of the paper clip, and assuming the string slides
over the smooth end of the glass tube with no friction, show from Newton’s law that when the
swinging mass moves at speed v in a circle of radius r, then
m
v2
r
 Mg
r
Explain as necessary for full credit.
12. Since you cannot easily measure r, you need an equation where this doesn’t appear, so use some
combination of the results above (explain which) to get
M  2 

m
gT2
2
Of course you must show your work.
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Circular Motion
Comment on force measured in “g”s: When we measure force on an object “in g’s” we are measuring the
force in terms of the object’s normal weight on earth, which is mg where m is its mass. So force in g’s
turns out to be a dimensionless number. Carnival ride designers, test pilots and astronauts know all about
this. If an object weighing 530 Newtons is subjected to a force of 1.06 kilo Newtons, we would call it a 2 g
force, since the ratio is 2.
Seated on the chair swing, the effective weight a rider would feel is the force the seat exerts on the rider,
which is the same as the tension. Notice that in the lab experiment, the ratio of the tension force (let’s call
it F) to the riders usual weight mg is just F/mg which in the experiment is Mg/mg=M/m. Notice that this
ratio is given in problem 9 above, so the right-hand side of that formula gives the “force in g’s” that a rider
on the chair swing would experience.
Force in g’s
 2 

2
T 2g
Equipment: Glass tube with smooth edges, string, paper clip, rubber stopper (for mass m) weights (for M
and for making measurements) a stop watch and a meter stick.
Problem 1
Think of the result for M/m above as a formula for force in g’s that the seat of the chair swing exerts on a
rider in terms of the length
and the time T it takes to go around once. Test this by an experiment to
see how rotation rate, T actually depends on the force ratio M/m at fixed .
1. Prediction question: Thus, from your pre-lab work, draw a force diagram for a rider on the swing and
explain why the ration of the seat force to the rider’s usual weight should be the same as M/m and
hence justify the formula given above for the g force.
2. Method question: If you have an equation like p (q-a) = b, and you took data for p and q, you could
plot them in such a way as to get a straight line. Thus you could perhaps plot y = q versus x = 1/p.
Then when you fit a straight line, the slope would give you b and the intercept would give you a.
Similarly, you will be taking data relating period T to the mass or force ratio M/m. How can you graph
your data so that the equation you expect to hold would give a straight line? What variable y that
depends on T might you plot versus what other variable x that depends on M/m? What would the
slope give you? To answer this, look at the expected relationship.
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Circular Motion
Plan: Write out a plan for making measurements of period T as you vary M/m. You need to figure out
what data you need and how your team members will need to cooperate to take it. All your team
members should get a chance at both swinging the stopper around in a circle and taking time
measurements with the stopwatch. It will be best to design a data table you can fill out. Then you need
to figure out the best way to make a graph. (It would be possible, according to your theory, to get a
straight line if you graph the right two things. What should the slope be? And so on…) Outline how you
would analyze the data. Let your TA see the plan with data table before you continue.
Implementation: Implement your plan and take the necessary data, filling out your table. Make notes too
of anything interesting that comes up or any additional data you’ll need for the analysis.
Analysis: Make the appropriate graph to show whether or not the experimental results agree with your
theory, as far as variation of the ride speed is concerned.
Conclusions: Was the theory correct?
1. Did you get the result you expected for your graph? Explain.
2. If the shank length in the actual ride were = 5 meters, how many rotations per minute would
correspond to an effective rider weight of 1.5 g’s ?
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Circular Motion
Problem 2
You have a formula expressing the force in g’s that the seat of the chair swing exerts on a rider in terms
of the length and the time T it takes to go around once. To test this further, you need to see how
period T depends on
if the mass ratio M/m is fixed.
1. Method question: You will find you cannot very well adjust the period holding M/m fixed, so you need
to take data for period T versus
and M/m fixed while changing . So if you want to plot some
variable involving T versus some variable involving at constant M/m and get a straight line, what
exactly should you plot against what? (Look at the equation.) What would the slope of the line be?
Plan: Write out a plan for making measurements of period T as a function of and checking the theory.
You need to figure out what data you need and how your team members will need to cooperate to take it.
It will be best to design a data table you can fill out. Then you need to figure out the best way to make a
graph. (Would it be possible, according to your theory, to get a straight line if you graph the right two
things? What should the slope be? And so on…) All your team members should get a chance at both
swinging the stopper around in a circle and taking time measurements with the stopwatch. Let your TA
see the plan with data table before you continue.
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Circular Motion
Implementation: Implement your plan and take the necessary data, filling out your table. Make notes too
of anything interesting that comes up or any additional data you’ll need for the analysis.
Analysis: Make the appropriate graph to show whether or not the experimental results agree with your
theory, as far as variation of the ride speed is concerned.
Conclusions: Was the theory correct?
1. Did you get the result you expected for your graph? Explain.
2. What was the major difficulty in making the measurements?
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Circular Motion
Problem 3 (Phys 211 and 251)
What happens if a rider drops an object
when the chair swing is in motion?
Develop and test a formula to estimate
far from the base of the support pole at
center of the chair swing an object
would land if dropped by a moving
rider.
how
the
1. Prediction question: From
Newton’s second law applied to
the mass m, relate the cosine of
the angle  (previous figure) to
the mass ratio, and hence to the
formula you got for the force
measured in g’s.
distance
2. Prediction question: Thus if the height of the support, corresponding to the top of the glass tube in
the experiment, is H, then show that the height above the ground of the circular orbit is
yo  H 
gT 2
4 2
3. Prediction question: Show that the speed can be written as
v
2
T
1
g 2T 4
16 4 2
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Circular Motion
4. Prediction problem: Now in order to answer the question for the Big-Top company, consider the
experiment: if you release the string while the mass m is swinging around, finding where it lands is
just a projectile motion problem. Calculate this distance and explain how this should solve the
original problem.
Plan: Write out a measurement plan that your group can use to verify the formula. As usual you need to
tell what you will do, what you will measure, how you will measure it and how these data need to be
analyzed. The point will be to see if your prediction formula works.
Implementation: When you actually carry out your plane it will involve a projectile, or a flying object. Thus
be sure to do it in some area where there is nothing to break and no windows or innocent bystanders to
be whacked by the flying object. Collect and record the necessary data.
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Circular Motion
Conclusions: What did you learn?
1. What general aspects of uniform circular motion does this problem relate to? What other topic you
have studied come up?
2. How accurate is your formula? That is to say that you should estimate the measurement error and
see whether the results you get fall within the margins of what you can explain by this limitation.
3. Finally there is an over-all observation you should make. (This one’s extra credit.) Going back to the
first problem, you found in essence that
M  2 
.

m
T 2g
2
Do you see anything strange or interesting about this when M = m ? Explain
what strange-looking result comes up when you do this. In the actual experiment, what happens
when M = m ? Can you explain what’s going on?
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