Question

Conservation of Angular Momentum

Lab Worksheet

Introduction

A non-rotating ring is dropped onto a rotating disk. The angular speed is measured (by the Rotary Motion Sensor) immediately before the drop and immediately after the ring stops sliding on the disk.

The initial angular momentum is compared to the final angular momentum, and the initial kinetic energy is compared to the final kinetic energy.

Theory:

When the ring is dropped onto the rotating disk, there is no net external torque on the system. Therefore, there is no change in angular momentum; angular momentum (L) is conserved.

L = I_i ω_i = I_f ω_f (1)

where I_i is the initial rotational inertia and ω_i is the initial angular speed. The initial rotational inertia is that of a disk about an axis perpendicular to the disk (see Fig. 2) and through the center-of-mass (c.m.) is

I_i = I_d = ½ MR² (2)

where M is the mass and R is the radius of the disk. The rotational inertia of the ring about an axis through its c.m. and parallel to the symmetry axis of the ring is

I_rcm = ½ M(R₁² + R₂²) (3)

where R₁ and R₂ are the inner and outer radii of the ring. If the rotation axis is displaced by a distance x from the c.m., the rotational inertia of the ring can be calculated from the parallel axis theorem and we have

I_r = ½ M(R₁² + R₂²) + Mx² (4)

Note that the final rotational inertia will be the sum of the initial disk plus the ring.

The rotational kinetic energy of a rotating object is given by

KE = ½Iω² (5)

Figure 1: Conservation of Angular Momentum Figure 2: Rotational Axis for Ring and Disk

Setup

1. Use the large rod base and the 45 cm rod to support the Rotary Motion Sensor as shown in Figure 1. Plug the sensor into the interface.
2. Measure the mass and radii for the disk and ring.
3. Attach the disk to the clear three-step pulley on the Rotary Motion Sensor using the thumb-screw.
4. Place a level on the disk and level the system using the adjustable feet on the base.
5. In PASCO Capstone, set the sample rate for the Rotary Motion Sensor to 20 Hz. Create a graph of Angular Velocity vs. Time.

Angular speed before and after collision

Position after collision with disc and ring

X max = ________ X min =_________

Analysis

1. Use Equation (2) to calculate the initial rotational inertia.

2. Use Equation (4) to calculate the final rotational inertia of the ring.

3. Calculate the final rotational inertia of the system.

4. Use Equation (1) to calculate the initial angular momentum of the system.

5. Calculate the final angular momentum of the system and compare with a % error. Was angular momentum conserved?

6. Use Equation (5) to calculate the kinetic energy before and after dropping the ring. Was energy conserved? Where did it go?

7. How can angular momentum be conserved and energy not be?

Answer 1

When the ring is placed on the disc,sliding occurs. Ring gain angular velocity but the disc looses it accordingly due to torque produced by kinetic friction.The sliding stops till both gain same angular momentum.