SAFETY: never turn on the power without the front/all side panels in place!
1
Centripetal Force
Introduction.
A centripetal force is any force (gravity, normal force, tension) causing an object to move in a
circle. Without it, the object would move in a straight line. Satellites orbit the Earth because of
the centripetal force exerted by gravity. If gravity were somehow “cut” a satellite would move
away from the Earth in a straight line.
For a mass M with tangential speed v at radius R, the centripetal force is F
c
= Mv
2
/R. With a
period of T, the linear speed is v = 2πR/T and the centripetal force is F
c
= 4π
2
MR/T
2
. You will
change Fc, mass, and radius to measure the effect on the period of rotation.
Study I Equipment: Meet “Bob”
1. Remove the spring and adjust the apparatus: check that the forces are balanced and it will
rotate freely. Check that the “flag” on the counterweight will pass through the photogate.
2. Measure the mass, M
B
, of Bob. There are triple-beam balances available at the back and
sides of the lab. Estimate δM
B
as you would estimate error of a ruler (or any analog device).
3. Select a constant radius R from the teeth marks below the hanging mass. You can adjust the
radius with the screw on the central post (vertical spindle). Check whether the counter-
weight needs to be moved in order to easily change the radius. Estimate δR.
4. Choose a spring, weak or strong, and attach it to Bob and vertical spindle. You’ll repeat the
experiment with the second spring.
5. Check that all screws (on Bob, the counter balance, and central spindle) are tight.
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Static Force: Bob at rest
Once everything is aligned/rotating freely, find the force,
Fs, the spring will exert on Bob when he’s stretched to the
chosen radius R. Since the spring pulls him inward when
system is static, you’ll be adding (easily measured) force
to balance the spring force and estimate Fs.
Measure the tension to balance the spring force, Fs, at R:
1. Pass a string from over the pulley and attach the
weight holder. Be sure the weight hangs straight down.
2. Add mass to the weight hanger until the string and the spring are horizontal (as pictured
above). Bob should be at R.
3. Record your results as m2 (kg). DON’T FORGET THE MASS OF THE WEIGHT
HANGER. Calculate Fs = m2*g.
4. Estimate the error δm2 using the small masses and calculate δFs.
5. Remove the string and weight holder. The spring will naturally pull Bob inward.
6. Close the panels.
Centripetal Force: Bob in motion
Next use the power supply to rotate Bob. He will want to fly out to the end of the string, and
beyond, but the spring will hold him in place. As you change the current, rotation speed and
radius will change. Adjust the current to get R back. Use DataStudio to measure the period of
rotation at your chosen radius:
1. Turn on the power supply. If you have a current dial, turn the dial up a bit (there is no
correct value, but keep it fairly low).
2. Use the coarse voltage adjustment to get the spindle rotating close to the proper radius. Use
the fine adjustment for “fine tuning”. Adjust the voltage until Bob to passes directly over R.
Equipment note: Sometimes the rotation of the spindle is unbalanced, resulting in strange
ticking noises and poor data. Adjust the counterweight if necessary.
3. Start up DataStudio and double-click the digital plug located in Channel 1 to connect the
photogate. From the pop-up menu, scroll down and select “photogate and picket fence”.
4. Check the box “Time between bands (tDelta).” This will be your measured period T.
5. Create the data table and graph by dragging each little image (in the lower left box) to “Time
between bands (tDelta)” (in the upper left box).
6. Start recording data once Bob has reached the desired radius. Record about 50 revolutions
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and then hit Stop. While recording don’t disturb the table. Slight shakes can affect the time
readings.
7. The mean value of the period (
!
T
), the count (N), and the standard deviation σ(T) may be
obtained directly from Data Studio using the Statistics Tool. Calculate σ(
!
T
).
Next run down the voltage on your power supply so that rotation stops. Make sure both
coarse and fine voltage controls are fully counterclockwise (the off position).
Complete Study I by changing the spring. Changing the spring will modify the centripetal force
on the hanging mass. Observe what happens to the period when you change the centripetal force.
For your analysis, you will calculate Fc and compare this force to the equivalent “static force” Fs
applied with tension.
Study II: Design an experiment to study the effects of mass on period. You must use this to
confirm the equation for centripetal force. Since mass, radius, and Fc all affect period, make
sure to isolate independent variables in your study.
1. First notice that you can rearrange the equation for centripetal force:
!
T
2
= 4
"
2
MR
F
c
2. Measure T while changing the mass. Bob is equipped with a screw on top so you can
attach slotted weights. You will need at least 5 independent data points.
3. Remember to use the total mass (Mass Bob + added mass) in any calculations or graphs.
When making adjustments, continually check your alignment.
4. Make a plot with the independent variable on the x-axis and the dependent variable T
2
on
the y-axis. If you get a linear relationship, you’ve qualitatively confirmed the centripetal
force equation F
c
= 4π
2
MR/T
2
.
5. Use the slope of your graph, quantify your results by comparing to the constant quantities
in your experiment.
Before you leave the lab:
• Turn the power supply off!
• Complete the data table for Study I (Constants, Bob in Motion, Bob at Rest).
• A plot for Study II that confirms F
c
= 4π
2
MR/T
2
. The slope of the graph should compare
with your measured values (within error). Be sure to include the usual labels and errors.
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Analysis for Study I: Calculate Centripetal and Static Forces
1. Calculate the static force Fs and its error (you’ve probably done this already).
2. Calculate the centripetal force F
c
.
3. Determine the greatest source of uncertainty in Fc: M, R, or T, and use it to calculate the
error in Fc (it will have the same relative uncertainty).
4. Calculate the difference between the calculated F
c
and the measured F
s
5. Calculate the error in the difference using the rules of error propagation (go back to the
“hands” lab if you need to).
Lab Diagram: Draw the free body diagram of the bob, for both the cases, while measuring the
static force Fs and when Bob is rotating.
Theory: Use these force diagrams to clearly show why you are comparing two different forces
Fc and Fs.
Discussion Questions
1. Do Fc and Fs agree within error?
2. While measuring the static force Fs, if the pulley is not completely frictionless, let’s say there
is a friction f present. Comment on the relationship between Fs, Fc and f.
3. Answer the following questions using your data to support your conclusions:
a. Given the same mass and radius, which spring (weak or strong) will have the greater
period? Does this agree with the theoretical equation?
b. Given the same spring, will a greater mass have a greater period? Will a greater radius
give a greater period?
A NOTE FOR LAB REPORTS: Treat Study I as a calibration (as long as Fc = Fs, or close) you
know your equipment is working properly. Include all data to support this and answer the
discussion questions. Write Study II up formally. Your theory section should show you
understand the essential physics used in both studies, referring to your force diagrams.
Attention: For safety, never turn on the power supply without the front panel
and all side panels in place.
5
STUDY I: Constants
Remember m, kg, and seconds allow you to calculate forces directly in Newtons
M
bob
(kg)
δM
bob
(kg)
Relative
Error in M
R (m)
δR (m)
STUDY I: Bob at Rest (remember to add the mass of the hanger to m2!)
Spring
Weak Spring
Strong Spring
m2
δm2
Fs
δ Fs
STUDY I: Bob in Motion
N (# periods)
!
T
(s)
!
"
(T)
(s)
δ T =
!
"
(T) =
"
(T)
N
Relative Error in T
Observation: What happens to the period when you change the spring from weak to strong?