Work done equals the area under a force-displacement graph

Work, Energy & Power — AQA A-Level Physics

When force is not constant, $W = Fs cannot be$ used directly.
Plot force against displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).; the area under the graph gives the total work doneEnergy transferred when a force moves an object. In electrical circuits, W = QV (chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). times potential difference)..
Split the area into rectangles, triangles and trapeziums to calculate it.
If the graph is curved, estimate the area by counting squares.
A variable force that increases produces increasing accelerationThe rate of change of velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. A vector quantity. Measured in m s⁻². (and vice versa).

Worked Example

A force varies linearly from 250 N to 100 N over a displacement of 80 m. Calculate the work done.

Show Solution

Split the area into a rectangle and a triangle

Rectangle (base): height = 100 N, width = 80 m.

Triangle (top): height = 250 - 100 = 150 N, $base = 80 m$.

Calculate each area

$$A_{\text{rect}} = 80 \times 100 = 8000 \text{ J}$$ $$A_{\text{tri}} = \frac{1}{2} \times 80 \times 150 = 6000 \text{ J}$$

Total work done

$$W = 8000 + 6000 = 14000 \text{ J}$$

Answer

$W = 14000$ J

Examiner Tips and Tricks

Always check the units on the axes.
Force might be in kN and displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). in km.
Convert to SI units (N and m) before calculating work in joules.