Tuesday, May 26, 2015

13-May-2015: Finding the moment of inertia of a uniform triangle

     The purpose of this experiment was to determine the moment of inertia of a uniform right triangle about its center of mass for two perpendicular orientations of the triangle.
     This experiment compares the theoretical and experimental inertia's of the triangle. First, the theoretical calculations were made. To find the moment of inertia about its center of mass, the parallel axis theorem was used. This made the calculations easier since it is easier to calculate moment of inertia about one end of a triangle and we already know where the center of mass is located on a triangle. Below is the derivation of the moment of inertia of the triangle about its center of mass.

     The setup for the experimental part had a triangle mounted on a disk. The disk "floats" on air making it nearly frictionless. A string it attached to a torque pulley, which is also mounted on the disk. A hanging mass is attached to the opposite end of the string and hung over another pulley. A picture of the setup is below. The hanging mass starts at the top and is released. It moves up and down as the disk and triangle spin.


     To find the experimental value, The inertia of the disk plus the triangle holder needed to be compared to the inertia of the disk plus the triangle. Taking the difference of the two values would give the inertia of the triangle. The equation I=mgr/(alpha)-mr^2 was used for the experimental portion. Alpha, angular acceleration, was found using the average of the up and down values of the slope of the angular velocity graph that was produced using Logger Pro. 
     Below is the graph of angular velocity of when the triangle was not mounted on the disk. In this graph, three slopes in both the up and down motion were used. The average was taken of those up and down slopes, and then the average of those two numbers were taken. 

     The triangle's dimensions were measured using a caliper and its mass was measured with a balance. The dimensions and mass of the disk and the torque pulley were also measured with the same tools. They were needed to calculate the inertia. 
     After the angular acceleration was found, simply plugging in the numbers into the equation I=mgr/(alpha)-mr^2 was the next step. The inertia of the disk was added to the inertia of the disk and holder using the angular acceleration measured in each of their trials. Then, the inertia of the disk was added to the inertia of when the triangle was mounted. The difference of the two final values was taken. 
     Using this experimental approach yielded a value for the inertia of the triangle (when it was upright) to be 2.482*10^-4 kg*m^2. Using the equation for the moment of inertia of a triangle about its center of mass, which was derived earlier, and the dimensions of the triangle found with the calipers, gave a theoretical value of 2.442*10^-4 kg*m^2. This resulted in a percent error of 1.6%. A really good result! The exact same approach was used for a second trial, but this time the triangle was placed sideways with its longer side as the base. That trial resulted in better data with an experimental value of 5.587*10^-4 kg*m^2 and a theoretical value of 5.634*10^-4 kg*m^2. A percent error of 0.83%. 
     The experiment was a success and the derived equation for the triangle's moment of inertia about its center of mass was accurate. The data shows that when the triangle is sideways its inertia value is larger. This means that it is harder to rotate as its base gets longer. That agrees with what our intuition would tell us. Sources of error included assumptions that the spinning disk was acting completely frictionless and errors and uncertainties in using calipers to measure dimensions.

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