Electrical Equivalent of Heat

30 Conservation of Angular Momentum30 - Page 5 of 5

Conservation of Angular Momentum

Introduction

A non-rotating ring is dropped onto a rotating disk. The angular speed is measured immediately before the drop and after the ring stops sliding on the disk. The measurements are repeated with a non-rotating disk being dropped onto a rotating disk. For each situation, the initial angular momentum is compared to the final angular momentum. Initial and final kinetic energy are also calculated and compared.

Equipment

Qty

Items

Part Number

1

Rotary Motion Sensor

PS-2120A

1

Rotational Inertia Accessory

ME-3420

1

Large Rod Base

ME-8735

1

Rod, 45 cm

ME-8736

Required, but not included:

1

Balance

SE-8723

1

Calipers

SE-8711

1

550 Universal Interface

UI-5001

1

PASCO Capstone software

Figure 1: Complete Setup

Theory

When the ring is dropped onto the rotating disk, there is no net torque on the system since the torque on the ring is equal and opposite to the torque on the disk. Therefore, there is no change in angular momentum (L = Iω), which we say is “conserved”:

Ii ωi = If ωf(1)

where Ii is the initial rotational inertia and ωi is the initial angular speed, and If is the final rotational inertia and ωf is the final angular speed. This assumes there is no torque due to friction in the rotational motion sensor. This is not true, but the effect can be minimized by operating over as short a time as possible. We also ignore the rotational inertia of the rotational motion sensor, which is quite small compared to that of the ring or disk.

The initial rotational inertia is that of a disk about an axis perpendicular to the disk and through the center of mass (c.m.):

(2)

The rotational inertia of the ring about an axis through its c.m. and parallel to the symmetry axis of the ring is

(3)

where R1 and R2 are the inner and outer radii of the ring.

The final rotational inertia If will be the sum of the initial disk plus whatever is dropped (either a ring or a second disk) on it.

The rotational kinetic energy of a rotating object is given by

(4)

1. Make predictions based on the equations: Will the final angular speed be more or less than the initial angular speed of the disk? Will the final angular momentum be more or less than the initial angular momentum of the disk?

2. What do you predict happens to the rotational kinetic energy of the system? (Hint: Is this an elastic or inelastic collision?)

Setup & Initial Measurements

1. Mount the Rotary Motion Sensor to the 45-cm rod, attached to the large rod base. See Figure 1. Connect the Rotary Motion Sensor to the interface.

2. Measure and record the mass of the ring using the balance, and the inner and outer radii of the ring using the calipers.

3. Measure and record the mass and radius of each of the two disks. Make one of these Disk 1 and the other Disk 2.

4. Attach the 3-step pulley and Disk 1 to the Rotary Motion Sensor. Place an Alignment Guide on top of the disk and tighten the screw.

5. To level the base: Place the ring (or other available mass) on top of the disk so it is off center. See Figure 2. This makes the disk unbalanced.

Figure 2: Offset Mass for Leveling

If the disk is not level, it will rotate when the offset mass is added. Adjust the leveling feet on the stand until the disk will stay in any position.

Once the disk is level, remove the offset mass.

6. In PASCO Capstone, create a graph of Angular Velocity vs. Time. Set the Sampling Rate to 25 Hz.

Procedure

1. Hold the ring centered with the disk and 2 to 3 mm above it.

2. Give the disk a spin (20-30 rad/sec) and start recording. After about two seconds of data has been taken, drop the ring onto the spinning disk. Notes: The Alignment Guide will center the ring on the disk. If the angular velocity is negative, you can either ignore the sign, or delete this run (lower toolbar) and redo with the initial spin in the opposite direction.

3. After another two seconds, stop collecting data.

4. Open Data Summary (left toolbar in Capstone) and label this run “Ring”.

5. Remove the Alignment Guide and repeat the procedure, dropping Disk 2 onto Disk 1. The square hole in the center of the dropped disk should be downward and must fit over the screw head sticking up from the lower disk. Label this run “Disk 2”. 

6. On the graph, select the “Ring” run. Select a coordinates tool and drag it to the last data point before the collision (still on the straight line). Record the value of the initial angular velocity.

Move the coordinates tool to the first data point after the collision. Record the value in the final angular velocity.

Repeat for the "Disk 2" run.

Calculations

1. Calculate the rotational inertias of the ring and each disk.

2. For the Ring dropping on Disk 1, calculate the following:

a. Total angular momentum before the drop

b. Total angular momentum after the drop

c. The percent difference between the angular momentum before and after:

3. Repeat these calculations for Disk 2 dropping on Disk 1.

4. For Disk 2 dropping on Disk 1, calculate the kinetic energy before and after the collision, and the percent of kinetic energy lost.

Conclusions

1. Do the experimental results support the law of Conservation of Angular Momentum? Explain why or why not.

2. Was kinetic energy conserved in the collision? Explain how you know.

3. How can angular momentum be conserved, but kinetic energy not be conserved?

Written by Chuck Hunt