Chapter 5
Uniform Circular Motion and
Centripetal Force
Lab Partner:
Name:
5.1
Section:
Purpose
In the experiment, uniform circular motion and centripetal force will be explored.
5.2
Introduction
For the purpose of this lab, all objects will be considered as rigid bodies. That is, an object
can be considered as consisting of many small point masses with constant distances between
them (no deformation).
Uniform circular motion is the motion of an object traveling at a constant (uniform)
speed in a circular path. The first thing to be noted about uniform circular motion is the
time it takes to make one complete trip around the circle. This is called the period. If r is
the radius of the circle, the distance around the circle is 2πr. the speed, v, is
v=
2πr
T
(5.1)
T =
2πr
v
(5.2)
so the period, T, is:
By the definition, the speed is constant in uniform circular motion. What about the
velocity? The velocity is tangent to the circle of motion so the direction changes as we go
around the circle. Thus the velocity changes. Since the velocity is changing, there must be
an acceleration.
The magnitude of the centripetal acceleration, ac , is given by:
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ac =
v2
r
(5.3)
where v is the velocity and r is the radius of the circle of motion. The units are the normal
units for acceleration (m/s2 ).
The direction of the centripetal acceleration is towards the center of the circle and perpendicular to the direction of the velocity. The centripetal acceleration has to continuously
change the velocity vector back towards the center of the circle to keep the object moving
in a circle.
When a body is caused to follow a curve instead of straight line, there is a force on the
body called ”centripetal” force. This force will produce a centripetal acceleration, a~c , given
by Newton’s Second Law:
F~ = ma~c
where m is the mass of the revolving object.
Now, consider a body with mass m revolving with a uniform speed around an axis at
constant distance r. The time to make one complete circle is the period, T. The linear speed
of a point at a distance r from the axis of rotation is given by equation 5.1. The centripetal
acceleration is then:
ac =
( 2πr )2
v2
r
= T
= 4π 2 2
r
r
T
(5.4)
Using Newton’s 2nd law, the centripetal force can be written as
F = 4π 2
5.3
mr
T2
(5.5)
Procedure
The centripetal force apparatus is shown in figure 5.2. It consists of an experiment mass
supported on a metal rod by a light string, which is in turn attached to a rotating metal
shaft. This shaft is mounted so that it will rotate freely and not wobble or bind. The cross
arm is counter balanced by the mass that is directly attached to the cross arm. The counter
balance mass should be positioned so the apparatus does not wobble or vibrate.
The experimental mass will be rotated at a constant angular velocity by one lab partner.
With some practice, it is easy to keep the angular velocity constant.
• Remove the experimental mass, weigh the mass and return it to the cross arm.
Experimental mass, m
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F1
F1
F2
a)
b)
c)
Figure 5.1: (a) Shows the proper adjustment of the indicator rod and the experimental mass
without the spring attached to the experimental mass. (b) Shows the spring attached to
the experimental mass. The other side of the experimental mass is attached to a weight
hanger and masses so Fmeasured can be determined. (c) When the vertical shaft is rotated
with constant speed and the indicator rod is directly below the experimental mass, the
centripetal force supplied by the spring is equal to Fmeasured .
2
Figure 5.2: The centripetal force apparatus.
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• Align the indicator rod (vertical rod under mass) with the experimental mass with the
spring disconnected. See Fig.5.1. Measure the radius of rotation taking into account
the thickness of the rotational shaft which is 12.5 mm.
Radius of rotation
• Re-attach the spring and connect a string to the experimental mass. Place the string
over the pulley. Attach a weight hanger and masses on the weight hanger until the
experimental mass is aligned with the indicator rod. (The apparatus will need to be
near the edge of the table so the masses can hang freely.) The weight hanging from
the pulley (in Newtons) is equal to a centrifugal force needed to balance a centripetal
force if you rotate the shaft. Record the value below as Fmeasured .
Fmeasured
• Practice rotating the shaft with your fingers. With a little practice, you should be able
to keep the experimental mass directly over the indicator rod. Rotate the system by
applying torque with your fingers to the ribbed portion of the shaft. Look from the
side and slow the shaft down a little bit if the tip of a experimental mass passes outside
of the indicator rod, and accelerate the shaft if the tip passes inside. With practice,
you can keep the mass passing directly over the indicator.
• Measure the time for fifty (50) rotations keeping the experimental mass directly over
the indicator shaft. Dividing this time by 50 gives the period.
Period
• Calculate the centripetal force from equation 5.5 and compare (percentage difference)
to the value Fmeasured .
Centripetal Force from equation 5.5
Percentage difference with Fmeasured
• Repeat the procedure with two different radii of rotation. Change the radius of rotation
by approximately 1-2 cm. Re-align the counter balance and indicator rod as necessary.
m
r
T
Fmeasured
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F (eq 5.5)
% diff
• Repeat the procedure two more times by adding a 50 g and 100 g slotted mass to the
experimental mass. The slotted mass can be added by fastening them with the wing
nut located on top of the experimental mass. Do not add more than 100 g to the
system. The radius should remained fixed.
m
5.3.1
r
T
Fmeasured
F (eq 5.5)
% diff
Questions
1. Confirm that the units are correct on equation 5.3 and equation 5.5
2. If the period increases, does the centripetal force increase or decrease? If the radius
increases does the centripetal force increase or decrease? If a stiffer spring (large spring
constant) were used with other things fixed, would the period increase or decrease?
5.4
Conclusion
Write a detailed conclusion about what you have learned. Include all relevant numbers
you have measured with errors. Sources of error should also be included.
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