Answer
Deliver a known amount of kinetic energy to the object, and measure its rate of rotation.
Work Step by Step
Attach the unknown object to the shaft of a motor, of known power P. Run the motor for a time t. The work done by the motor is Pt, and by the work-energy theorem, this is the increase in the object’s kinetic energy.
The kinetic energy is $\frac{1}{2}I\omega^2$.
$$Pt=\frac{1}{2}I\omega^2$$
After measuring the angular speed $\omega$, one may then solve for the unknown moment of inertia.
$$I=\frac{2Pt}{\omega^2}$$
