Angular acceleration: Jose Rodriguez: Lab Partners: Kevin Tran; Kevin Nguyen: Lab Completed: 5/15/17

The purpose of this lab was to apply a known torque to an object that can rotate, and measure the angular acceleration as well as the inertia of the rotating object to find the relationship between the two.

For this experiment we needed to measure several variables in order to calculate the inertia of the rotating object per each experiment.  In order to achieve these objectives we used an apparatus that has the potential to spin one or two disks simultaneously, had different pulleys, and were tied to a string which was attached to a hanging mass.  The apparatus also had a sensor with the capacity to measure angular velocity, angular position, and angular acceleration vs. time of the system.  We focused on measuring the tension in the string because that value is the value for the torque due to the string at the point of action on the pulley, so we used Newton's second law to derive force and torque equations that we used to solve for the inertia of the disk.    

To begin with, we measured the mass of each lab equipment used except for the main frame of the apparatus with a digital balance.  Then, we connected the sensor to the computer and Logger-pro and began the six experiments that we had to complete.  For the first three experiments we used two steel disks and a small pulley, but the hanging mass (which began with 24.6 grams) increased by one each time.  The fourth experiment changed because the large pulley was used instead of the small pulley and the only hanging mass that was used was the initial hanging mass for experiment one.  As for the fifth experiment, the only change was that we used an aluminum top disk instead of the steel one that we used for the first four experiments.  Lastly, the six experiment was different because we set the apparatus to rotate the two steel disk simultaneously while the initial hanging mass descended and ascended.  Next, we used Newton's second law to speculate a value for the inertia of the spinning disk(s) per each trail and investigate the relationship between angular acceleration and inertia.

This is a table graph that has the values for the initial measurements

Measurements for the different lab equipment that we used for the experiments
equipment used	mass in kilograms	diameter in Meters	radius in Meters
top steel disk	1.357	0.1270	0.0635
bottom steel disk	1.348	0.1260	0.0630
top aluminum disk	0.466	0.1270	0.0635
smaller torque pulley	0.0100	0.0251	0.0126
larger torque pulley	0.0363	0.0489	0.0246
hanging mass	0.0246	N/A	N/A

This table illustrates the effects of the varying experimental variables.  

Effects of various changes in the experiment on the angular acceleration of the system
experiment #		kilograms actually hanging	torque pulley	Disk spinning	acceleration down (rad/2(sec))	acceleration up (rad/2(sec))	average acceleration (rad/2(sec))
1	hanging mass only	0.0246	small	top steel	1.061	-1.235	1.148
2	2x hanging mass	0.0446	small	top steel	2.150	-2.33	2.240
3	3x hanging mass	0.0746	small	top steel	3.238	-3.536	3.387
4	hanging mass only	0.0246	large	top steel	2.074	-2.310	2.192
5	hanging mass only	0.0246	large	top aluminum	5.802	-6.477	6.140
6	hanging mass only	0.0246	Large	top steel + bottom steel	1.176	-1.373	1.275

Image 1 illustrates the apparatus that allowed us to complete the experiments.  Note the sensor box next to the disks.  Also, the larger pulley is on in this picture.

Image 2 illustrates the velocity vs. time graph for the first experiment.  Note slope of the graph is the angular acceleration.

Image 3 illustrates the velocity vs. time graph for the second experiment.  Note slope of the graph is the angular acceleration.

Image 4 illustrates the velocity vs. time graph for the third experiment.  Note slope of the graph is the angular acceleration.

Image 5 illustrates the velocity vs. time graph for the fourth experiment.  Note slope of the graph is the angular acceleration.

Image 6 illustrates the velocity vs. time graph for the fifth experiment.  Note slope of the graph is the angular acceleration.

Image 7 illustrates the velocity vs. time graph for the last experiment.  Note slope of the graph is the angular acceleration.

  Image 8 shows the derivation of the formula for the inertia of disk without friction. 

Image 9 illustrates the formula and calculations for the inertia of the spinning disk(s) for trial 1-4.
Note the inertia is less for experiment number four when the aluminum disk was rotating with the large pulley.

Image 10 illustrates the formula and calculations for the inertia of the spinning disk(s) for trials 5 and 6 as well as the frictional Torque for experiments 1 and 2.
Note that experiment 6 has a greater inertia for the entire lab, as well as the fact that the lowest inertia for the disk was experiment 5.

Image 11 shows the calculations for the frictional torque for the rest of the experiments.  Note that experiment number 6 has the highest value for frictional torque.

 In conclusion, I noticed some relationships for angular acceleration and the inertia for the disk(s) that was or were spinning.  For example, I noticed that for experiments 1-3 the inertia's were similar because all of the values were within 0.0003 kg * (m)^2, but the average angular accelerations differed considerably as seen in image 9.  It seems like the inertia of the disk is not affected greatly by the added massed, even though think it should since the added weight would create a larger value for tension in the string that is the cause of the torque that motivates the disk to move.  Although, the added mass did cause the angular acceleration to increase.  For experiment four, the inertia of the disk was a smaller value than previous trials as seen in image 9.  This may have been due to the added radius of the larger pulley since torque is equal to radius multiplied by force.  Interestingly, experiment 5 had the smallest value for inertia as seen in image 10.  The cause to this is probably because the frictional force is due to the mass of the spinning disk and since the aluminum disk is a lesser mass than the steel disk, it would imply that there was less force to stop the disk from rotating.  As for the last trial, it produced the largest value for inertia of the disks as seen in image 10.  As for the cause, I think it was due to an added force of friction from the lower disk that was spinning as well.  That would imply that there was opposing force towards the inertia of the disk, and image 11 shows that the frictional force for that trial was larger than any other during the entire experiment.  The only errors for the experiment I can think of are mathematical and fundamental physics understanding by my part, as well as possibly have some force of friction from the pulley that detoured the string towards the floor.