11.7:
Work-Energy Theorem for Rotational Motion
Consider a rigid body, for instance a fidget spinner. If an external force is applied on it such that it rotates from θ1 to θ2, about a fixed axis, then the total work done on the body equals the integration of the product of the net torque on it and its angular displacement.
Since the net torque on any rigid body is equal to the moment of inertia times angular acceleration, substituting the value of torque in the work expression and using the chain rule, dω/dt can be expressed as dθ/dt multiplied by dω/dθ.
Now, dθ/dt is the angular velocity, ω. Therefore, the integrand in the equation equals Iω.
Integrating within the limits of initial and final angular velocities, the net work done by the external force to rotate the rigid object about a fixed axis equals the change in the object’s rotational kinetic energy.
This is the expression of the work-energy theorem for a rotating rigid body.
11.7:
Work-Energy Theorem for Rotational Motion
The work-energy theorem for rotational motion is analogous to the work-energy theorem in translational motion. It states that the net work done by an external force to rotate a rigid body equals the change in the object's rotational kinetic energy. The power delivered is simply the time derivative of the work done; therefore, power is the dot product of torque and angular velocity. This relation is analogous to power in translational motion, which is given by the dot product of force and velocity. It is assumed that frictional force is absent here; however, this is not always the case for a real system. For example, an airplane's engine does work to set the propeller into a spinning motion. However, air friction and the friction between the mechanical parts of the engine lead to inevitable losses, as the work done by the engine translates into the change in rotational kinetic energy of the propeller. Therefore, even after the propeller gains the final desired angular velocity (and hence the desired rotational kinetic energy), the engine still needs to work to balance the opposing forces, which could otherwise slow down the spinning propeller.
This text is adapted from Openstax, University Physics Volume 1, Section 10.8: Work and Power for Rotational Motion.