Lab 16: May 15,2017: Angular Acceleration

Lab 16: Angular Acceleration

Amy, Chris, and John

May 15, 2017

In today's lab we figured out the factors that make angular acceleration in our rotating disk with a Pasco air sensor set up, and also to apply our knowledge using calculations of torque. With this data we measured the values for moments of inertia for the second part of the lab where we factor in minute values for torque. 

Theory: Torque is calculated as the moment arm from the axis of rotation, I is the moment of inertia of a rotating object, and alpha is angular acceleration. 

Where the delta r is the moment arm from the axis of rotation, I is the moment of inertia of the rotating object, and alpha is the angular acceleration. Our setup for the lab allows us to apply a known force (gravitational force from hanging mass) which rotates our pulley and disk system. Friction is reduced significantly by using air between the rotating tracks.
For the first part of our experiment, we use LoggerPro to record the angular velocity of the object, we are able to find the slope of this value to find angular acceleration. We find both the angular acceleration going up and going down with a couple of variables changed in each trial run to see how these changes affect the angular acceleration (changes in hanging mass, change in radius of the pulley and the change of the rotating mass itself). By finding the pattern in the change of our average angular acceleration value, we can determine what factors affect the torque of the system.

With the data we collected in part 1 of our experiment, we are able to numerically calculate the torque in the system of different combinations of disks and pulleys. There is a small factor we neglected in part 1 of our lab, that is the frictional torque in the system. Although our setup uses air to reduce friction to a minimum, this is still a quantity we cannot ignore in our experiment as well as some small mass in the pulley we used.
Using the derivations outlined in our Lab handout (shown below):


We can calculate for different moments of inertia for disk/disk combinations in our experiment.

Derivation of inertia of disk for a case with no friction and |alpha up| = |alpha down| :

For the up direction:

Multiply both sides by r:

For the down direction:

Again, multiply both sides by r:

Somethings to take into consideration at this point in our derivation, while the magnitude of alpha up is equal to magnitude of alpha down, the signs are different:

Now we set up our torque equations by plugging our torque up and down values into our general sum of torque equations (expressed I alpha up as a negative quantity since clockwise rotation is considered negative):

subtracting the torque down equation by the torque up equation gives us:

isolate I disk to get:

substituting alpha up with the relationship we established above (alpha down = -alpha up):

and alpha down is just alpha in this case since both alpha has the same magnitude which brings us to our expected result:

Apparatus and Procedure:
We use our Pasco rotational sensor hooked up to our LoggerPro equipment to read the rotation of the disk. There are 200 different markings on the disks so we have the equipment read 200 marks per rotation. We make sure the hose clamp below is open so the two disks rotate freely (except for the last trial run where the two disks rotate together) and turn on the air valve to get the disks rotating smoothly. We wrap our string around the torque pulley and allow the hanging mass to accelerate up and down to the find the angular velocity and derive the angular acceleration from the slope of our graphs. We take the magnitude of these measurements and average them for each of the 6 experimental trial runs to see how changes in mass, pulley size and disk(s) affects the value of angular acceleration.
Once we have obtained these values, we carry out the calculations outlined above in Theory/Introduction.

Data:

Measured mass and radii of the disks and pulleys:

	Radius (m)	Mass (kg)
Steel	0.0632	1.358
Aluminum	0.0683	0.465
Small Pulley	0.0125	0.0101
Large Pulley	0.02745	0.0362

Note: The radius measurement has a uncertainty of +/- 0.001 m since we measured with the digital caliper that shows values to the millimeters place. The mass measurement has a uncertainty of +/- 0.00001 kg since the digital scale we used gives measurements to the hundredths place in grams (Fig. 2). 

Fig. 2. The digital scale reading for the small pulley. Shows the hundredths decimal place.

We will use these values later when we calculated for the actual value of inertia to compared with our experimental values.

The data we collected for the first part of the experiment (fig. 3 and 4) and an example of the data collection graphs:

Experiment #	mass (kg)	torque pulley	Disk	|αdown|	|α up |	|α avg|
1	0.02461	small	Top steel	1.204	1.062	1.133
2	0.04961	small	Top steel	2.34	2.187	2.2635
3	0.07461	small	Top steel	3.509	3.279	3.394
4	0.02461	large	Top steel	2.313	2.12	2.2165
5	0.02461	large	Top aluminum	6.659	5.853	6.256
6	0.02461	large	top steel + bottom steel	2.363	2.051	2.207

Fig. 3. The positive slope of the graph represents our angular acceleration going up.

Fig. 4. The negative slope of the graph represents our angular acceleration going down.

We consider the counter clockwise direction as positive and the mass is descending when the torque pulley spins counter clockwise. 

Calculated Data:

Following the equations derived in our lab handout, taking trial 1 for example:

plugging in the values we get:

We apply this same calculation for each of the trials and obtain these values:

Experiment #	Idisk (kg*m^2)
1	0.002656989
2	0.002677128
3	0.002681249
4	0.002968299
5	0.001039694
6	0.002981155

Analysis of Graph and Data:

From our initial data table, we can see what affects our angular acceleration. Increasing the hanging mass increased the angular acceleration, increasing the moment arm (which is the increase in the size of the torque pulley in our experiment, distance from rotational axis to the point of contact) caused the angular acceleration to increase (comparison of trial 1 and 4 shows us this result; same value of mass and disk type but a change in the size of the torque pulley changed the magnitude of the average angular acceleration from 1.133 to 2.2165). Using a much lighter disk (steel vs aluminum, the latter being much lighter) also increased the angular acceleration. When we used two steel disks instead of one, the mass of the disk was greater than any of the trial runs and as expected the angular acceleration decreased. 

Conclusion:

We can compare these calculated experimental values with the moment of inertia equations derived using calculus:

and to find percent error between experimental and theoretical:

Experiment #	Experimental Value	Inertia Equation Value (Theoretical)	% Error
1	0.002656989	0.00273226	2.754898875
2	0.002677128	0.00273226	2.017816752
3	0.002681249	0.00273226	1.866989232
4	0.002968299	0.002784385	6.605192888
5	0.001039694	0.001169021	11.0628466
6	0.002981155	0.005496474	45.76241059

Note: for experiment #6, The pulley system mass was two of the steel disks + the large pulley. 

We see that our derived values for Inertia are very close to the true value for most of the trials. The percent error for the first couple of calculations are very minimal while the last two trials are rather high in error. For trial 6, a source of uncertainty was the amount of air we had coming into the system. The lab did mention that the air had to be enough to get the system running smooth but an excess would cause unwanted results. We did notice that initially our two steel disks were not rotating together but still separate which we fixed by turning the air down. Even after we did this though, the disks didn't completely rotate together which explains the discrepancy in data above. 

Other usual sources of uncertainty stem from correlation in graph data, while our correlation values we obtained were all greater than ~0.97, ideally we'd want the highest correlation possible and in some case data collection wasn't as precise as we would have liked it to be. This can be from data collecting at such a rapid pace and possibly because of the fact that the apparatus we were using was very old.