2015-May-04 Lab 16: Angular Acceleration

PURPOSE:  To observe how various factors affect angular acceleration, and then use data collected from a system of angular acceleration with a hanging mass to verify our experimental moment of inertia with our known equation for the moment of inertia of a disk.

SET UP:

Compressed air blows into a system of two large disks rotating one on top of the other, independently of each other.  On top of these disks is a pulley (of interchangeable sizes), with a string wrapped around it with a hanging mass tied to the other side, hanging over a frictionless pulley at the edge of the system (see below).

When the system is in motion, a sensor counts marks on the sides of the disks to graph angular position and angular velocity.

PROCEDURE:

We ran 6 experiments, with various changes in the experiment in order to observe how these changes affected angular acceleration.

• In the first 3 experiments, we used a top steel disk, bottom steel disk, small pulley, and only varied the hanging mass.  Experiment 1 used 25 g, 2 used 50 g, and 3 used 75 g.
• In the 4th experiment we hung the 25 grams back on, and only changed pulley from a small one to a large one.
• In experiment 5 we hung 25 grams, kept the large pulley, but changed the top disk from steel to aluminum (smaller mass).
• In experiment 6 we hung 25 grams, kept the large pulley, and replaced the steel disk back on the top, but this time allowed the top and bottom steel disks to rotate together.

For each experiment we ran the sensor to graph an angular position vs. time graph and an angular velocity vs. time graph.  We used the angular velocity graph to measure angular acceleration as the mass moves up and down by finding the slopes of angular velocity as the mass moved up and down. 

Our graph from experiment 2:

Slope of 5.715 rad/s^2 is angular acceleration while the mass is moving down, and -7.460 rad/s^2 is angular acceleration while the mass is moving up.

We take the average of the magnitudes of the two angular accelerations because there is some frictional torque in the system, which causes all the downward accelerations to be slightly less than the upward accelerations.  We take the average to balance this out a little more.  After finding the slopes of the angular velocity vs. time graphs for each experiment, we came up with the following data:

CONCLUSIONS from Part 1:

In the first 3 experiments when we only changed the hanging mass, acceleration increased by the same factor that we increased the mass.  Experiment 2 had two times the mass of Experiment 1, and angular acceleration was around 2 times as large.  Experiment 3 had 3 times the mass of Experiment 1, and angular acceleration was around 3 times as large.

In Experiments 1 and 4 when we only changed the size the pulley but had the same hanging mass of 25 grams, angular acceleration decreased with a larger pulley.  This makes sense because it would take longer for the pulley to make a full rotation.

In Experiments 4, 5, and 6, we changed the mass of the rotating body.  In Exp 4 with a top steel disk of 1.361 kg, angular acceleration was 2.228 rad/s^2.  Exp 5 had a top aluminum disk which was about 1/3 the mass of the steel disk (0.466 kg), and angular acceleration was about 3 times as much as with the top steel disk (6.261 rad/s^2).  This also makes sense because a larger mass would not rotate as quickly, which leads us to Exp 6 using both the top steel and bottom steel disks together with a combined mass of 2.709 kg.  This is roughly double the mass of the disk rotating in Exp 4, and its angular acceleration is roughly 1/2 of that in Exp 4 (0.922 rad/s^2).

*** One other part we modified to Part 1 of this lab was to add a motion sensor to record the linear velocity of the hanging mass as it moved up and down (shown below):

Only ran the motion detector for Experiment 1.

By doing this, we were able to compare angular acceleration of the system to linear velocity of the system, by multiplying angular acceleration by the radius of the small pulley (used in Exp 1).  This gave us the following graphs:

The first two graphs are angular position and velocity vs. time, and the second two graphs are linear position and velocity vs. time.  We found the slopes of each graph with the mass moving down and then down.  If we multiply the magnitude of angular acceleration in the down direction (3.301 rad/s^2) by the radius of the small pulley (0.013 m), we get a linear acceleration of 0.043 m/s^2, which is close to the experimental linear acceleration from the motion detector of 0.042 m/s^2.  We can do the same for the magnitude of angular acceleration in the up direction (3.620 rad/s^2) multiplied by 0.013 m, to get a linear acceleration of 0.047 m/s^2, while the experimental value was 0.046 m/s^2.

PART 2:

We can use our data collected and measurements to calculate moment of inertia for the system.  We used Newton's second law to find the tension, T, in the hanging mass.  We also used the equation for torque and plugged in tension to this equation (also changing linear acceleration to angular acceleration):

Moment of inertia using data we collected in the experiment from Part 1.

For Experiment 1, we plugged in our data:  on the right hand side is the equation for moment of inertia of a disk, using the measurements of the disk.  

However, our percent error shows we were extremely far off when our answers should have been the same. This could have been from a misread from any of the measurements in the disk or the pulley.

Continuing with the calculations for the rest of the experiments, we see that our percent error is much less in the last 3 experiments.  What's different about the last 3 experiments is the use of the large pulley instead of the small pulley, so our error may have been caused from a misreading in the measurements of the small pulley.

CONCLUSION:

By using Newton's second law, the torque equation, and conversion of linear to angular acceleration, we were able to come up with an equation for the moment of inertia of a rotating system with a hanging mass.  This equation used the mass of the hanging object, radius of the pulley in which it was spinning, and the angular acceleration of the mass.  This should have given us the same moment of inertia if we used the moment of inertia for a disk, 1/2MR^2.  However, our results were off by a significant value.  Our errors were much too large to have been caused by a small source of uncertainty.  We must have had made a huge blunder in measurements, possibly by reading the vernier caliper wrong.  Other small sources of error could have been caused by fluctuations of air in the compressed air tube, causing the disks to rotate at a not completely constant speed.  Also, there was friction in the pulley which caused our angular accelerations to vary.  An error could have been caused when the mass was hanging, and if we taped multiple masses in a way that made the mass sway just a little bit.

Despite our errors, our group understood clearly the way we came about deriving an experimental equation for the moment of inertia of our system and how it can compare to the moment of inertia of a disk.