May-16-2015 Lab 17: Moment of inertia and frictional torque

For this lab, Given a large metal disk on a central shaft:

To do this lab:
1.  Make appropriate measurements of the rotating part of the apparatus and determine its moment of inertia.

here is our measurements.

w(1) and w(3) are widths of the cylinders. w(2) is width of large disk.

R(1) and R(3) are radius of the cylinders. R(2) is radius of large disk.

Two cylinders have same mass. Total mass is 4.615 Kg.







Before to determine moment of inertia of the rotating part, we have to find the mass of each rotating parts.    Here is calculation for finding mass of each rotating parts:

Then, we could calculate the moment of inertia of whole system:

2. Next step, we need to spin the apparatus and Use video capture to determine its angular deceleration as it slows down. Calculate the frictional torque acting on the apparatus.(because friction exists and can affect the result.)

Before using video capture to take video, we need to stick a tape on the edge of large disk. Because we have to track the apparatus to determine its angular deceleration when it's spinning. 




















After we got the video, we need to set the center of the large disk is the origin first and follow the mark position and point it out.

Then we got the graph.

Red point shows us the mark position on X axis. V(x) could be found.

Blue point shows us the mark position on Y axis. V(y) could be found.



To find the angular deceleration, we have to set up a new column which called Total Velocity to be (V(x)^2 + V(y)^2)^0.5.
There is the graph of V(total) vs. time. After we linear fit it, we will get the linear deceleration a = -0.0533 m/s^2.

Then, we can calculate the frictional torque right now. Here is calculation:

 Using formula, we got angular deceleration alph = - 0.533 rad/s^2

the frictional torque = moment of inertia of system *
angular deceleration

we got the frictional torque = -0.01063 N*m







3. we are going to be connecting this apparatus to a 500-gram dynamics cart. the cart will roll down an inclined track for a distance of 1 meter. Calculate what the time for the cart which from rest to travel 1 meter should be with the actual angle. 

We measured the actual angle is 48 degrees.

we measured three times (7.15 s, 7.19 s, 7.26 s) for how long cart should take for traveling 1 meter from rest, and we got the average time is 7.2 seconds.

After Drawing a force diagram:

We could know three equations. then plug the data we got into those three equations: 

We calculated the acceleration a = 0.0366 m/s^2.
plug this acceleration into the formula  d = 0.5 * acceleration * time^2 by newton laws. (d = 1 m.) then we got t = 7.4 s.
Then we calculated the uncertainty of time between the time we calculated and the time we measured is 2.7%.

Conclusion : 

we found the moment of inertia and frictional torque by step 1 and 2. then we were going to test the data that we found and compare the result we calculated with the result we measured.  we did this lab carefully. However, we have to say that the 2.7% uncertainty is little big for us. Even our average time is not more than 4% off from what we calculated.  I think all measurements are good but the cart, because the mass of cart is not perfectly 500-gram. it may be around 500-gram. But we used 500-gram to calculate.