# How to Calculate Rotational Work

In physics, one major player in the linear-force game is *work*; in equation form, work equals force times distance, or *W* = *Fs*. Work has a rotational analog. To relate a linear force acting for a certain distance with the idea of rotational work, you relate force to torque (its angular equivalent) and distance to angle.

When force moves an object through a distance, work is done on the object. Similarly, when a torque rotates an object through an angle, work is done. In this example, you work out how much work is done when you rotate a wheel by pulling a string attached to the wheel’s outside edge (see the figure).

*Work* is the amount of force applied to an object multiplied by the distance it’s applied. In this case, a force *F* is applied with the string. Bingo! The string lets you make the handy transition between linear and rotational work. So how much work is done? Use the following equation:

*W* = *Fs*

where *s *is the distance the person pulling the string applies the force over. In this case, the distance *s *equals the radius multiplied by the angle through which the wheel turns,

so you get

right angles to the radius. So you’re left with

When the string is pulled, applying a constant torque that turns the wheel, the work done equals

This makes sense, because linear work is *Fs,* and to convert to rotational work, you convert from force to torque and from distance to angle. The units here are the standard units for work — joules in the MKS (meter-kilogram-second) system.

You have to give the angle in radians for the conversion between linear work and rotational work to come out right.

Say that you have a plane that uses propellers, and you want to determine how much work the plane’s engine does on a propeller when applying a constant torque of 600 newton-meters over 100 revolutions. You start with the work equation in terms of torque:

Plugging the numbers into the equation gives you the work: