May 22, 2017 Lab 18: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque

Chris Ceron, Amy Chung, John Choi

Goal of Lab:
Derive the frictional torque of a large metal disk and evaluate this value is by timing the movement of a dynamics cart down a near frictionless track.

Theory/Introduction:

When we give our large metal disk an initial velocity ,the frictional torque is what brings the disk to a stop. Sum of torques are expressed in the general format of:



I = Inertia of the large metal disk
 = angular acceleration or deceleration

That means once we find the inertia of the disk, we can find the frictional torque since this is the only torque acting on the system. The large metal disk can be modeled as three different shapes, two small radius cylinders on the side and one big metal disk in the middle.

Apparatus used in lab

The summation of these values gives us the total inertia of the system. Inertia for a solid cylinder spinning about its center axis is given by:

To find the mass of each piece of the system, we took the rations of the total mass:



Once we have this calculated data, we spin the apparatus and use our phone's slow motion capture to derive the angular deceleration of the system. The method we used to derive angular deceleration involved finding out the tangential deceleration first and using the relationship between tangential acceleration and angular acceleration.

To evaluate our deceleration value, we  a cart to the disk with some string and allow the cart to roll down a frictionless track going a distance of 1 meter. We timed this and compared it to the time we got using our derived value of angular deceleration to compare how accurate our value was.

Apparatus and Procedure:

Calipers used for messurements

For the first part of the lab, we calculated the total inertia of the system by using the procedure outlined in our introduction. The total mass of the disk was given on the side of the disk, and using our ratio of volume to the total mass we were able to find the mass of the 3 different pieces. We use different precision vernier calipers to find the appropriate measurements

For the side thickness, we used a smaller caliper measurement that protrudes from one end.

The larger caliper we used for the diameter measurements are even more precise than the smaller caliper.

For the second part, we gave the disk an initial spin and used Amy's phone recording at 240 frames/sec to record the deceleration of the disk. Once we had this video, we used LoggerPro to analyze the video. We chose to advance every 8 frames so the video capture on LoggerPro was at 30 frames/sec.

The dot plots of the rotation slowing down.

Our method of deriving angular acceleration:

1. Use LoggerPro's video analysis to setup different aspects needed for calculation.
2. Plot a dot for every frame on the blue tape portion of the disk.
3. Find the linear distance between each dot using the distance formula.

         

4. Find midpoint velocity between each delta x:

5. Plot this midpoint velocity against time and do a linear fit on the graph.

6. The slope of this line is the tangential deceleration.

7. The relationship between tangential acceleration and angular is:

      

Using the equation above, we divided the tangential acceleration by the radius to find alpha.

The setup for the cart running down the track connected to the disk.

For the third part of this lab, we test how accurate our derived value of angular deceleration is. We set up our apparatus in a manner that allows for the string to always be parallel to the track, otherwise the torque from the tension force changes as the cart rolls down. We allow the cart to go a distance of 1 meter and time how long it takes. We then compare this value to our calculated value for time.

We found two different ways to calculate for time. 

Method one:
 Use rotational kinematics where theta was found using the relationship:

Here delta x is 1 meter, so theta came out to be 63.69 rad. Since we know the initial angular velocity is 0, we can use the formula:

Another method involves using forces and torques. From our setup we can set up two equations:

Manipulating the first equation we can obtain:

Adding this new equation with the sum of forces equation cancels out the value of T which brings us to:

Using this tangential acceleration value, we can use kinematics again to find the time:

We can compare this time value to the one we found by just timing the descent of the cart 1 meter down the track.

Data and Calculated Data:

The mass of the disk system we used was 4.928 kg.

The mass of the cart we used was 0.5249 kg.

We can find the volume of each part using this data:

where r is radius and h is the thickness. 

Examples of mass and inertia calculations:

Data we obtained/measured:

	Thickness (m)	Diameter (m)	Radius (m)	Volume (m^3)	Mass (kg)	Inertia (kg*m2)
Cylinder 1	0.0507	0.0314	0.0157	1.2497 * 10-5 π	0.3416	4.210*10-5
Cylinder 2	0.0511	0.0316	0.0158	1.2757 * 10-5 π	4.139	0.01973
Disk	0.0159	0.19528	0.09764	1.5158 * 10-4 π	0.3487	4.352*10-5

*Note* the uncertainty in the diameter measurement and for disk thickness was +/- 0.00002 m (from the bigger caliper). The uncertainty for the thickness of the two cylinders was 0.0001 m (from the smaller caliper). 

So Inertia of the system will be 4.210*10^-5 + 0.01973 + 4.352*10^-5 = 0.01982.

The graph we obtained by plotting the midpoint velocity with time. 

The tangential deceleration value we found was -0.1485 m/s^2 (not the -0.01077 on the graph above).

With the relationship between tangential deceleration and angular deceleration we find angular deceleration to be -1.5209 rad/s^2.

For the timed descent down the track, we record the time to be about 7.1 seconds.

The angle of our track was 48.0 degrees from the horizontal.

The time calculated using our rotational kinematics approach was:

and for the force/torque approach:

So the calculated times are very close to each other regardless of method (small discrepancy between the two values comes from uncertainty in the different measurements).

Conclusion:

We had an issue in our lab where the time we recorded for the cart was different from the calculated time above. This would be fine if our method of finding angular acceleration was incorrect or presented flaws, but even the professor said our method for finding angular acceleration and frictional torque was valid. The error calculation for these values comes out to be:

Which is a rather large % error. Of course the 7.1 second value we have for expected is a rather rough estimate (due to human error when observing when the cart has moved a distance of one meter. Our value of angular acceleration could have been more accurate if we advanced the videos by a number less than 8 frames. Our inertia value matched up very closely to the expected value. This is largely due to the precision in the equipment used for our measurements. The only potential source for this great and error seems to be from our data gathering for finding angular deceleration. It could potentially be the video capture wasn't aligned as well as possible with the rotational axis, we used too little frames. and potential human error from our plot of the dots being as very rough eyeballed estimate for the location on the video.

May 15, 2017 Lab 16: Angular Acceleration

Lab 16: Angular Acceleration

Chris Ceron, Amy Chung, John Choi

Goal of Lab:
Find out what factors affect angular acceleration using our rotating disk with Pasco air sensor setup and apply this knowledge to our calculation of torque. 

Theory/Introduction:
Torque is a quantity defined by the equation:



Where delta r is distance 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 can then find find angular acceleration by taking the slope of the graph. 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). 

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. We neglect the small frictional torque in our 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.


We can calculate for different moments of inertia for disk/disk combinations in our experiment using the derivations outlined in our Lab handout:

Apparatus and Procedure:
We connect our Pasco rotational sensor to LoggerPro in order to read the rotation of the disk. There are 200 different markings on the disks we use, so we ensured that LoggerPro accurately records one rotation of the disk by counting the markings. We also ensured that both disks in our apparatus can move freely by checking that the hose clamp on the bottom of the apparatus was open. 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+/-.0001m	1.358+/-.0001m
Aluminum	0.0683+/-.0001m	0.465+/-.0001m
Small Pulley	0.0125+/-.0001m	0.0101+/-.0001m
Large Pulley	0.02745+/-.0001m	0.0362+/-.0001m

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

The data we collected for the first part of the experiment 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 and increasing the torque pulley both caused the angular acceleration to increase . Using a much lighter disk from steel to aluminum (aluminum 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

We see that our derived values for Inertia are very close to the true value for most of the trials. The percent error found in the first experiments were minimal in comparison to the last one. For trial 6, one of the factors that contributed to the percentage of error was the air we had coming into the system. The introduction of the lab stated to use only enough air so the disks can rotate. Initially, our two steel disks were not rotating together as they were supposed to. We fixed this issue 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.