Lab 18 moment of inertia and frictional torque

Purpose: In this laboratory we are trying to determine the moment of inertia of the apparatus, and figure out a way to use video capture to determine the angular deceleration as it slows down. And calculate the frictional torque as it slows down.

The purpose of the apparatus is to use it to find the angular deceleration by taking a video of the apparatus spinning, and eventually use the ramp to find the time it takes the cart to travel one meter.

Procedure:
In this lab, the first thing we did was measure the diameters and lengths of the different cylinders that made up the rotating object on the apparatus. Then we found the volume of the two smaller cylinders and of the big disk to find the percent of volume the big disk was compared to the three put together. We then used that information to find the mass the large disk and the smaller cylinders because the mass would be the percent of the volume of each. We found the moment of inertia of the three parts together after finding the mass of each cylinder and the disk. Then we took a video of the apparatus rotating to a stop after giving it an initial push. Using the video analyzer we to marked an initial point on the apparatus and added another point every time the apparatus made a revolution so all our points were on the same spot and marked one revolution of the apparatus. Using the marked points we were able to make a position versus time graph of the points. Then we added a power fit line to get an equation of the position with respect to time. To get the deceleration we took the derivative twice of the position vs. time equation. Using newton's second law and angular acceleration we found the linear acceleration of the cart going down the ramp at the measured angle of the ramp. Plugging the acceleration in a kinematics equation we solved for the time it would take the cart to go a distance of one meter. Then we compared that value to the value we got by measuring the time it took the cart to go down the ramp a distance of one meter.

We measured the diameters and heights of the cylinders and the disks and used the measurements to find the volumes of each. Then we calculated the percent of volume the disk was so that we could find the mass because the mass percent is the same as the percent in volume. So we multiplied the percent to the mass of the whole that read on the three objects together and got the mass of the disk. And we got the mass of the smaller cylinders by subtracting the mass of the disk to the entire mass and divided it by two. The calculations for this are as follows:

To find the moment of inertia of the whole object we had to add the moment of inertia of the individual parts(the two cylinders and the disk). Individually the moment of inertia of a solid disk is 1/2mr^2 and since all three parts were solid disks we added this formula three times with the mass of the individual parts being m and r being the radius of each part. Our calculations were as follows:

Next we recorded a video of the apparatus spinning and coming to a stop, and plotted the points every time it made a revolution and put it on a distance versus time graph on logger pro. We power fitted the line to get an equation.

Using the equation of the line we took the derivative twice so that we could get the angular acceleration of the apparatus.

Using this calculated angular acceleration and converting it to linear acceleration using the equation alpha = a/r where a is the linear acceleration and r is the radius of the large disk. We used newton's second law to calculate the linear acceleration of the cart and the frictional torque of the disk.

Then we calculated the time it would take the cart would take to travel 1 meter using a kinematics equation:

This was our theoretical value so then we found the experimental value by letting the cart go down the ramp 1 meter and timing how long It would take. We did this three times and took the average time of the three.

Conclusion:
According to the data the experimental value for the time it would take the cart to travel 1 meter down the ramp was 7.98 seconds and the theoretical value that was calculated for the cart to travel 1 meter was  8.57 seconds. The values are pretty close but there is some error that could have came from the method that we came up to calculate the time theoretically. For example, when we were marking the point of one revolution the disk would take sometime it was hard to see the mark we made on the disk so we had to guess a little when the mark was at the point of revolution. this could have altered the acceleration a little. Also, the time we calculated experimentally of 7.98 seconds I feel was a little off we should have ran more trials to get a value that was more accurate.