Background theory for calculating moment of inertia

Physics 1AA3 Team

Background theory for calculating moment of inertia

In this experiment, the gravitational potential energy ( $PE$ ) of a falling mass is converted into the translational kinetic energy ( $KE$ ) of this mass and the rotational kinetic energy of the crossbar system. The conservation of energy principle requires that the initial total energy ( $PE$ plus $KE$ ) be equal to the total final energy ( $PE$ plus $KE$ ), provided frictional forces are negligible.

The initial conditions for the system just before the driving mass is released are:

The system is at rest, so $KE = 0$ .
We define the final (lowest) position of the driving mass $M$ as the zero point for the $PE$ , so the initial $PE$ is $Mgh$ , where $h$ is the initial height of the mass above the final height.
Combining the two points above means the initial total energy is $Mgh$ .

Laying out the final conditions for the system we have:

The driving mass has translational $KE$ , $KE_{trans} = \frac{M v^2}{2}$ , where $v$ is the final speed.

The rotating apparatus has rotational KE, $KE_rot = \frac{I \omega^2}{2}$ , where $\omega$ is the final angular speed.
The driving mass has $PE = 0$ .
Tallying up the energies gives the final total energy: $\frac{M v^2}{2} + \frac{I \omega^2}{2}$ .

Conservation of energy requires

(5) $\begin{equation*} Mgh = \frac{M v^2}{2} + \frac{I \omega^2}{2} \end{equation*}$

be true. If $v$ and $\omega$ can be determined for the system, then Equation (5) can be used to calculate the moment of inertia. To calculate $v$ from $t$ , the time it takes for $M$ to fall the distance $h$ , we use the constant acceleration formulae