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Unformatted text preview: 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. This force is
called centripetal force. The reaction, which is equal
and opposite, is the pull of the body on the restrain THEORY ing medium and is called centrifugal force. The pur pose of this experiment is to study uniform circular
motion and to compare the observed value of the cen—
tripetal force with the calculated value. 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 and that its magnitude is given by ”2
r a: where o 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,
and this force is called centripetal force because it is
always directed toward the center of rotation. By
Newton’s second law of motion, the magnitude of
this force is given by the relation F=m—
r F=ma or where F is the force in newtons, m is the mass 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 because of the inertia of the body. This reac
tion is called centrifugal force. The centripetal force can also be expressed in
terms of the angular speed, since w=27rf where v is the linear speed and r the radius of the path as before, to is the angular speed in radians per second, and f is the number of revolutions per sec
ond. Thus 0 = rm,» and F = mrw2 or F = 4772f2rm where F is the centripetal force in newtons. In the apparatus used for this experiment (see
Fig. 6.1) a stretched spring provides the centripetal
force necessary to keep a mass rotating ina circle of
a particular measurable radius. The rotator is adjusted
to a rotational speed such that the back of the mass
just touches the sensing probe of a sensitive indica
tor. The indicator’s pointer then rises to stand oppo Rotating mass figa 57' Eye for
1 er hanging
. L: ; apparatus on hook L—«av Indicator
Index Figure 6.1 Centripetal force apparatus '41 42 UNIFORM CIRCULAR MOTION site an index ridge at the end of the shaft on which
the apparatus tums. As this index and the pointer are
near the center of rotation, they are easily visible re
gardless of the rotational speed. This speed is kept
constant at the value just described and is measured APPARATUS 1. Centripetal force apparatus 2. Motor with coupler and means for counting
turns 3. Stopwatch or stop clock PROCEDURE 1. When you come into the laboratory, examine
the rotating assembly carefully in order to understand
how it works. Do not change the spring tension ad
justment; it has already been set by your instructor.
Note and record the value of the rotating mass. This
value is either marked on the mass itself or will be
given to you. 2. Set up the bench clamp, a vertical rod, right
angle clamp, horizontal rod, and hooked collar so as
to suspend the apparatus vertically. Hook the weight
hanger to the eye on the rotating mass and add
weights carefully until this mass is pulled down just enough to touch the sensitive indicator and bring the I pointer to the index. Record the mass required to do
this. Don’t forget to include the mass of the weight
hanger. . 3. With the weights still on the hanger so that
the rotating mass is in the position established in Pro
cedure 2, use the vemier caliper to measure the
mass’s radius of rotation. The measurement is made from the point on the shaft above the index (the axis of rotation) to the line engraved around the mass,
which marks the location of its center of gravity in
that direction. 4. Unhook the apparatus from the weight hanger
and supporting rods and mount it on the motor shaft
by means of the coupling provided. Mount the motor
unit securely on the bench using the bench clamp and
check that all mounting clamps and screws are tight. 5. Make sure the speed control is at its lowest
position and turn on the motor. As you gradually in
crease the speed, note that until the rotating mass has
moved out to the radius at which it engages the indi
cator, the apparatus is highly unbalanced and will DATA Value of the rotating mass
Mass hung on the spring Total mass necessary to stretch the
spring by timing some convenient number of turns. The
force required to stretch the spring by the same amount is subsequently measured by hanging weights
from it. ~ ' 4. Supporting rods, hook, right—angle clamp, and
bench clamp 5. Weight hanger and weights 6. Vernier caliper shake quite badly. Increase the speed until you see
the pointer rise, but be careful not to increase it too
much. CAUTION: It is dangerous to let the apparatus
rotate at excessive speeds. Best results will be ob
tained if you get the pointer to rise just above the in
dex and then slow the rotation very carefully, making
very small speed changes and watching the results.
Your aim is to keep the motor speed at the value for
which the pointer stands exactly opposite the index,
but this adjustment is very critical, and you may want
to practice a while before attempting to take data.
Holding a sheet of white paper behind the apparatus as you watch the pointer and index will help you see
them more clearly. 6. When you have the speed set at the correct
value so that the pointer is opposite the index, mea—
sure this speed by timing a counted number of turns.
If you are using a revolution counter, note its reading
and then engage it and start the stop clock simulta
neously. Disengage it and simultaneously stop the
clock after about a minute and record the number of
turns and the elapsed time. The setup shown in Fig.
6.1 uses a variable speed motor with an 18 : 1 reduc
tion gear on one end and a pointer mounted on the
slowly turning shaft thus provided. Revolutions of
this shaft are easily counted visually. In performing
this experiment it is a good plan to have one observer
pay strict attention to the proper adjustment of the
speed while the other counts turns and operates the
stop clock. 7. Repeat Procedures 5 and 6 three more times
so as to have a total of four measurements of the rota
tion speed. Record the number of revolutions counted
and the elapsed time in each case. Radius of rotation Centripetal force calculated from
the theory Name __—__.__..__—____ Section __ Date __ EXPERIMENT 6 43 Measured value of the centripetal Percent discrepancy
force Difference between calculated and
measured value CALCULATIONS 1. From the data of Procedures 5—7, calculate the speed of rotation for each set of readings. 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 277 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.). 44 UNIFORM CIRCULAR MOTION _ 3. Calculate the centripetal force from the theory using the average speed of rotation obtained in
Calculation 2. 4. Find the directly measured valueof the centripetal force by adding the value of the rotating
mass given you in Procedure 1 to the mass found in Procedure 2 and multiplying by the acceleration of gravity to get the force stretching the spring in newtons. Record this result in the space provided in the
data sheet.  ' 5. Compare the calculated value of the centripetal force with the value measured directly by sub
tracting one from the other and computing the percent discrepancy. Name —_______—__—___ Section ___ Date __ EXPERIMENT 6 45 6. Using the ad. found in Calculation 2 as the error in your rotational speed measurement, ﬁnd the resulting error in your calculated value of the centripetal ferce and see whether the directly measured
value falls within the limits of error so established. ' QUESTIONS ' 1. State what the experiment checks. 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? 46 UNIFORM CIRCULAR MOTION ' 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 the earth would have to rotate in order that objects at the equator would
have no weight. Assume the radius of the 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 con
ditions?  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 1). Taking 6 ft as a typical diameter for a propeller of a light airplane and 1100 ft/s as the speed of sound, ﬁnd the upper
limit on the rpm (revolutions per minute) of the propeller shaft. ' , ...
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 Spring '08
 ERSKINE/TSOI

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