5/4 Angular Acceleration

Part One

Purpose: To note the difference in acceleration of a system when we change the hanging mass, radius of the torque pulley, and the rotating mass. 

Procedure:
We started with the setup in FIGURE (1) and took the measurements of the masses and diameters of the disks, pulleys, and hanging mass.

FIGURE (1)

FIGURE (2)

The values we measured are listed in FIGURE (2). Once we set up the apparatus, connected it to the computer, and set up the sensor settings, we turned on the compressed air so that the disks can rotate separately. We started with just the hanging mass attached to a string that was wrapped around the pulley. In the next trial, we added twice the hanging mass and then three times the hanging mass. In the fourth through sixth experiment we changed the size of the pulley to a larger one. In the last two experiments we also changed the disk. So in the fifth experiment we had only the top aluminum disk rotating and in the sixth experiment we has the top steel disk and the bottom steel disk both rotating. The data we recorded was the angular velocity. We took the derivative of that velocity to get the angular acceleration. (FIGURE (3)-(8)).

FIGURE (3)

FIGURE (4)

FIGURE (5)

FIGURE (6)

FIGURE (7)

FIGURE (8)

Once we recorded all the data, we were able to find the average acceleration for each trial, simply by adding the up and down acceleration and dividing by two (FIGURE (9)).

FIGURE (9)

We can notice some patterns in FIGURE (9). When we increased the mass of the hanging mass, the acceleration increased as well. It increased about one radian per second when we doubled the mass, and another radian per second when we tripled the original mass. The acceleration increased with a larger hanging mass because the torque on the pulley increased from the initial trial. In trail four, the first trial with the larger pulley, we noticed that the acceleration increased. While keeping the larger pulley and changing the top steel disk to an aluminum one we can notice that the acceleration drastically increases. This is because the inertia of the lighter disk is much smaller than the steel disk, since the aluminum disk is lighter. In the last trial, when we added the rotating bottom steel disk, is slightly slower than the initial acceleration. This could also be due to the amount of inertia that is needed to move two disks instead of just one. 

PART TWO

In class, we derived an equation to solve for the inertia of the system for each trial. We plugged in the measurements we took for each trial to the equation and came up with each inertia (FIGURES (10) & (11)).

FIGURE (1)

FIGURE (11)

We also calculated the expected value of inertia for each system using the equation: I = 1/2*m*r^2

For the first four trials, the expected value of inertia was .00276, for trial five the expected value was .000929, and for the last trial it was .1162. 

Conclusion:

The first five trials were very close to the expected value of inertia. They were all within 4% of the expected value (most fell under 2%). There is an error in the sixth one though. I think our group might have measured the value of acceleration wrong- maybe we did not have the lab pro calibrated correctly. Other sources of error in this lab would come from measuring incorrectly. The relationships between hanging mass, radius, and rotating mass are evident in our data though. When the hanging mass was increased to twice as much the acceleration also doubled. And when we tripled the hanging mass we saw the same pattern. This does not affect the inertia though because the mass and radius of the pulley and rotating disk stayed the same. In the fourth trial, it is shown that the size of the torque pulley does not affect inertia, because inertia is dependent on the size and mass of the rotating disk. In trial five we changed the rotating disk to a lighter, aluminum one and, as expected, the acceleration increased dramatically. The inertia decreased in this trial since the mass was lighter. In the last trial, we expected to see that the acceleration decreased from the original acceleration. We saw a slight decrease but not as big as we expected. That is why I believe this is where our error was.