26-Nov-2014: Period of Oscillation of semi-circle

Purpose:
The purpose of this lab is to determine the period of oscillation of a semi circle when the pivot point is located in the center on its flat side and on its round side.

Apparatus:
Our apparatus consisted of a semi circle cut from a Styrofoam board suspended from a bar, tape, and a photo gate. The experiment would be conducted twice, changing the pivot points for each part.

Experiment:
Before conducting the actual experiment we need to make predictions of the outcome. In order to do this we need to use various concept that we've used in the past which include: Moments of Inertia, the parallel axis theorem, and simple harmonic motion.

Our first step was to find the moment of inertia of the semi circle. In order to find its moment of inertia we must use the parallel axis theorem. We derived the moment of inertia as follows:

Ishape = Icenter mass + MR^2

After integrating our function we find that the moments of inertia are:

Flat side: I = (4/3π)MR^2

Round side: I = π M (R- (4/3)R)^2

After finding the moment we can proceed to find the angular speed of the oscillating semi circle at any given time using:

Torque = Inertia * angular acceleration = Force * radius

[ (1/2) MR^2 ] α = mass * gravity * radius cos θ

α = ω / t

ω^2 = (2*g*r) / (R^2)

Period = 2π / ω

Here are our calculations:

 Flat side: Period prediction = 0.857 seconds
Experimental Period = 0.843 seconds

Round side: Period prediction = 0.830 seconds
Experimental Period = 0.821

Conclusion:
When changing the pivot of our semi circle we see a slight change in its oscillation. This caused  because of the motion of the center mass.