Physics 4A Lab 11: MOMENT OF INERTIA Parts List

Part Quantity Rotating cross-arm table 1

Large iron disk 1

Large iron ring 1

50 grams hanger 1

Weights 1 set

Table clamp 1

1-meter long rod 1 Pulleys 2

Rght-angle clamps 2

Spirit level 1 Stopwatch 1 Vernier caliper 1

Two-meter stick 1 String As Needed Paper clips As Needed Balance 1

Physics 4A Lab 11: MOMENT OF INERTIA The object of this experiment is to make a dynamic measurement of the moment of inertia of a ring about its cylinder axis, and to compare the value obtained with the theoretical value calculated from measurements of the ring’s mass and inside and outside radii. Method The setup for this experiment is shown in the figure below. When the mass m is released from the position shown, it falls, causing the table’s cross arms C to rotate. Since the acceleration of the falling mass is constant, the tension F in the string will be constant, which means that the torque applied to the rotating table will be constant, and this will produce a constant angular acceleration of table. As the mass m falls, its initial gravitational potential energy, mgh, is gradually converted into kinetic energy, part of which is the translational kinetic energy of the mass itself, 1/2mv2, and the rest of which is the rotational kinetic energy of the table and whatever might be on it, 1/2 Iω2. Since the falling mass and the rotation table are connected by a string, the speed v of the mass at any instant is related to the angular speed w of the table by v = ωr, where r is the radius of the table’s axle. The instant before the falling mass strikes the floor,

mgh = 1/2mv2 + 1/2 Iω2 (1)

The average speed of the falling mass is vavg = h/t, therefore the final velocity will be

v = 2h/t. (2) And the final angular speed of the rotating table will be

ω = v/r = 2h/(rt) (3)

It can be seen that the only measurements necessary to calculate I are the time t required for mass m to fall a distance h, and the radius r of the table’s axle. In order to determine the rotational inertia of the ring IR, it is necessary to determine that of the table IT first, and to subtract this from the value obtained when the ring is on the table. Before you begin Lab, you must show your derivation of I in terms of measured quantities to your lab instructor!!!

Procedure

A. Level the rotating cross-arm table so that the axle is vertical. B. Suspend enough paper clips from the string so that when the unloaded table is given a

slight start it continues to rotate at a constant speed. This weight counterbalances the frictional forces in the system and should not be included with the driving mass that accelerates the system.

C. Add a driving mass of 20 grams to the paper clips, record the height of fall h, and make five runs on the unloaded table recording the time t for each run.

D. Place the iron ring on the cross arms and again suspend enough paper clips from the string so that the table just barely continues to turn after it is given a slight push.

E. Add a driving force of 150 grams and make five runs with the ring on the cross-arms. Record t and h as before.

F. Using the vernier caliper, measure and record the diameter of the table’s axle. G. Place the ring on the heavy-duty balance and record its mass. H. Using a meter stick, measure and record both the inside and the outside diameter of the

ring. I. Repeat procedures D through H for the large iron disk.

Calculations and Results

1. Using the average time and height of fall, calculate the moment of inertia of the table, IT. 2. Repeat for the table with the ring on it, then determine the moment of inertia of the ring

alone. IR = IT+R - IT. 3. Calculate the moment of inertia of the ring using the formula IR = (1/2) M (R2

2 + R12).

4. Compute the percentage by which the value calculated in step 2 differs from the theoretical value calculated in step 3.

5. Repeat calculations 2 through 4 for the iron disk (no need to calculate errors for the disk). 6. You will receive additional instructions in the laboratory.