5-13-2015: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque

Purpose: The purpose of this experiment was to experimentally determine the moment of inertia and the frictional torque of a large metal disk rotating about a shaft in its center.

Apparatus:

 

 

 The apparatus for this experiment consisted of a large metal disk on a central shaft. The disk and central shaft rotate about a center axis. The disk and shaft are held up by metal stands attached to a metal plate. Calipers were used to measure the diameter of the large disk and the central shaft. The mass of the disk and shaft was stamped on the side of the large disk.

In addition, a camera was used to calculate the deceleration of the rotating part of the apparatus. The video was then analyzed on LoggerPro.

Lastly, we used the setup shown above to verify the accuracy of the calculated moment of inertia and frictional torque of the system.

 

Procedure (Part 1): Calculating the Inertia of the apparatus

 

The dimensions of the large disk and central shaft were measured using calipers. In this case, we treated the central shaft as two separate cylinders. To recap, now we have three cylinders: a large disk and two identical cylinders. Since the total mass of the system was given, we calculated the mass of each cylinder by first calculating each cylinder's volume. We then calculated the percentage of volume it occupied compared to the total volume of the three cylinders and we determined the appropriate ratio for each cylinder. We then used that ratio compared to the mass of the system to determine the mass of each individual cylinder.

 

Below you can see the dimensions and volume calculations of each cylinder.



As you can see, we determined that the total volume of the three cylinders was 0.0005982 m^3
Next, we found that the large disk represents 86.82%  of the total mass. So if the total mass was  4.887 kg, this means that large disk weighs 4.243 kg. Now, the two cylinders represent 13.18% of the total mass. This means that each cylinder weighs 0.322 kg.

Finally, we can find the Inertia of the system since we now have the mass and radius of each cylinder. The moment of inertia was found using the equation shown above.

• Inertia of the system: 0.0214784 kg*m^2

Procedure (Part 2): Determining the Angular Deceleration and the Frictional Torque of the system
In this part of the lab, we calculated the angular deceleration of the disk. First, we put a piece of green tape on the side of the disk to use as a marker. Next, we captured video of the spinning disk. Then, we analyzed the video on Logger Pro. We tracked the marker as it completed one rotation. We also set the scale of the video by inputting that diameter of the disk on LoggerPro. From here, LoggerPro calculated the velocity of the marker in the x and y direction.

Video Capture

Next, we created a calculated column on LoggerPro to find Vtangential. Then, we created another calculated column to find omega. Finally, we graphed omega vs. time and took a linear fit of the graph; as the slope of the graph equals the angular deceleration.

• Vtangential: sqrt( (Vx^2) + (Vy^2) )
• Omega: Vtan/radius of disk

Angular Deceleration =  -0.07691 rad/sec^2

 

Lastly, the Frictional Torque of the system can be determined using:                                                     Tf = Inertia*Angular Deceleration=  (0.0214784 kg*m^2)*-0.7691 rad/sec^2

In this case, the frictional torque is -0.0165 N*m

 

Procedure (Part 3):

To determine the accuracy of our values for angular deceleration and frictional torque, a 500-gram cart was attached to the central shaft of the apparatus using a long string. The cart was then placed on an inclined slope and we timed how long it took for the cart to travel one meter.

 

First, we solved the problem symbolically. Then we inputted the following data:

• mass of cart= .5 kg
• Angle of incline= 40 degrees
• Frictional torque= -0.0165 N*m
• Inertia= 0.0214784 kg*m^2
• radius= 0.0156 m

In this case, the cart traveled 1 meter in 6.71 seconds. We used this as a benchmark for our actual experiment.

The setup for our experiment was the same but our angle of incline changed.

All the necessary values can be seen below.



 

As you can see, first we drew and FBD of the cart. Then we setup our Fx ,Fy and Torque equations. We solved symbolically for a and then inputted the data. Ultimately, we predicted that the cart would travel 1 meter in 6.2 seconds.

Next, we actually timed the cart as it traveled 1 meter. We conducted three trials and the average time came out to be 7.7 sec.  As you can see our answers are ways off, we assume that the difference can be the result of slow reaction time when using the stopwatch in the experiment.

If we compare our theoretical value of 6.2 seconds to our benchmark, our value seems accurate. The benchmark experiment had an inclination of 40 degrees and the time it took to travel 1 meter was 6.71 sec. In our theoretical experiment, the inclination was 51.7 degrees and the time it took to travel 1 meter was 6.2 sec. It makes sense that a steeper incline would make the cart travel faster, which would result in a faster time.

Conclusion:
In this lab, we determined the inertia, angular deceleration, and frictional torque of a rotating system. We then tested these values for accuracy and came to the conclusion that the uncertainties are very small. However, the experiment in calculating time resulted in a 19.85% error. After going over our work multiple times to catch any mistakes, we came to the conclusion that the biggest source of error is the result of slow reaction time when using a stopwatch. Other than that the methods used in calculating the inertia, angular deceleration, and frictional torque of a rotating system are systematically correct and any uncertainty is very minimal to the point that it will not affect the final result.