Calculating Work

Most work calculations in mechanics fall into three families: forces that never change, springs, and forces that push sideways. Learn these and the integral almost never needs doing by hand.

A constant force#

If $\vec F$ is constant in size and direction, it comes out of the integral: $W = \vec F\cdot\int d\vec r = \vec F\cdot\vec r$, where $\vec r$ is the total displacement.

Gravity near the Earth

The weight $mg$ is constant near the surface. If a particle moves from A to B, with AB making angle θ with the vertical, then $W = mg(AB)\cos\theta = mgh$, where $h$ is the height descended. Climbing a height h gives $-mgh$. Remarkably, the curve followed between A and B doesn’t matter, and a round trip gives zero.

The spring force#

A block on a smooth table is attached to a spring; x = 0 is the natural length. The spring’s force on the block is $-kx$, which changes as the block moves, so we must integrate. Over a small step dx the work is $-kx\,dx$, and

The pattern: the spring does negative work while you deform it and positive work as it relaxes. A round trip nets zero.

A force perpendicular to the velocity#

If $\vec F \perp \vec v$ at every instant, then $\vec F\cdot d\vec r = \vec F\cdot\vec v\,dt = 0$ throughout. The tension on a ball whirled in a circle, the normal force on a body sliding over a fixed surface and the magnetic force on a moving charge all do zero work.

Work as an area#

For a force along a line, $\int F\,dx$ is the area under the F–x graph, just like distance under a v–t graph.