The purpose of the apparatus used is to measure the angular velocity so that we then can get the angular acceleration.
Procedure:
First thing we did in this experiment was measure the diameters and the masses of the disks and pulleys and the mass of the hanging mass. Then we set up the apparatus on logger pro and changed the equation settings to 200 counts per rotation. We attached the air hose to the apparatus so that the top disk could move but the bottom disk wouldn't. We ran the trials and collected data on logger pro for the angular velocity versus time for each trial. We took the derivatives of the angular velocity graphs (for each part whether it was ascending or descending) to get the angular acceleration up or down. Then we took the average of the angular acceleration up and down to get the average acceleration overall for the trial. For the second part of the experiment we used the data from the first part of the experiment to determined the experimental values for the moments of inertia for the different combination of disks used in each trial.
The diameters and masses of the various disks and pulleys were as follows:
For the first trial we used the hanging mass, the small torque pulley and the top disk was the steel one. The data we got on logger pro is as follows:
The measured value that we got was in rotations per second squared so we multiplied it by 2pi to convert it to radians per second squared. We did this for the first three trials then for trials 4,5 and 6, we added an equation into logger pro to convert the angular acceleration to rad per sec squared for us.
For the second trial we doubled the hanging mass using the same small torque pulley and same top steel disk and collected the following data:
I did not take a pic of the third trial, but it is safe to assume it looks similar to the trials above. For the third trial we tripled the hanging mass and used the small torque pulley and the steel disk on top. The acceleration down for the third trial is 1.33 rad/sec^2, and the acceleration up is 1.43 rad/sec^2.
For the fourth trial we went back to the original hanging mass by itself and changed the torque pulley to the larger pulley and kept the top disk the steel disk. The data we collected was as follows:
For the fifth trial we kept the hanging mass the same the torque pulley was the large pulley, but the top disk was the aluminum disk. The data we collected for this trial is as follows:
For the sixth trial we kept the hanging mass the same as the previous trial and kept the large torque pulley, but we changed the apparatus to allow the two disks to move and we used both steel disks on top and bottom. The data collected for this trial was as follows:
We copied the accelerations up and down from the computer and got the average for each trial and put everything on the following chart in our lab manual.
For the second part of the experiment we derived an equation in class to get the moment of inertia if the disk that is spinning using newton's second law(derivation is in the lab manual) and we got the equation:
Where m is the mass of the hanging mass, r is the radius of the torque pulley, g is acceleration due to gravity, and alpha is the average of the angular acceleration up and angular acceleration down.
Using the equation above we calculated the moment of inertia for the disk or disks combination for each trial.
Conclusion:
According to the data in the first part of the lab we can see that changing the hanging mass by making it bigger and keeping torque pulley and the top disk the same makes the average angular acceleration increase from .454 rad/sec^2 in trial 1 to .926 rad/sec^2 in trial 2 and 1.38 rad/sec^2 in trial 3. Changing the radius of the torque pulley from the small one in trial 1 to the larger one in trial 4 makes the acceleration increase. In trial 1 the average acceleration was .454 rad/sec^2 and in trial 4 the average acceleration was .8719 rad/sec^2. This makes sense because a larger radius equals a larger torque force. The effect of changing the rotating mass in trial 4 just letting the top steel disk spin the average angular acceleration is .8719 rad/sec^2, while in trial 5 making the top disk the aluminum disk and only allowing that disk to spins gives an angular acceleration of 2.4885 rad/sec^2. In trial 6 allowing both steel disk on top and steel disk on the bottom to move gives an angular acceleration of 1.033 rad/sec^2. It looks like making the top disk a lighter mass makes the average acceleration higher, but when it comes to trials 4 and 6 the two disks spinning together makes the acceleration value greater than just the top steel disk moving alone. For the second part of the experiment we calculated the moment of inertia for the different trials. The first three trials the moment of inertia was pretty much the same for all three trials being .0135,.0132, and .0133. Which makes sense because the top steel disk spinning was the same for all three trials. For the fourth trial the moment of inertia was a little larger than the first three trials being .0150 which should be so because we change the torque pulley from the smaller to the larger adding a little more mass to the rotating system. For the fifth trial the moment of inertia has the smallest value being .00522 which makes sense because mass of the spinning disk was the smallest. And for trial 6 the moment of inertia was .0127 which is the smallest value out of all the trials which makes sense because the mass was the greatest with both steel disks spinning. some sources of uncertainty that could have effected our calculations were the fact that we could have measured the wrong diameter on the large vernier caliper because sometimes it was hard to read the correct line that aligned the best. There also could have been sources of uncertainty from the machine itself because it is a very old machine and worn down from multiple use over the years.
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