Lab 16: Angular Acceleration

The term angular acceleration inspires in my imagination a drifter tackling sharp corners while carrying tofu, but today we're going to be doing something even more exciting. We're going to be intercepting Morse code from the Communists.

 

Here's an apparatus manufactured some time in the 1970s. The large steel cylinder contains a toroid that will pick up electromagnetic waves...

 

Okay, back to reality. Actually, the cylinder is a pair of polished disks with a tiny hole that will accept air from the machine, reasonably eliminating friction and allowing the disks to turn with minimal force. We will use a light mass with a string tied around a pulley connected to the disks to provide torque, then measure the angular acceleration experimentally.

Note: That's not a pizza box.

This process will be repeated in future labs to derive inertia. For this lab, we will determine the factors which affect angular acceleration. The relevant formula is τ = Iα, but what is that in terms of actual objects? First, we need to accurately measure the diameter and mass of the top steel disk, bottom steel disk, an alternative aluminum disk, a large and small torque pulley, and the hanging mass. We will swap these items out to see their effects. We use a caliper for the diameters and, obviously, a scale for the masses.

A few minutes later and carpal tunnel, we've recorded all our data:

Item	Diameter (mm)	Mass (g)
Top steel disk	126.2	1361
Bottom steel disk		1348
Top aluminum disk		464.5
Small torque pulley	24.9	10
Large torque pulley	49.8	36.3
Hanging mass	N/A	25

 

Since we need air to be pumped through the disks, we need to plug power into the machine, made by Pasco, according to the lab worksheet--their motto (on the website) is amusingly "Science is served!" This is all connected back to Logger Pro. Everything is connected to Logger Pro. We set the sensor settings to correspond to the disks, so now it will measure the θ that the disk moves. For some reason, the acceleration graph isn't reliable, so we use measure the upward and downward slope of the velocity graphs, corresponding to the descent and ascent of the mass on the string, and average the downward and upward accelerations. We repeat this 6 times, once for each condition. Experiments 1-3 measure the effect of different mass. Experiments 1 and 4 measure the effect changing torque pulley diameter or weight. Experiments 4-6 measure the effect of changing the weight of the disks. Here are our graphs:

Ex 1: Hanging mass, small torque pulley, steel disk

Ex 2: 2X Hanging mass, small torque pulley, steel disk

Ex 3: 3X Hanging mass, small torque pulley, steel disk

Ex 4: Hanging mass, large torque pulley, steel disk

Ex 5: Hanging mass, large torque pulley, aluminum disk

Ex 6: Hanging mass, large torque pulley, steel + aluminum disk

We are able to use the bottom disk by clamping the air hose on the side of the machine:

Using a pair of linear fits, we record downward and upward accelerations, and calculate average acceleration for each. Here's the resulting chart data:

Expt #	Hanging mass (g)	Torque pulley	Disk	αdown (rad/s2)	αup (rad/s2)	αaverage (rad/s2)
1	25	small	top steel	1.0830	1.1920	1.1375
2	50	small	top steel	2.0830	2.4990	2.2910
3	75	small	top steel	2.5970	4.3280	3.4625
4	25	large	top steel	2.1100	2.3160	2.2130
5	25	large	top aluminum	5.9550	6.4720	6.2135
6	25	large	top steel + bottom steel	0.7944	1.5610	1.1777

Ignoring minor variation, we observe a linear relationship between the hanging mass weight and the average acceleration. We also notice a linear relationship between the diameter of the torque pulley, but not its weight--the larger pulley is roughly 3.5 times heavier than the smaller. Finally, there is an inverse relationship between the weight of the disk and its acceleration. The top steel disk is about 3 times heavier than the aluminum disk, and acceleration with the steel is about 3 times slower.

This seems consistent with what we know about angular acceleration. If:

τ = rT = Iα

We see that, if all else is equal, increasing the radius and tension (represented by the hanging mass) of the torque pulley should proportionally increase acceleration. Since inertia is mr² of the disks, increasing the mass of the disk should have an inversely proportional effect on the acceleration.