Work from a force–displacement graph

Intermediate

Area under the F–x graph equals work.

On an F–x graph the area under the curve is the work. Triangles and rectangles give simple results; areas below the axis count as negative.

Worked example

F rising from 0 to 10 N over 2 m: W = ½·10·2 = 10 J.

Common mistake

Plot against displacement, not time.

In more detail

The work done by a variable force equals the area under a force-displacement graph. This works for both constant and changing forces.

Spring testing uses force-extension graphs to find the work stored in the spring.
Engineers estimate energy needed to deform a material from its stress-strain graph.
A physicist estimates the energy stored in a spring by finding the area under its force-extension graph.

What if the area is below the axis?

The work is negative, since the force acts opposite to the displacement.

Why is the area the work?

Because the work is the integral of force over displacement, which is the area under the curve.

What if the force graph is curved?

Count squares under the curve, or integrate the force over the displacement, to get the work done.