Experiment 8 Uniform Circular Motion
Whenever a body moves in a circular path, a force directed toward the center of the circle must
act on the body to keep it moving in this path. That force is called centripetal force. The reaction,
which is equal and opposite, is the pull of the body on the restraining medium and is called
centrifugal force. The purpose of this experiment is to study uniform circular motion and to
compare the observed value of the centripetal force with the calculated value.
THEORY
If a body moves with constant speed in a circle, it is said to be moving with uniform circular
motion. Even though the speed is constant, the velocity is continuously changing because the
direction of the motion is continuously changing. Thus, such a body has an acceleration. It can
be shown that the direction of the acceleration is always toward the center of the circle (because
it is only the direction and not the magnitude of the velocity that is changing) and that its
magnitude is given by
v2
r
where v is the speed of the body in meters per second and r is the radius of the path in meters.
A force is necessary to produce this acceleration. Because it must be in the same direction as
the acceleration-namely, toward the center of the circle as noted above-it is called centripetal
force. Newton's second law now requires that the magnitude of this force be equal to the mass
times the acceleration produced, so that in our present case
a
v2
r
where F is the force in newtons, m is the mass of the rotating body in kilograms, and v and r are
the same as before. But by Newton's third law of motion, an equal and opposite force is exerted
by the body on the restraining medium. This reaction is called centrifugal force.
The centripetal force can also be expressed in terms of the angular speed, since
F m
v = rand f
where v is the linear speed and r the radius of the path as before, is the angular speed in
radians per second, and
f is the number of revolutions per second. Thus,
F = mrw2 or
F = f2rm
where F is the centripetal force in newtons as already described.
Experiment ()
Uniform Circular Motion
Figure () Centripetal force apparatus, manual form
Apparatus for the experiment is shown in Fig. (). A stretched string provides the centrifugal
force to keep the rotating mass moving in a circle, but the complete assembly is mounted on a
horizontal bar (the rotating platform) mounted on a vertical shaft carried in very good bearings.
A knurled portion of the shaft provides a convenient grip for the experimenter's fingers, and
you'll find with a little practice that you can keep the assembly rotating at a constant speed quite
accurately. Note that at the bottom of the spring there is a small pink disk, and because the spring
and disk are at the center of rotation, the position of the latter is easily observed even with the
apparatus in motion. The mass to be kept rotating at a particular radius is suspended by threads
from a post mounted on the rotating platform at that radius. Weights hung on a thread passing
over a pulley at the end of the platform are used to bring the rotating mass to the selected radius
with the apparatus at rest, and the marker ring on the central post is moved to the position now
occupied by the pink disk. The weights are then removed and the apparatus rotated to a speed
that again brings the pink disk into the marker ring. The centripetal force exerted by the spring
must then be the same as that exerted by the weight when the apparatus wasn't turning, and the
rotating mass must be back at the selected radius. The apparatus parameters have been chosen so
that no rotational speed will be high enough to make counting the turns difficult. Be careful,
however, to count only one end of the platform as it comes around.
APPARATUS
1. Centripetal force apparatus
2. Motor with coupler and means for counting turns
(Procedure A only)
3. Stopwatch or stop clock
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Experiment ()
Uniform Circular Motion
4. Supporting rods, hook, right-angle clamp, and bench clamp (Procedure A only)
5. Weight hanger and weights
6. Vernier caliper (Procedure A only)
7. Level (Procedure B only)
8. Triple-beam balance (Procedure B only)
PROCEDURE
1. When you come into the laboratory, examine the rotating assembly carefully in order to
understand how it works. Make sure that it is level by placing your level on the rotating platform
and adjusting the leveling screws in the base until the platform's top surface remains exactly
horizontal no matter which way it is turned. Observe that the rotating mass consists of three parts
and is suspended by two threads from a supporting post that may be set at any point along the
rotating platform. A centimeter scale ] on this platform and a vertical line on the post allow the
distance of the post from the center of rotation to be measured quite accurately. Unhook the
complete rotating mass from the threads, weigh it, and record your result. Then remove the two
outer sections, weigh the central § section alone, and record this result also. You will thus have
two possible values for the rotating mass.
2. Replace the central piece of the complete rotating mass on the supporting threads and note that
in addition to the hook by which you hang this mass, it has two other hooks, one facing toward
the center of the apparatus and the other facing outward. A piece of thread about 30 cm long
should run from the inner hook, under the pulley on the central post, and up to the bottom of the
pink indicator disk. Check that this thread is in place (install it yourself if it is not) and that the
run from the hook to the pulley is horizontal. If it is not, the height of the bracket holding the
supporting threads may be adjusted until it is.
3. Run another thread from the outer hook on the rotating mass over the pulley at the end of the
platform and down to a weight hanger. Place 150 g on the hanger and record the total mass
(weights plus the hanger mass) so suspended. This will pull the rotating mass out and stretch the
spring.
4. Carefully loosen the clamp fixing the supporting post to the platform and slide it along its
track until the threads supporting the rotating mass are exactly vertical. You can do this by lining
up the two threads with each other and with the vertical line scribed on the post. Be sure the post
itself is vertical by sliding it so that its bottom edge bears on the track on the platform's lower
edge. Clamp the post in position and read the radius at which you have set it from the scale on
the platform. Record
this value. Finally, move the indicator ring on the central post so that the pink indicator disk is
centered in it and fix the ring in this position.
5. Now remove the weight hanger, weights, and thread from the outer hook on the rotating mass.
The spring will pull this mass in toward the center of the apparatus, and your plan is to spin the
assembly up to the speed at which the rotating mass returns to the position in which it was held
by the weights with the apparatus at rest. Do this by carefully turning the knurled portion of the
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Experiment ()
Uniform Circular Motion
supporting shaft with your fingers, gradually increasing the rotational velocity until the pink
indicator disk is pulled back down into the indicator ring. Practice for a while until you can keep
this velocity constant at the value for which the pink disk is centered in the ring. Then start the
stop clock as one end of the platform comes around and stop it after ten turns. Measure the time
for ten turns in this manner four times and record your results. Note that it is a good plan in
making these measurements to have one observer pay strict attention to keeping the speed
constant at the proper value while the other counts turns and operates the stop clock.
6. Repeat Procedures 3-5 with 100 g on the weight hanger and again using 50 g.
7. Reassemble the two outer portions of the rotating mass to the central piece and repeat
Procedures 3-6 using this larger rotating mass.
DATA
Value of the rotating mass ----------
and measured value ---------------------------
Mass hung on the spring -------------------
Radius of rotation -----------------------------
Measured value of the
centripetal force ------------------------
Centripetal force calculated
from the theory ------------------------
Difference between calculated
Percent discrepancy ----------------------
Trial
1
Number
of turns
Time
interval
Rotation
speed
Deviation
2
3
4
Averages
Value of the rotating mass ---------Mass hung on the spring -------------------
Difference between calculated and measured
value ---------------------------
Measured value of the centripetal force -----------------------
Radius of rotation ----------------------------4
Experiment ()
Uniform Circular Motion
Percent discrepancy ----------------------------------
Centripetal force calculated from the theory
------------------------
Trial
1
Number
of turns
Time
interval
Rotation
speed
Deviation
2
3
4
Averages
Value of the rotating mass ---------Mass hung on the spring -------------------
Radius of rotation ----------------------------Centripetal force calculated from the theory
-----------------------Percent discrepancy ----------------------------------
Measured value of the centripetal force ----------------------Difference between calculated and measured
value --------------------------Number
Time
Trial
ofturns
Interval
1
Rotation
speed
Deviation
2
3
4
Averages
Value of the rotating mass ---------Mass hung on the spring -------------------
Difference between calculated and measured
value ---------------------------
Measured value of the centripetal force -----------------------
Radius of rotation ----------------------------5
Experiment ()
Uniform Circular Motion
Percent discrepancy ----------------------------------
Centripetal force calculated from the theory
------------------------
Trial
1
Number
of turns
Time
interval
Rotation
speed
Deviation
2
3
4
Averages
Value of the rotating mass ---------Mass hung on the spring -------------------
Radius of rotation ----------------------------Centripetal force calculated from the theory
-----------------------Percent discrepancy ----------------------------------
Measured value of the centripetal force ----------------------Difference between calculated and measured
value ---------------------------
Trial
1
Number
of turns
Time
interval
Rotation
speed
Deviation
2
\
3
4
Averages
Value of the rotating mass ---------Mass hung on the spring ------------------Measured value of the centripetal force ----------------------Difference between calculated and measured
value --------------------------Radius of rotation ----------------------------6
Experiment ()
Uniform Circular Motion
------------------------
Centripetal force calculated from the theory
Percent discrepancy -----------------------------------
Trial
1
Number
of turns
Time
interval
Rotation
speed
Deviation
2
3
4
Averages
CALCULATIONS
1. From the data of Procedures 5-7, calculate the speed of rotation for each reading you
have taken. Note that dividing the number of turns by the elapsed time gives you
revolutions per second, that if you used a reduction gear you must multiply by the gear
ratio to get the rotational speed of the apparatus, and that you must then multiply by 2
to get your result in radians per second.
2. Compute the average of your four speed measurements, the deviation of each from the
average.. and the average deviation (a.d.) for each set of readings. Note that you will have
six sets if you used the manual apparatus of measurement.
3. Calculate the centripetal force from the theory using the average speed of rotation obtained
in Calculation 2. You will have to do this and the next three Calculations six times.
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Experiment ()
Uniform Circular Motion
4 The mass whose weight stretches the spring is just the mass hung on the weight hanger in
Procedures 3, 6, and 7 plus the mass of the weight hanger. Multiply by the acceleration of
gravity to get the force stretching the spring in newtons in each case and record your three
results in the appropriate spaces.
5. Compare the ca1culated value of the centripetal force with the value measured directly by
subtracting one from the other and computing the percent discrepancy.
6. Using the a.d. found in Calculation 2 as the error in your rotational speed measurement,
find the resulting error in your calculated value of the centripetal force and see whether the
directly measured value falls within the limits of error so established.
QUESTIONS
1. State what the experiment checks.
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Experiment ()
Uniform Circular Motion
2. (a) How does the centripetal force vary with the speed of rotation for a constant radius of the
path?
(b) How does it vary with the radius of the path for a constant speed of rotation?
3. Distinguish between centripetal force and centrifugal force. Explain in what direction each
force is acting and on what it is acting.
4. Calculate at what speed Earth would have to rotate in order that objects at the equator would
have no weight. Assume the radius of Earth to be 6400 km. What would be the linear speed of a
point on the equator? What would be the length of a day (time from sunrise to sunset) under
these conditions?
5. Engines for propeller-driven aircraft are limited in their maximum rotational speed by the fact
that the tip speed of the propeller must not approach the speed of sound in air (Mach I). Taking 6
ft as a typical diameter for a propeller of a light airplane and 1100 fils as the speed of sound, find
the upper limit on the rpm (revolutions per minute) of the propeller shaft.
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