M-6  Uniform Circular Motion

OBJECTIVE: To measure the centripetal force, $F_{c}$, and compare to

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

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 you measure first the frequency of rotation required to make the bob just cover the optical light pipe, and if you then measure the force required to pull the bob out the same distance when the system is not rotating, you can determine $F_c$ and compare your result to

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

where $r$ is the distance from the axis of rotation 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 slowly increased to the correct value and and as the speed is decreased 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 and the 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 to its proper length (item 3 above), measure the distance r from the axis of rotation to the center of mass of the bob. The center of mass is marked on the bob.

   Figure 2: Static measurement of the force using hanging weights

   
   
   
5. Compare the measured centripetal force to the computation $F_c = m\omega^{2}r$. 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.