Work and Energy

Work is done when a force moves the body it acts on. The precise statement is that work is the scalar product of the force and the displacement, $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$, where $\theta$ is the angle between the force and the displacement. Because it is a scalar product, work is a scalar quantity, so it has no direction. Its SI unit is the newton metre, renamed the joule, $1\text{ J} = 1\text{ N m}$.

The sign of the work follows directly from the cosine. If $\theta < 90^\circ$ the work is positive and the force is helping the motion; if $\theta = 90^\circ$ the force does no work at all; if $\theta > 90^\circ$ the work is negative and the force is opposing the motion. At $\theta = 0^\circ$ the force and displacement are parallel and the work is maximum, $W = Fd$; at $\theta = 180^\circ$ it is maximum negative. This is why a centripetal force never does work: it always acts at $90^\circ$ to the velocity.

When the force varies along the path, the product cannot be taken in one step. The displacement is split into many small intervals, the force is treated as constant inside each, and the contributions are added: $W = \sum F_i\cos\theta_i,\Delta d_i$. As the intervals shrink this sum becomes the area under the graph of $F\cos\theta$ against $d$. The area under a force-displacement curve is therefore the work done. A force is called conservative when the work it does depends only on the end points (gravity, electric and magnetic fields all qualify), and non-conservative when the path matters. Friction, air resistance, tension and the thrust of a rocket are non-conservative, and the work done by a non-conservative force around a closed path is not zero.