May-17-2015 Lab 18: Finding the moment of inertia of a uniform triangle about its center of mass.

Purpose of this lab:  We need to determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.
Theory:

Part 1: 

For this kind of perpendicular orientations of the triangle, we mount the triangle on a holder and disk. The upper disk floats on a cushion of air.

A string is wrapped around a pulley on top of and attached to the disk. The tension in the string exerts a torque on the pulley-disk combination. By measuring the angular acceleration of the system we can determine the moment of inertia of the system.


Note : Because there is some frictional torque in the system.(the rotating disk is not truly frictionless and there is some mass in the frictionless pulley, which we can approximate as a frictional torque.)

So we have to measure the angular acceleration of disk without triangle to determine the moment of inertia of disk without triangle I'.


Here is the angular velocity vs. time graph of disk without triangle:

Here is the angular velocity vs. time graph of disk with triangle:

After we collected data, we got :

Newton's second law would lead us to get the formula:

Then we could calculate the moment of inertia. Here is calculation:

We calculated the I(disk with triangle) = 0.0028551 N*m.
We calculated the I'(disk without triangle) = 0.0026006 N*m.
We calculated the I(around cm) = I(disk with triangle) -  I'(disk without triangle) = 0.0002545 N*m by collecting data.

Then we measured the mass of triangle M = 455 g. 

a = 9.8 cm.

b = 14.85 cm.


For this kind of perpendicular orientations of the triangle, a is radius.

By definition, We got I(parallel axis) = 1/6 * M * a.



From approach-theory:   
I(parallel axis) = I'(around cm) + M*(d(parallel axis displacement))^2.

We got the I'(around cm) = 0.000242768 N*m by measuring. 

We calculated the uncertainty between the moment of inertia what we calculated and what we measured is 4.8%.

Part 2: 

For this kind of perpendicular orientations of the triangle,
repeat all of steps from part 1.





After we collected data, we determined the moment of inertia. Here is calculation:

For this kind of perpendicular orientations of the triangle, b is radius. 

By definition, We got I(parallel axis) = 1/6 * M * b.

From approach-theory:   
I(parallel axis) = I'(around cm) + M*(d(parallel axis displacement))^2.

We got the I'(around cm) = 0.000557432 N*m by measuring. 

However,
We got the I'(disk without triangle) = 0.0026006 N*m from part 1.
We calculated the I(disk with triangle) = 0.00314948 N*m.

We calculated the I(around cm) = I(disk with triangle) -  I'(disk without triangle) = 0.0005488 N*m by collecting data.
For part 2, We calculated the uncertainty between the moment of inertia what we calculated and what we measured is only 1.53%.

Conclusion: 

For each part (different perpendicular orientations of the triangle) , we found the moment of inertia of a right triangular thin plate around its center of mass by using formula and our measurements. we compared the results between the what we measured and what we predicted to test the data we collected. For part 1, 4.8% uncertainty is higher than 1.53% uncertainty from part 2. That's mean our measurements is good.  I think the only reason to make different result is that we may made big mistake for determining the angular acceleration with triangle on part 1.