Rotational work and power

A torque does work when it turns an object through an angle. For a constant torque $$T$$ acting in the direction of rotation, the work done through an angular displacement $$\theta$$ is:

Here, $$W$$ is work in joules, $$T$$ is torque in newton metres, and $$\theta$$ must be measured in radians. A larger torque or a greater angular displacement means more work is done.

Rotational power

Power is the rate at which work is done. At angular speed $$\omega$$, the instantaneous power transferred by a torque acting in the direction of rotation is:

Angular speed $$\omega$$ is measured in radians per second, and power $$P$$ is measured in watts. This relationship applies instantaneously: use the torque and angular speed at the same moment.

Power dissipated by friction

A frictional torque opposes rotation. Work must be done against it, transferring mechanical energy to thermal energy. If the frictional torque has magnitude $$T_f$$ and the angular speed is $$\omega$$, the rate of this energy transfer is:

A driving torque must therefore provide for friction as well as any other torque required by the motion or the load. Some of the power supplied to the machinery is dissipated by friction rather than transferred to the load.