PHY 131/221
Centripetal Force
(revised 4/26/08 J)
Name ___________________
Date: ______________
Lab Partners ___________________________________________
Introduction: As you now know, a force is required to change the velocity of an object. Velocity can change
in two ways, a change of direction and/or a change in magnitude. As you have learned, if the net force on an
object, and thus the acceleration, is perpendicular to the velocity at an instant in time then only the direction
of the velocity will change and the speed will remain constant. If the net force, and thus the acceleration, is
parallel, or anti-parallel, to the velocity then only the speed will change and not the direction. If the net force
is somehow created such that it is always perpendicular to the velocity then the particle will move at constant
speed while the velocity continually changes direction. The path of the object will be a circle if the force is
constant.
The force, or combination of forces, that causes an object to move in a circle, or an arc, is called the
centripetal force. The centripetal force is not a new force, but rather it is the name given to the various forces
that cause the object to move in a circle. Examples are tension in a string as you whirl a ball over your head,
static friction acting on the wheels of a car as it navigates a turn on a road, gravity that acts on the Moon to
keep it in an (almost circular) orbit around Earth, etc.
In this lab you will investigate how the centripetal force depends on such things as the radius of the circular
path, the mass of the object, and the speed of the object. You will use the apparatus shown in the picture
below.
The apparatus consists of a nearly frictionless vertical rod that is free to rotate. Mounted on top of this rod is
another horizontal rod that serves to support the rotating mass and provide counterbalance so the whole
system does not wobble. The mass that you will be forcing to travel in a circular path is the black object with
the pointy bottom and the eye-bolt on top. We’ll call this mass the bob. (A bob is often the name given to a
hanging, or swinging mass.) A spring is connected between the vertical rod and the bob. This spring will be
1
providing the centripetal force. If the spring is not there then when you rotate the vertical rod, the bob will
swing outward and a component of tension in the support string would be providing the centripetal force. Do
not try this! The support string could break sending the bob off at a high speed possibly resulting in injury
or death! Ok, probably not death to you, but certainly death to your lab grade.
For all steps in this lab you will want the support string to be in a vertical plane as you rotate the vertical rod
so that the centripetal force is provided solely by the spring. To help you rotate the rod while making sure the
bob’s path is indeed circular a vertical metal pointer is provided. You would first adjust the pointer so that it
is a specified distance from the center of the vertical rod. Then you would disconnect the spring from the bob
and adjust the horizontal rod so that the bob hangs directly over the pointer. Before you do any rotating you
must be sure the horizontal rod and the counterweight are secure. It is a good idea to wear safety goggles in
the event that something comes loose. Once everything is secure you may reconnect the spring.
Now that you have established a radius for the bob’s circular path you can directly measure the force that
the spring will exert on the bob when the bob is revolving in its circular path. To do this, loop the string
attached to the bob, opposite the spring, over the pulley and add hanging weights until the bob is directly
over the pointer. The weight of the hanging weights will be a direct measure of the force the spring is
providing when the bob is revolving in its circular path.
Another measure of the centripetal force can be done while the bob is rotating. To do this you will need to
determine the speed of the bob as it revolves. The easiest way to do this is to measure the amount of time that
it takes for the bob to revolve 20 to 30 times, divide this time by the number of revolutions to get the time for
one revolution, the period, T, and then divide the circle’s circumference by T to get the speed.
Part I: The First Trial:
Remove the bob from the supporting string, and from the spring, and measure its mass. Record this in the
Data section. Reattach the bob. Adjust the eye-bolt on the vertical rod by loosening the wing nut so that the
eye-bolt is centered on the vertical rod. Adjust the vertical pointer to a position midway between its limits
and secure it in place. Adjust the horizontal rod so that the bob hangs freely above the pointer without the
spring attached. Directly measure the spring force (as described above) that will exist when the bob is
revolving in a circle with this radius. Record this data in the Data section for Part I.
Decide on the number of revolutions that you will be timing and record this number in the Data section. One
lab partner should practice rotating the vertical rod so that the bob moves at a constant speed and passes over
the pointer every time. Once you are satisfied that this can be done precisely do three trials to determine the
total time for the number of revolutions that you decided on earlier. Record these times in Table 1.
Using these times calculate the average time per revolution and then the speed of the bob. Record these
numbers in the appropriate spots below Table 1.
What you have now done is established a starting point that we can use to compare how the centripetal force
changes as we change variables such as the mass of the bob, the speed of the bob, and the radius of the
circular path.
2
Part II: Dependence of Centripetal Force on Speed:
Do not change the position of the pointer, or the mass of the bob. We only want to change the speed of the
bob as it revolves in the same circular path as in Part I. How do we change the speed? Try spinning the
vertical rod slower than you had in Part I. Does the bob revolve in the same circular path? What should you
change so that the bob does revolve in the same circular path? Discuss this with you lab partners then with
your instructor. Make the necessary change and do three trials as in Part I where you measure the time for a
certain number of revolutions, find the average time per revolution, find the bob’s speed, and measure the
centripetal force directly. Enter this data in Table 2 and in the area below Table 2. (Hint: The change(s) you
make may involve a significant change in how “things” are connected. Be creative.)
Do the above procedures again, but try to give the bob a speed greater than you had in Part I. What change is
necessary (without changing the bob’s mass, or the radius of the circle) to keep the bob traveling in the same
circular path but with a higher speed? Enter this data in Table 3 and the space below Table 3.
Part III: Dependence of Centripetal Force on Radius of path:
Setup the apparatus as you had in Part I. Move the pointer out to its maximum distance and lock it in place.
You will now need to loosen the horizontal rod and disconnect the spring from the bob to be sure it is
hanging freely above the pointer. Secure the horizontal rod after making the appropriate adjustment.
Your goal now is to make the bob travel in a larger circular path, but with the same speed as in Part I. This is
the most difficult part of this lab. You know speed is determined by dividing circumference by period. So the
speed in Part I equals 2  r1 T1 , where r1 is the radius of the circular path in Part I and T1 is the time per
revolution in Part I. This speed should be equal to 2  r2 T2 , where r2 is the new radius that you just set, and
T2 is the time per revolution that you will need to have so that the speeds from Part I and this part are equal.
Setting these speeds equal to each other and solving for T2 , we have T2  r2T1 r1 . Calculate T2 and enter it
into the data section for Part III. As before, you will want to time more than one revolution so let’s agree that
you will do 20 revolutions.
What adjustments do you have to make (without changing the mass of the bob, or the radius of the path) to
make the time for 20 revolutions equal 20T2 ? You will need to fine tune your adjustment to get the total time
for 20 revolutions to be as close to 20T2 as possible. Once you are satisfied that the bob revolves around a
circular path of radius r2 in time T2 , as calculated above, make a direct measurement of the centripetal force
and record this in the data section for Part III.
Now change the radius to the smallest radius allowed by the pointer. Call this radius r3 . Calculate the time
per revolution, T3 , that you will need to ensure the speed of the bob is equal to what it was in Part I. Again,
make fine adjustments so that the total time for 20 revolutions is equal to 20T3 . Measure and record the
centripetal force.
Proceed to the Questions section.
3
Data
Mass of bob with eyebolt
Total number of
revolutions
Data For Part I:
Trail #
Total Time (s)
Time per Revolution
(s)
1
2
3
Table 1
Average Time per Revolution, T1
Radius of circular path, r1
Speed of bob
Mass of Hanger and Masses
Direct measurement of centripetal Force
Show calculation for the speed of the bob:
4
Data For Part II:
Slower Speed
Same Radius as
Part I
Trail #
Total Time (s)
Time per Revolution (s)
1
2
3
Table 2
Average Time per Revolution
Speed of bob
Mass of Hanger and Masses
Direct measurement of centripetal Force
Show calculation for the speed of the bob:
5
Greater Speed
Same Radius as
Part I
Trail #
Total Time (s)
Time per Revolution (s)
1
2
3
Table 3
Average Time per Revolution
Speed of bob
Mass of Hanger and Masses
Direct measurement of centripetal Force
Show calculation for the speed of the bob:
6
Data For Part III:
Larger Radius
Same Speed as
Part I
Radius of circular path, r2
Desired time per revolution for larger radius, T2
Desired time for 20 revolutions, 20T2
Direct Measurement of Centripetal Force for r2
Smaller Radius
Same Speed as
Part I
Radius of circular path, r3
Desired time per revolution for larger radius, T3
Desired time for 20 revolutions, 20T3
Direct Measurement of Centripetal Force for r3
Show calculations for Part III below:
7
Questions
1. What change did you make to give the bob a slower speed than in Part I without changing the mass of
the bob, or radius of the circular path? Explain how you did this.
2. What change did you make to give the bob a greater speed than in Part I without changing the mass
of the bob, or radius of the circular path? Explain how you did this.
3. Make a graph of Measured Centripetal Force as function of speed (with constant radius and mass) for
the three speed changes you made in Part I and Part II. (Be sure to properly label the graph, and its
axes.) What kind of function fits this data best? Is it linear, quadratic, or exponential? What does this
tell you about how centripetal force depends on speed? Explain.
8
4. Make a graph of Measured Centripetal Force as a function of radius (with constant speed and mass)
for the three radius changes you make in Part I and Part III. (Be sure to properly label the graph, and
its axes.) What kind of function fits this data best? Is it linear, quadratic, exponential, or reciprocal
relationship? Or is it something else? What does this tell you about how centripetal force depends on
radius? Explain.
5. What were the most significant sources of error in this lab and what did you do to minimize their
affect?
9