Work Done by a Variable Force

The simple formula $W = Fd\cos\theta$ applies only when the force stays exactly constant, in both magnitude and direction, throughout the whole displacement. In a great many real situations this is not true -- the restoring force of a stretched spring grows steadily larger the further it is stretched; the gravitational pull between two masses grows weaker with increasing separation; the drag force on a fast-moving body depends on its instantaneous speed. To handle a force $F(x)$ that varies with position along a straight line, work is built up from many small steps.

Imagine dividing the total displacement, from $x_1$ to $x_2$, into a very large number of very small steps, each of width $\Delta x$, so small that the force can be treated as very nearly constant over any one such step. The work done over one such small step is then approximately $F(x)\,\Delta x$, and the total work done over the whole displacement is the sum of all these small contributions:

As the width $\Delta x$ of each step is made smaller and smaller, this sum becomes, in the limit, exact, and turns into a definite integral:

Geometrically, this integral is exactly the area under the graph of force plotted against displacement, between $x_1$ and $x_2$ -- a fact that is often the fastest way to estimate the work done by a variable force whose exact functional form is not known, but whose graph is (for instance, a graph of force read off from an experiment).

Worked illustration -- the spring force. An ideal spring obeying Hooke's law exerts a restoring force of magnitude $F(x) = kx$ when stretched (or compressed) a distance $x$ from its natural length, where $k$ is the spring's force constant; to stretch the spring, an external agent must apply a force that is, at every instant, equal in magnitude to this restoring force, so the work done by the external agent in stretching the spring from its natural length to an extension $x_0$ is …