December 1 2014: Period of Oscillation

Purpose: To find the period of oscillation for two objects: a semi-circle and a triangle pivoted at different locations.

These two pictures show the apparatus that we used. First is a semicircle pivoted in the middle of the long side, we also did from the top of the curved side. Second is a triangle pivoted from one edge, we also did from the middle of one of the sides.

Explanation: First we are going to find the moment of inertia of each object pivoted at different locations by hand. From that we are going to calculate the period of the system from small oscillation angles. We are going to use two objects: a semi-circle and a triangle, and we are going to have two pivot points for each object.

 This picture shows the calculations to find the center of mass of the semi-circle, we found it to be 4R/3pi.

 This picture shows the calculations that we used to find the moment of inertia of the object pivoted at the two points. We got the long side to be 1/2MR^2, and the curved side to be 1/2MR^2+(1-24/9pi)MR^2.

This picture shows the calculations that we did in order to find the period of oscillation for each pivot point. From the long side we got 0.603s and from the curved side 0.59s

 This picture shows the period of oscillation for the long side by using LoggerPro. We found it to be 0.5994s.

This picture shows the period of oscillation for the curved side by using LoggerPro. We found it to be 0.5991s.

Here, we compared the results that we got by hand and by actually performing the experiment. The % error in both were less than 1%, which was the expected.

 This picture shows the calculations that we did to find the moment of inertia for the triangle at different points. We found it to be 1/24MB^2+1/2MH^2. The bottom part of the picture shows the period that we calculated when the pivot is at one end. We found it to be 0.77s.

This picture shows the moment of inertia when the triangle was pivoted at one of the sides. We found it to be MB^2/24+1/6MH^2. We also calculated the period of oscillation for this case to be 0.74s.