The Rotational Work-Energy Theorem

A free wheel with a fixed axis can't move its center of mass, so it can't store energy as translational kinetic energy. Any work a motor does on it has nowhere else to go, it all becomes rotational kinetic energy. That's the rotational twin of the work-energy theorem you already know for sliding blocks, just swap force for torque, distance for angle, mass for moment of inertia, and linear speed for angular speed.

The mechanism mirrors the linear case exactly. Net work equals force times distance in a straight line problem; here net rotational work equals torque times angular displacement, W = tau times theta, for a constant torque. That work changes rotational kinetic energy: W_net = (1/2) I omega_f² - (1/2) I omega_i^2. Because I plays the role mass played before, a large I needs more work to reach the same omega, just like a heavy object needs more work to reach the same speed. The theorem works for varying torque too, as long as you use the correct general definition of work as the integral of tau d(theta).

This theorem connects directly to power and to angular momentum, the next ideas in this unit. Rotational power is P = tau times omega, the exact analog of P = F dot v, and it tells you the rate work is being delivered to a spinning object. Where this topic cares about energy, the next one cares about angular momentum, L = I omega, which is conserved when net external torque is zero, even while kinetic energy might change. Keeping these two conserved quantities separate, energy from work and angular momentum from torque impulse, is what the rest of the unit builds on.