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

Lab 18: Moment of Inertia and Frictional Torque

Amy, Chris, and John

May 22, 2017

In today's lab, we will determine the time for the cart as it travels down a ramp one meter. The cart will experience tension from the metal disk as the metal disk rotates. 

Theory: There are many components happening in this lab. In order to find the time the cart travels one meter, our theory is to relate the tangential acceleration of the metal disk is equal to the acceleration of the cart. Let's first look at the metal disk. As the metal disk rotates from the pull of the cart, the metal disk is experiencing two different types of torques. The first torque is the tension force from the cart. The second torque the metal disk will be experiencing is the frictional torque. We will sum these torques to equal moment of inertia of the disk and the angular acceleration. Instead of the spring applying torque on the outer rim of the disk, it will be applying, it will be applying torque in the inner radius of the disk. Since the angular acceleration is the same all around the disk, linear acceleration will vary in different points of the radius. 

Apparatus and Procedure: We first have to find a way to measure the frictional torque before we calculate the angular acceleration. For this lab, we did a video capture of the metal disk spinning until it comes to a stop. 

We measured the dimensions of our apparatus to input in our theoretical calculations. The disk already had the total mas given. We used the formula to calculate the density of the apparatus. Once we found the density of the apparatus, we calculate the mass of the smaller metal disk in the middle of our set up. 

While the disk is spinning until it stops, the only torque acting on the disk is the frictional torque. There is a piece of masking tape applied on the outer rim of the disk. We recorded the rotational motion in slow speed from a phone. Then we uploaded the video on Logger Pro and plotted dots following the tape we had placed on the disk until the rotational motion came to a rest. The graph would determine the slope of angular velocity. That slope is the angular acceleration of friction. We will use this value for our theoretical calculation to solve time. 

For the second part, we gave the disk an initial spin and used my phone recording at 240 frames/sec to record the deceleration of the disk. We made sure to record as parallel to the axis of rotation as possible so the plots for distance were as accurate as possible. Once we had this video, we used LoggerPro to analyze the video (video analysis shown in figure 6). We chose to advance every 8 frames so the video capture on LoggerPro was at 30 frames/sec.

We also had to calculate the time it took a cart we were given to go down a track using the disk we had figured out the angular acceleration from, 

Ont the experimental lab, we use a timer to calculate the time it takes for the cart to descend one meter down the ramp. We used three trials to average our times. 

Data: We differed from other groups as to how we derived the angular acceleration. 

1.We used Logger Pro video to analyze to set up different aspects needed for calculation (correct frames, have a reference measurement, we used the diameter of the disk) 

2. Plot a dot for every frame on the blue tape portion of our disk. 

3. Find a linear distance between each dot using the distance formula. 

4. Find a midpoint velocity between each delta x:

5. Plot the midpoint vs. time and do a linear fit on the graph (make sure linear fit on the later part of the graph when the disk is slowing down, initially there is some erroneous data from the force applied to get the disk to rotate. 

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

7. The relationship between tangential acceleration and angular is

      

taking the initial acceleration from the linear fit and divide it by the radius of the large metal disk in order to find alpha. 

With these results giving us the needed values, we are ready to compare our theoretical values with our experimental values. The calculations we make assume the string is parallel to the track so make sure the cart is indeed parallel to the ramp (if not the torque from the tension force acting on the disk changes as the cart rolls down and this throws off our calculations). 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 involved using 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.

The mass of the disk system we used was 4928 grams or 4.928 kg.

The mass of the cart we used was 524.9 grams or 0.5249 kg.

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

where r is radius and h is the thickness. 

Mass can be found using the ratio of volume to total mass outlined above. Inertia for each system can also be found using the equation outlined above in theory/intro and adding up those quantities will gives us inertia of the whole system.

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.

Fig. 8. The graph we obtained by plotting the midpoint velocity with time. The graph above had the wrong time parameter so the value of slope does not match the correct graph we got but the method is the same.

The tangential deceleration value we found was -0.1485 m/s^2 

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 (this value wasn't exact since the 100 meter measurement was approximated by eye).

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 some initial issues with our recorded times on Logger Pro, the time on our data was not close to the calculated time on our derivation of angular acceleration. Normally this would be fine if our method for finding angular acceleration had some errors, but even Professor Wolf said our derivation was alright. The errors of calculations measured out to be

This is a big percent error, but the second value of 7.1 seconds we have is a very rough estimate. 

the derivations we made could have been more accurate if we had chosen to use plot more dots on the Logger Pro frames of our video. Our inertia values matched up rather closely to the expected value as expected since the equipment we used to measure these quantities had very good precision. One of our major sources of error can be narrowed down to the data we kept deriving to find angular acceleration. 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. I'd guess this is the biggest potential source of uncertainty as plotting dots on LoggerPro using a rather small video size and the trackpad is bound to have error on every data point.