22-May-2017: Lab 17: Physics 4A Lab

Lab 17: Physics 4A Lab
Kevin Nguyen
Lab Partners: Kevin Tran, Jose Rodriguez
Date of Lab Performed: 22-May-2017

Statement: The purpose of this lab is to find the moment of inertia of the right triangular thin plate around its center of mass, for both of its perpendicular orientations.

Theory/ Introduction:
For this lab, we used two approaches to find the moment of inertia of the triangular masses.

For the first approach, we used the idea that we can subtract the inertia of both the triangle and disk from the inertia of the disk in order to find the inertia of the triangle.

Each moment of inertia is defined by this equation.

In this equation, "mg" represents the weight of the hanging mass (in Newtons), "r" represents the radius of the torque pulley (in meters), and alpha average represents the sum of the absolute value of the angular acceleration of an ascending and descending hanging mass.

In the second approach to find the moment of inertia for the right triangular thin plate around its center of mass, we used the parallel axis theorem, which states that

Since it is easier to find the moment of inertia of one vertical end of the triangle rather than the inertia around the center mass of the triangle, we do this

Summary of the apparatus:

First, we used the apparatus already set up in order to find the inertia of the top disk since it is the only one that is spinning in the experiment. 

We measured the mass of the hanging mass, the radius of the torque pulley, and angular acceleration ascending and descending since those measurements are crucial for solving for the inertia of the disk and the inertia of the disk and triangle. We took measurements of the angular acceleration by running the logger program. We found those values by using an angular velocity vs time graph and finding the slope of the positive line and negative line.

Graph of the disk alone
Angular acceleration up = 5.775 rad/sec^2
Angular acceleration down = -6.494 rad/sec^2

After finding the inertia of the disk, we placed the thin triangular plate in two orientations shown below and found the moment of inertia for both of them.

The longer side of the triangle is the base

Graph where the longer side is the base
Angular acceleration up = 3.698 rad/sec^2
Angular acceleration down = -4.142 rad/sec^2

The shorter side of the triangle is the base

Graph where shorter side is the base
Angular acceleration up = 4.789 rad/sec^2
Angular acceleration down = -5.357 rad/sec^2 

After taking necessary measurements, we plugged the values into our equations.

For the moment of inertia of the shorter base, we got the value 2.059 x 10^-4 kgm^2. For the moment of inertia of the longer base, we got the value 4.169 x 10^-4 kgm^2.

For the second approach, we first need to derive an equation for the inertia of a right triangle around the vertical end. Before calculations, we measured the dimensions of the triangular plate since we need those values to calculate the moment of inertia of the perpendicularly oriented triangles. The derivation can be found here. 

After finding that the moment of inertia of the oriented triangles is (1/18)MB^2, we use this equation to solve for the inertia of the perpendicularly oriented triangles.

The moment of inertia of the triangle with the short base is 2.528 x 10^-4 kgm^2 and the moment of inertia of the triangle with the long base is 5.840 x 10^-4 kgm^2. 

Conclusion

In the end, the values of inertia for the triangles did not match each other although they both do agree that the triangle with the longer end as the base had a larger moment of inertia. The moment of inertia for the triangle with the short base found in the first and second approach differ by nearly 20%. The moment of inertia for the triangle with the long base found in the first and second approach differ by 33%. This was found by using the equation 

Equation for percent difference

There are multiple sources of uncertainty that may have gave us a large percent difference between the results. 

First, there is air resistance acting on the hanging mass as it ascends and descends, which could have lowered the angular acceleration of the apparatus. 

Second, there is also air resistance acting on the triangle as the apparatus spun, which also lowered the angular acceleration of the system,

Third, since we assumed that there was no friction in this experiment, the values of angular acceleration may have been lower than what it should have been.