M-6  Uniform Circular Motion

OBJECT: To verify experimentally that the centripetal force, F_c is

$$F_{c}=\frac{mv^{2}}{r}=m\omega^{2}r$$

APPARATUS:


  Fig. 1 is a schematic of the equipment. The bobs and springs are removable for weighing. Not shown are table clamp and pulley, slotted masses and weight hanger.

 

Figure 1: The UCM apparatus.

INTRODUCTION:


  A variable speed motor drives the rotating system which has two slotted bobs which slide on a low friction bar. One adjusts the speed until one bob just covers the optical light pipe and thus reduces the signal seen at the center of the rotating system to zero. A revolutions counter is on the shaft. The counter operates by sensing the rotating magnetic poles and electronically reads out directly the frequency of revolution in rpm. A spring (plus any friction) supplies the centripetal force required to keep the bob traveling in a circle.
If one measures first the frequency of rotation required to make the bob just cover the optical light pipe, and if one then measures the force required to pull the bob out the same distance when the system is not rotating, one can check whether

$$F=m\omega^{2}r$$

where r is the distance from the axis of rotations to the center of mass of the bob.

Pre-lab Quiz

You should be able to complete this brief quiz before proceding.

• Expt. M06 Quiz

SUGGESTIONS:

 

1.

Find the mass of the nickel plated brass bob; also the aluminum bob.

2.

Dynamic measurement of the force: Attach the brass bob to the spring. Replace the lucite cover, and adjust the motor speed until the light from the light pipe at the center of the rotating system goes to zero. Record the rotation frequency.

To correct for frictional effects of the bob on the bar, record the frequencies both as the speed is increasing to the correct value and and as the speed is decreasing from too high a value. Since the direction of the frictional force reverses for the two cases, the average should eliminate the frictional effect.

Repeat several times so you can estimate the average or standard deviation of your values.

3.

Static measurement of the force: Use the string, pulley and weight holder plus slotted weights to measure the force required to stretch the spring so that the optical light pipe is again just covered. Devise a way to avoid error caused by the friction at the pulley and of the sliding bob on the bar.

4.

While the spring is stretched in item 3 to its proper length, measure the distance r from the axis of rotation to the center of mass of the bob. The c. of m. of the bob is marked on it.

 

Figure 2: Static measurement of the force using hanging weights

 

5.

Compare the force as computed from $F=m\omega^{2}r$ and as measured by stretching the spring. In computing the centripetal force, also take into consideration the mass of the spring. One can show (Weinstock, American Journal of Physics, 32,p. 370, 1964) that $\approx (1/3)$ of the spring mass should be added to the mass of the bob to obtain the total effective mass.

6.

Repeat the above item 1 through item 5 but for the aluminum bob.

QUESTIONS:

 

1.

Estimate the reliability of your measurements. How well do the measured and computed forces agree? Try to account for any discrepancy.

2.

How would the experiment be affected if gravitational and inertial mass were not equal? [Gravitational mass refers to that appropriate for Newton's principle of universal gravitation ($F=G\frac{m_{1}m_{2}}{r^{2}}$) while inertial mass refers to the mass appropriate for Newton's second law (F=ma).]