Work, energy and torque in circular motion

This page has two parts. First, energy: what work does the centripetal force do? Then rotation: what makes something start turning, and how do we measure the turning effect of a force?

Work done in uniform circular motion

Work is $W = Fs\cos\theta$, where $\theta$ is the angle between the force and the displacement. In uniform circular motion, the centripetal force points to the centre while the object moves along the tangent. The angle between them is always 90°, and $\cos90° = 0$, so:

$W_{\text{centripetal}} = 0$

That fits what we see. The centripetal force changes the direction of motion but not the speed, so kinetic energy, $K = \tfrac12 mv^2$, stays constant. By the work–energy theorem, no change in kinetic energy means no net work. A satellite in a circular orbit can go round for years with gravity doing no work on it.

In a horizontal circle, the total energy is constant: $K = \tfrac12 mv^2$, unchanged, and height doesn't change either. If friction or drag acts (a car on a roundabout), the engine must do work against it to keep the speed constant, but that is a tangential force, not the centripetal one.

Energy in a vertical circle

In a vertical circle, gravity has a component along the path, so it does do work: the object slows going up and speeds up coming down. With no friction, mechanical energy is conserved, so between the top and bottom of a loop of radius $r$ (a height difference of $2r$):

$\tfrac12 mv_{\text{bottom}}^2 = \tfrac12 mv_{\text{top}}^2 + mg(2r)$

Torque: the turning effect of a force

A force makes an object rotate when its line of action doesn't pass through the pivot (the axis of rotation). The turning effect is called torque, $\tau$ (Greek "tau"):

$\tau = r_\perp F = rF\sin\theta$

Here $r$ is the distance from the pivot to where the force acts and $\theta$ is the angle between the force and the line from the pivot. There are two ways to think about the $\sin\theta$:

• only the component of the force perpendicular to the spanner, $F\sin\theta$, turns it; or
• the perpendicular distance from the pivot to the line of action of the force is $r_\perp = r\sin\theta$.

Both give the same answer. Torque is greatest when you push at 90° to the handle and zero when you push straight along it (towards or away from the pivot).

The unit of torque is the newton metre (N m). That's dimensionally the same as the joule, but torque is not energy, so never write J for a torque. Torque has a direction too: clockwise or anticlockwise around the pivot.

Torque explains everyday design: door handles are on the edge away from the hinges; long-handled bolt cutters and wheelbarrows let small forces produce big torques; a bike's gears and pedal cranks are all about torque. And the reason a pipe "cheater bar" helps with a stuck nut is that it increases $r$.

Balanced torques

If the clockwise and anticlockwise torques about a pivot are equal, there is no net torque and the object doesn't start to rotate (or keeps rotating at a constant rate). This is the principle behind a seesaw, a crane's counterweight and a set of balance scales.

Torque is to rotation what force is to straight-line motion. A net force causes a linear acceleration; a net torque causes an angular acceleration. You don't need the rotational version of Newton's second law for the HSC, but you should be able to explain qualitatively that a bigger net torque makes something spin up faster.