Work in rotational motion

The work of a constant torque in an angular displacement \(\Delta\phi\) is

\[W = \tau \Delta\phi.\]

If the torque changes with the angular position, we have to divide the motion into very small rotations, in which the torque can be approximated as constant and add the work values \(\Delta W_i\) in such small rotations up:

\[W \approx \sum\limits_i\Delta W_i = \sum\limits_i\tau(\phi_i) \Delta\phi.\]

To be exact, we have to use integration:

\[W = \lim\limits_{\Delta\phi \to 0}\sum\limits_i{\tau(\phi_i) \Delta\phi} = \int\limits_{\phi_1}^{\phi_2}\tau(\phi)\mathrm\phi.\]