11.6:
Work and Power for Rotational Motion
Consider a waterwheel in a pond. The falling water from the pipe exerts a force on the waterwheel making an angle Φ with the position vector, then the work done by the force to rotate the wheel through a small angle, dθ, is FsinΦds.
Since the arc ds is equal to r times dθ, the work done is the product of FsinΦ and rdθ.
Recall that the magnitude of torque equals rFsinΦ. Therefore, on substituting, the expression for work done equals to τ times dθ.
In general, if a rigid body rotates from θ1 to θ2, the total work done on the body equals to the integration of the product of net torque and angular displacement.
The instantaneous power delivered by a force is the rate at which the work is done by the force to rotate an object about its fixed axis having a constant torque. Therefore, power is expressed as τ times dθ/dt. Since dθ/dt is the angular velocity ω of the wheel, power equals to torque times angular velocity.
11.6:
Work and Power for Rotational Motion
Work and power in rotational motion are completely analogous to work and power in translational motion. The total work done to rotate a rigid body through an angle 'θ' about a fixed axis is the sum of the torques integrated over the angular displacement. Hence, torque and angular displacement in rotational motion are analogous to force and linear displacement in translational motion, respectively.
Similarly, the power delivered to a system that is rotating about a fixed axis is given by the torque multiplied by the angular velocity. For example, when a person tries to turn a merry-go-round, torque must be generated by applying force at a distance away from the central axis. Hence, the power delivered depends on the torque generated due to the applied force and the angular velocity of the merry-go-round.
Work in rotational motion is measured in joules. Since power is the rate at which work is done, it can be measured in units of joules per second, which is more commonly referred to as watt. These are SI units of work (J) and power (W or J/s).
This text is adapted from Openstax, University Physics Volume 1, Section 10.8: Work and Power for Rotational Motion.