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Uniform Circular Motion

Introduction

Uniform circular motion is the motion of an object traveling at a constant (uniform) speed in a circular path. Besides the speed, there are several other variables that are used to characterize the motion. They are the radius of the motion r, the angular speed ω, the period T, and the rotational frequency f. The period is the time required for the object to complete one revolution of motion. The angular speed is the angular displacement per second and it is related to the frequency by:
( 1 )
ω = 2πf 
with ω in rad/s.
The rotational frequency is the number of revolutions per second and is given by:
( 2 )
f =
1
T
 
with f in Hz or sec-1.
The velocity of the object is tangential to the circle with a magnitude v = . The acceleration, a, is directed toward the center of the circle (centripetal) with a magnitude given by:
( 3 )
a =
v2
r
= rω2 
with a in m/s2.
For an object of mass m to move around the circle with constant speed, there must be a net centripetal force acting on the object. The magnitude of the net force, F, must be constant and is related to the centripetal acceleration by Newton's Second Law:
( 4 )
F = ma = mrω2 
with F in N.
This centripetal force may be provided by tension (as in this lab), friction (as for a car rounding a curve), a normal force (as in a looping roller coaster), or gravity (as for satellite motion). In this experiment, you will measure the period of an object undergoing uniform circular motion with a fixed radius, but with various values of F. From the period, you can calculate the angular speed. With these known values and the equations above, you can find an empirical mass of the rotating object and compare with the mass value found by directly weighing it with a balance.

UCM Apparatus

The UCM apparatus consists of a variable-speed rotating platform. A distance r from the center of rotation is the side post assembly from which hangs an object of mass m, referred to as the rotating mass (not to be confused with the static mass). The rotating mass is attached to a spring on the center post by way of a string and small pulley. When the platform rotates, the rotating mass will travel in a circular path due to the force exerted on it by the string (by way of the tension in the spring). Since it is not possible to have an instantaneous readout of this tension force while the platform is rotating, an indirect measurement of this force will be made using the weight of the static mass as shown and explained below.
Figure 1

Figure 1

When the platform is not turning, the rotating mass does not hang vertically from the side post but rather is pulled inward by the tension in the string and spring. When performing the experiment, you will adjust the rotational speed of the platform until the rotating mass is hanging vertically at radius r. The orange indicator disk will help you determine when the rotating mass has reached this position.
Figure 2

Figure 2

The period of rotation is measured with a stopwatch. The platform can be rotated by turning the knurled rod by hand.

Procedure

Leveling the Apparatus

If the platform is not level, it will adversely affect your results. Students in the first lab of the week should have leveled the apparatus already. Hopefully, the apparatus has not been moved since then and will not need to be leveled again. Check to see if your apparatus is level by turning on the speed control motor and observing the orange indicator disk to see if it bobs up and down as the platform rotates. If your apparatus needs to be leveled, perform the following steps.

Setting the Radius

Setting the Centripetal Force Magnitude

In this first part of the procedure, you will use a static (non-rotating) method to adjust the apparatus for a known value of centripetal force. At this point, stop and analyze all of the forces acting on the rotating mass. What are their magnitudes and directions? Are any of the forces not quite vertical or horizontal? If so, how will this affect your results?
Checkpoint 1:
Have your TA check your apparatus before continuing.

Measurement of the Period

Checkpoint 2:
Have your TA check your data and calculation results before continuing.

Vary the Centripetal Force

Repeat the above procedure with at least five different static masses (and therefore, five different forces) that span the widest possible range of values (usually 40 to 150 g).

Analysis

Discussion

Compare the value of m obtained from your curve fit to the measured values of the rotating mass m and radius of motion r. Is there agreement to within the uncertainties? If you perform this experiment carefully, you should be able to obtain results with less than 3% error. Compare the value of the y-intercept with the expected value. Do they agree? Why is it important for this (and most other) experiments to obtain data over the widest possible range of values? What is the potential consequence of having data points that are close together? Explain how you would take data with this apparatus in order to test the following hypothesis: For a given value of centripetal force magnitude F, the radius of motion r is inversely proportional to the square of the angular speed ω.