Nov-14-2016: Lab 17 - Moment of Inertia of a Uniform Triangle

LAB 17 - Moment of Inertia of a Uniform Triangle

Tony Wu, Michell Kuang

November 14, 2016

Lab Goal: 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/Introduction: In this lab, we use the parallel axis theorem states to find the moment of inertia about two different orientations of the triangle.  We used the same apparatus as Lab 16.  The metal circular disks float on a cushion of air, similar to the air track but for rotation.  A hanging mass wraps around a "frictionless" pulley.  The tension on the string exerts a torque on the pull and disk combination.  Unlike Lab 16, this lab has a holder that attaches to the floating disks.  The holder secures the triangular plate at its center of mass.  We know that the center of mass of a uniform triangle is (1/3 x base) and (1/3 x height) from the right angle of the triangle.

Apparatus:

Experimental Procedure: We first measured the angular acceleration of the metal disk and holder combination.  We take the average of the alpha values, alpha_up and alpha_down.  This first run is without the triangular plate.  For the second run, we attach the triangular plate with the long side of the triangle parallel to the horizontal.  The last run has the triangular plate oriented with the long side perpendicular to the horizontal.  We use the alpha values to calculate the moment of inertia of each setup.  The equation we use is the same as the one from Lab 16, I = mgr/alpha - mr^2.  m is the mass of the hanging mass and r is the radius of the pulley.  At the end of the lab, we compare the expected inertia values, I = 1/18 * MB^2, with the experimental values, I = mgr/alpha - mr^2.

The inertia of the triangular plate by itself is derived from calculating the inertia of the whole system, the triangle + holder + disks, minus the inertia of the system without the triangle, holder + disks.

I_everything = I_triangle_by_itself + I_everything_except_triangle

I_triangle_by_itself = I_everything - I_everything_except_triangle

Data and Calculations: 

Conclusions: Our expected and experimental values ended up very close.  Our results were within 0.089% for the triangle oriented with the long side parallel to the horizontal.  We were within 2.73% for the triangle oriented with the long side perpendicular to the horizontal.  

Some sources of error:
- There is some frictional torque in the system, the rotating disk isn't truly frictionless.
- There is some mass in the frictionless pulley that we did not consider.
- The angular acceleration of the system when the mass is descending is not exactly the same as when it is ascending.