Angular Motion and Torque

By applying a known torque to a freely rotatable disk and measuring the resultant angular acceleration, we can compute the moment of inertia, 퐼, of the disk. The known torque will result from a falling mass, 푚, causing the disc to rotate via a string wound to a pulley of radius 푟푝 attached to the disk.

Since the known torque is 휏 = 퐹 푟푝

2 As long as 푚푟푝 ≪ 퐼, we will have 퐹 ≈ 푚푔

The torque is related to the angular acceleration by 휏 = 퐼 훼,

푚𝑔푟 so the moment of inertia will be given by 퐼 ≈ 푝 훼

To perform this experiment, we will use a rotational motion apparatus, a vertical pulley, a photogate, digital caliper, and a Smart Timer. To ensure that the acceleration of the falling mass is small, we will ensure that the falling mass is very much smaller than the mass of the rotating disk. The apparatus should be leveled before beginning the experiment.

Name:

Lab Partners:

Procedure: 1. Measure the mass of the disc:

M = g

2. Measure the mass of the weight hanger:

mh = g

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3. Measure the diameter of the step pulley with the digital caliper (make sure to measure its diameter on the inside track) and divide it by 2 to get its radius:

rp = cm

4. Measure the outer diameter of the primary disc of the rotational motion apparatus and divide by two to gets its radius. Then, flip it over and measure the diameter of the hole in its center, divide by two, and record its radius:

rout = cm

rin = cm

Place the disc back on the apparatus. Give it a turn to ensure it rotates freely.

5. Run the thread, with the weight hanger attached, over the vertical pulley. Place a 20 g mass in the weight hanger and rotate the disc until the weight is near the vertical pulley. Make sure to secure to disc, so that it does not begin to rotate.

6. Record Time. Turn on the Smart Timer. The photogate should be connected to input channel 1. Set the timer to “Time/Fence” mode. Press “Start/Stop” before the first trial.

When the disc is unsecured, the thread will generate a torque on the disc causing it to rotate. In “Time/Fence” mode, the Smart Timer will start the clock when the card first intercepts the beam of the photogate. It then records the elapsed time since this first trigger for every subsequent interception of the beam.

When the timer is set, unsecure the disc and allow the mass to fall. Just before the weight hanger hits the floor, press the “Start/Stop” button on the Smart Timer. At this point, pressing the “Select Measurement” button will allow you to see the times measured by the Smart Timer. Record these data below in order of increasing time. Repeat this procedure for a total of 5 trials.

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Data:

n t θ nav 풕풂풗 ω α

0 0 0 1

1 2π 2 2 4π 3 3 6π 4 4 8π 5 5 10π 6 6 12π 7 7 14π 8 8 16π 9 9 18π

Analysis

1. Angular Velocity. The average angular velocity, 휔𝑖 during each of these intervals is given by the angular distance traveled by the disc, 2π radians, divided by the elapsed time, ∆푡𝑖, for the full rotation. Calculate each 휔𝑖 for all trials and record the data in the table below.

2휋 2휋 휔𝑖 = = ∆푡푖 푡푖+1−푡푖

In each interval, when is the angular velocity precisely equal to 휔𝑖

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2. Determine the midpoint of each time interval. 3. Assign time to each angular velocity. The Smart Timer takes readings each time the disc rotates—thus at equal distances, but changing time intervals. In order to determine angular acceleration, we need to assign a time to each angular velocity. 4. Angular Acceleration. Calculate the angular acceleration between each pair of readings for each trial

∆휔 휔 − 휔 훼 = = 𝑖+1 𝑖 ∆푡푎푣𝑔 푡𝑖+1 푎푣𝑔 − 푡𝑖 푎푣𝑔

5. Compute the average angular acceleration (the sum of all the α’s in every trial divided by the number of α’s in every trial.

α = rad/s2

6. Calculate the moment of inertia of the disc for each trial using the formula

(푚ℎ + 20)푔 푟푝 퐼 = 훼

7. Calculate the average value of 퐼 for all trials

2 퐼av = g cm

7. Calculate the theoretical moment of inertia of the disc using the formula:

1 퐼 = 푀(푟2 + 푟2 ) 푡ℎ 2 𝑖푛 표푢푡

2 Ith = g cm

8. Compute the percent difference between the theoretical and measured moments of inertia for the disc.

|퐼 − 퐼 | ∆% = av 푡ℎ × 100% 퐼푡ℎ

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