2015-May-11 Lab 18: Moment of Inertia and Frictional Torque

PURPOSE:  To find the moment of inertia in a system and calculate its frictional torque using anguelar acceleration; to set up a system that can use tension in both a torque equation and a force equation to predict linear acceleration of a cart.

PROCEDURE: 

We first found the inertia of a system made up of three circular disks, one large metal disk on a central shaft with two smaller disks on each side, by finding the moments of inertia of each individual disk and adding them together.

• By finding the width and diameter of each disk, we were able to calculate the volume of each one, and then add the three to find total volume.
• We then took the ratio of the volume of one disk to total volume, and set it equal to the ratio of the mass of that disk to total mass (given), to find the mass of that particular disk.

By finding the masses of each individual disk, we were able to calculate moments of inertia for each disk, then add them to get the moment of inertia for the system.  Inertia for a solid cylinder around its axis is 1/2mr^2:

Total inertia for the system is 0.02048 kg m^2

Next, we found the frictional torque of the system by setting frictional torque equal to inertia of the system times its angular deceleration.  To find angular deceleration, we used video capture on the spinning system, and marked each time a piece of tape made one full rotation.

Set up of video capturing angular acceleration.

Set the origin to the axis of rotation and marked points (at the top) when piece of tape rotated each time around.

These points gave us a graph of position (theta) vs. time (seconds), which we placed a curve fit to to give us the equation of position, theta = -0.516t^2 + 10.59t - 26.84.

By taking the derivative of this twice, we were able to find angular acceleration:

Then, multiplying inertia for the system by angular acceleration (0.02048 kg m^2 x -1.0322 rad/s^2), we get a frictional torque of -0.0211 N m.

We then used this data to connect a cart to the system by a string, and predicted how much time it would take for the cart to travel 1 meter from rest.

Set up for the second part of our experiment.

We did this by setting up a free body diagram of the cart attached to the system.  We found the sum of forces in the x-direction, assuming the track was angled at 40 degrees with the horizontal.  We also found the equation for torque, which was torque of the string (distance r from the axis), minus frictional torque, which was equal to inertia times angular acceleration.

Solving for T (tension) in each system, we then set the two equations equal to each other to find linear acceleration.  *Note: we replaced angular acceleration in the torque equation with linear acceleration divided by radius.  *Since the string was wrapped around the smaller cylinder, we used that radius.  *Mass of cart was measured as 0.497 kg.

Linear acceleration came out to be 0.01788 m/s^2, which we then plugged into a kinematics equation, with distance as 1 meter, to find the time.

Prediction for the time it takes the cart to travel 1 meter down a ramp angled at 40 degrees.

After finding these calculations, we ran the experiment by releasing the cart from rest and timing (with a stopwatch) how long it took for the cart to travel 1 meter down the ramp.  We also made sure the ramp was angled at 40 degrees with the ground.  

We ran our experiment twice, with the first time resulting in 11.86 seconds, the second in 10.58 seconds, and an average of 11.22 seconds.

Our error of uncertainty was (10.576-11.22)/11.22 x 100% = 5.74%

CONCLUSION:

This experiment allowed us to find the total inertia of a system by adding up individual moments of inertia.  We then found the angular acceleration of the system, and used the equation Torque = Inertia x angular acceleration to calculate frictional torque.  By connecting a cart rolling down a ramp to the system, we solved for the tension force in two equations: one for the sum of forces in the x-direction of our cart and one for the sum of torques in the rotating system.  This allowed us to solve for linear acceleration, and then predict the time it took the cart to move 1 meter.

Our error of uncertainty came out to be 5.74%.  The second trial was a lot closer to the theoretical time, and perhaps if we had done a third trial our results may have been close again. making our average time more accurate.  Error could have resulted from timing the experiment, because it was difficult to stop the stopwatch at the exact moment the cart passed 1 meter.  It also could have been from any slight error in measuring the initial dimensions of each cylinder.  Another source of error was from marking the position of the tape each time it made a revolution in the video, because the frames that the video stopped at did not always match up to exactly one full rotation.