Moment of Inertia Questions

1. Trucks can be run on energy stored in a rotating flywheel, with an electric motor getting the flywheel up to its top speed of $200\pi$ rad/s.  One such flywheel is a solid, uniform cylinder with a mass of 500kg and a radius of 1m.  if the truck uses an average power of 8kW, for how many minutes can it operate between charging?   (answer: 10^2 min)   Please explain.
2. Please find the rotational inertia of a wheel that has a kinetic energy of 24000J when rotating at 602rev/min.   (answer: 12.3kg.m^2). Please explain.

Answers
1.

the momentum of inertia I of a uniform cylinder of radius R and mass m is

$I = m*R^2/2 =500*1*1/2 =250 kg*m^2$

The power is equal to the mechanical work over the total time, or the variation of kinetic energy over the total time.

$P = W/t = (Eci-Ecf)/t$

The final Ecf is zero, the initial Eci is

$Eci = I*\omega^2/2$

time $t = Eci/P = I*\omega^2/2/P = 250*(200*\pi)^2/2/8000 =6168 sec =102.8 minutes$

2.

By definition the kinetic energy Ec of a solid having a moment of inertia I and an angular velocity $\omega$ is

$Ec = I*\omega^2/2$

$\omega = 602 *2*\pi/min = 602*2*\pi/60 =10.03*2*\pi = 63 1/s$

$I = 2*Ec/\omega^2 =2*24000/63^2 =12.09 kg*m^2$