22-May-2017: Lab 18: A Lab Problem - Moment of Inertia and Frictional Torque

Lab 18: A Lab Problem - Moment of Inertia and Frictional Torque

Author: Tian Cih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: May 17th & 22th, 2017

Purpose of the lab: 
The purpose of this lab is to determine the moment of inertia given a large metal disk on a central shaft. After we have the results, we calculate how long it should take for the cart to travel 1 meter from rest. Compare predicted values with experimental values and hopefully we will see that these figures match.

Introduction/Theory:

Part 1:

What we are given is the total mass of disk plus shaft. Since the effectively rotating part is the disk only without shaft, we need to make measurements to calculate the actual volume of the rotating part (see how much it matters comparing to the total volume) so that we know how much mass the disk occupies. In order to determine angular acceleration, we look at any arbitrary point at the edge of the disk.
Frictional torque
Part 2:
We connect the apparatus with a 500-gram(hopefully) dynamics cart. Before we carry out part 2, we calculate how long it should take for the cart to travel 1 meter from rest along the track assuming that it is leaned 40 degrees above the ground (predicted value).

Apparatus/Experimental setup:
Apparatus: Metal disk on a central shaft, calipers, iPhone for video capture, tape for labeling any arbitrary point chosen at the edge of the disk, MacBook with LoggerPro, cart, air track, string, books or something else to stabilize the angle of the track leaning above the ground.

For part 1, we need to make measurements to the disk and the shaft (diameter/length). By calculating the volumes separately (we see shaft as solid cylinders and the rotating part as a solid disk) and distributing the total mass equally, we can have a great approximation of MoI of the rotating part of apparatus.
A video is captured for future analysis on angular acceleration of the disk as it slows down. We trim the video and only need the last part of the video. but when we plot out the dots, it is pretty obvious that the distance between the dots starts to increase, which is an indication of a slow-down motion.

Then we get a graph of omega vs. time and we linear fit it to get the angular acceleration.
Part 2: Angular acceleration (alpha-f) is obtained by applying linear fit to omega vs. time graph. Since total torque minus frictional torque leads us to the "actual" acceleration (see calculations ini the following sections), we can thus have a great idea of how long it will take for a 0.5 kilogram cart to slide along the track for 1 meter.

Measured data:

Mass of the disk + shaft = 4.802 kilograms
Diameter of the disk = 19.594 centimeters
Width of the disk = 2.37 centimeters
Length of the shaft for each side = 5.65 centimeters, 5.54 centimeters
Diameter of the shaft for each side = 3.13 centimeters
Volume of the disk = pi * (19.594/2)^2 * 2.37 = 714.64 cm^3
Volume of the shafts = pi * (3.13/2)^2 * (5.65 + 5.54) = 86.1 cm^3
Calculated "actual" mass for the rotating part = 4.802 * 714.64/(714.64+86.1) = 4.286 kilograms

Analysis:

Here is the calculation of the prediction:

Conclusion:
In this lab, we use camera to find out the angular acceleration, then use the calculated acceleration, we compute the predict time. Then we use cart to find the real time and average it, which is very close.