Moment of Inertia

­­­Moment of Inertia
1. Abstract
The goal of this study is to understand the transfer of potential energy to kinetic energy of rotation and kinetic energy of translation. The moment of inertia of the cross arm my group measured with the conservation of energy equation is: 0.01044 kg/m2 (with the mass of 15g), 0.01055 kg/m2 (with the mass of 30g), which is kind of similar to the standard magnitude of the moment of inertia of the cross arm: 0.0095 kg/m2 (Gotten by measuring the radius and the mass of the cross arm and use the definition equation of moment inertia). And we also get the moment of inertia for disk: 0.00604 kg/m2, and the ring: 0.00494 kg/m2.
2. Introduction
For that experiment, we use the conservation of energy equation to find the moment of inertia of the cross arm, and then use the definition equation of the moment of inertia to get the exact magnitude for the cross arm, the disk and the ring.

For the first step, finding the moment of inertia of cross arm, we need the conservation of energy equation for the transferring of the energy from the potential energy to the sum of the kinetic energy for both the mass strikes and the cross arm, like equation 1:

Since the mass is falling with uniform acceleration, its final velocity, v, after having fallen through height h can be found by using equation 2:

And the angular velocity of the cross arm is shown by equation 3:

Then we can use the equation 1,2 and 3 to get equation 4:

The M means the mass of the mass strikes, the R means the radius of the cross arm, the g means the acceleration of the gravity, t means the time it cost for the mass strikes to reach the floor, the h means the height of the mass strikes’ start point compared to the ground.
Then, we can use the physical method to measure the moment of inertia of the cross arm and disk with equation 5, and the ring with equation 6:

The R1 and R2 are the inside and outside radius of the ring.
3. Experimental Setup