18-May 2015 Parallel Axis Theorm

Purpose: To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle, and the "holder."

Theory: The Parallel axis theorem states that the moment of inertia (I) around a parallel axis (Ipa) is equal to the moment of inertia around the center of mass plus the mass of the object multiplied by the displacement from the parallel axis (d) squared. By determining the moment of inertia of the disk, pulley, and "holder" we can subtract that from the moment of inertia calculated when we have the triangle included to fine the moment of inertia of just the triangle. If we compare the two calculations they should be fairly close with minimal difference error

Procedure: Have the apparatus spin without the triangle, but with the holder, to discover the moment of inertia of the system (use a 25 gram hanging mass). Average three angular accelerations (both up and down) and use the formula I= (Mass(hanging)*gravity*radius/avg angular acceleration)-Mass*radius^2 to determine the moment of inertia. Run the experiment twice more, with the triangle included, alternating the base and height

Ex: Triangle Max Height (Left) Triangle Max Length (Right)

Calculate this moment of inertia, and then subtract the previous calculated moment of inertia to find the moment of inertia of just the triangle. Finally calculate the moment of inertia of the triangle with the parallel axis theorem (Ipa=Icm+md^2), and compare it with your previous calculation.

Apparatus

Graph of angular momentum.

Results of Angular acceleration

	1 rad/s^2	2 rad/s^2	3 rad/s^2	Average
No Triangle up	-3.421	-3.416	-3.475
No Triangle down	3.059	3.038	3.081	3.248333
Triangle Max height up	-2.803	-2.824	-2.803
Triangle Max height down	2.514	2.488	2.514	2.657667
Triangle Max Length up	-2.156	-2.137	-2.175
Triangle Max Length down	1.914	1.912	1.92	2.035667

*Moment of inertia no triangle, 0.000939 kg*m

Sample calculation for Moment of inertia Triangle

*Moment of inertia of Triangle Max height, 0.00021 kg*m

*Moment of inertia of Triangle Max length, 0.000562 kg*m

Dimensions of the triangle

*Since the center of mass of a right triangle is always 1/3 of the base from the 90 degree angle, there is no need to calculate center of mass.

Sample calculation for Moment of inertia Triangle Using parallel axis theorem

*Triangle max height 0.000005056 kg*m, error 97%

*Triangle max length 0.000001137 kg*m, error 99%

Conclusion 

*Because of the large error, I believe I must have measured something wrong.

*Perhaps I was mistaken in my assumption that the distance from the center of mass was 1/3 of the base, or where the point of rotation was on the triangle (again a mistake in measurement).

*I did not take into account friction.