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Uniform Circular Motion©98
Experiment 7
Objective: To verify the expression relating centripetal force to the mass of an object,
the radius of the circle on which it moves, and the speed it possesses on the circle.
DISCUSSION:
Solving the problem of the relationship between force and motion along a curved
path was a crucial milestone in the development of physics. The problem arose in an
attempt to understand the motion of the planets around the Sun. Isaac Newton's study of
the orbital question led him to the invention of the calculus, the mathematical description
of the physical world which today has become the foundation of the physical sciences.
However, Newton, when writing for a world untutored in calculus, chose to present his
proofs in the language of geometry.
The simplest form of orbital motion is motion in a circle at constant speed. The
geometric method by which Newton analyzed this problem is both simple and elegant. It
is worth reproducing here not only as an introduction to the laboratory study of uniform
circular motion but also as an illustration of Newton's genius.
vcos
It is clear that to cause the direction of motion of an object to change requires
some interaction with the world beyond the object. A car turning a corner does so only
because it has a road with which to
vsin P
interact. Remove the mechanism of that
interaction and the car fails to make the
corner, traveling instead in a straight line.
v
Anyone who has attempted to navigate a
car on an icy road knows this lesson well.
Nothing is wrong with the steering
mechanism. The wheels turn as they


should. What is lacking is the
v
horizontally directed force, friction, with
which the road acts against the car,
m
turning the car into the curve.
r
Fig. 1 shows a body of mass m
bouncing in a plane around a circular
container in such a way that its path is a
polygon. For the sake of simplicity, the
polygon chosen is a triangle. Whenever
the body collides with the wall of the
container, say at point P, the angle of
incidence, , equals the angle of
reflection. Consequently, the path traced
out by the body is a regular polygon.

rcos
s
Figure 1. Centripetal force.
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The component of the velocity v tangent to the circle is vsin. This component is
unchanged at P. But the component parallel to the radius of the circle is reversed at P
because just before contact with the wall, the body is traveling away from the center of
the circle, and then, just after contact, it travels toward the center. This radially directed
component is vcos.
The change in the velocity at P arises because there is a reversal in the direction,
not in magnitude, of the velocity. This change is twice the magnitude of the radial
component, namely 2vcos. Furthermore the change in direction occurs in the radial
direction, toward the center of the circle. That is, the change is centripetal. Calling the
change v, we write
(1)
v  2v cos 
Next we let s be the length of one side of the triangle. Then, from the figure,
s  2r cos 
(2)
Finally, we let t be the lapse of time during which the body travels along s. Then
Δt 
s

2r cos 
(3)
v
v
If we increase the number of sides of the polygon to some number n, the value of
 changes, but we do not change the geometric relationship given in Eqs. (1) and (2).
The value of  for the triangle is 30, for a square it is 45°, and for a pentagon it is 54. It
is not difficult to show that for a regular polygon of n sides,   90  180 n . As n
approaches infinity without limit,  approaches 90 and the polygon approaches a circle.
Increasing the number of sides of the polygon decreases the distance s and
shortens the lapse of time t, for a given speed v. For a polygon whose number of sides
increases without limit, the lapse of time t becomes the time during which the velocity
of the particle changes. The acceleration a of the particle is therefore
a
Hence,
Δv
Δt

2v cos 
 2r cos  
v

(4a)
2
a v
r
(4b)
Since the change in the velocity is directed towards the center of the circle, so also is the
acceleration. Equation (4b) is the well known expression for centripetal acceleration.
The force on the body producing the change in the velocity is provided by the
wall of the container. Each time a body strikes the wall it pushes the body toward the
center of the circle. This centripetal force is the net force on a body and is given by the
product of the mass of the body and its acceleration. Thus
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F
mv 2
(5)
r
In this experiment, the centripetal force which causes the body to move in a circle
is a spring. The apparatus is designed so that the force F needed to elongate the spring an
amount sufficient to produce the circular motion of the body is determined by hanging
weights from the spring. The radius r of the circular motion is found by direct measure,
and the mass m of the rotated body is to be found inscribed on the body. The speed v is
determined by timing a certain number of rotations of the apparatus and using the
equation:
2πr
v
(6)
T
Here 2r is the circumference of the circle and T is the lapse of time in which the body
makes one revolution.
Since the central centripetal force, mass, speed and radius can be determined
separately, Eq. (5) can be tested experimentally.
EXERCISES:
1. Set the spring tension on the apparatus to some arbitrary value.
2. Determine the force F necessary to effect the critical elongation of the spring by
suspending known weights from the spring. (Include the weight of the rotated body.)
3. Measure the radius r of the circular motion.
4. Rotate the apparatus, adjusting its angular velocity  so that the spring just maintains
the critical extension while the counter is recording the number of revolutions of the
apparatus.
5. Measure the lapse of time t during which the apparatus makes a recorded number n of
revolutions.
6. From these data, t and n, calculate the period T of a revolution of the apparatus.
7. Calculate the speed v of the body.
8. From the measurements of m, v, and r and using Eq. (5), calculate the centripetal
force F on the body.
9. Find the percent deviation of this force from the force F measured in Exercise 1.
10. Repeat this experiment several times with different spring tensions.
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