Significance of the area under a force–displacement graph.

The idea

The area under a force–displacement graph represents the work done by the force. Work is the energy transferred when a force moves its point of application through a displacement in the direction of the force. For a constant force, work W = Fs cos θ, and the area of a rectangle on an F–s graph gives F × s. For a variable force, the area under the curve must be found by splitting the graph into strips, counting squares or integrating F with respect to s. For example, a spring obeying Hooke's law has F = kx, so the area under its straight-line graph is ½kx², the elastic potential energy stored. The area is a signed quantity: area below the displacement axis represents negative work, meaning energy is transferred away from the object. Always check the axes and units before calculating area.

What earns the marks

• States that the area under a force–displacement graph equals the work done by the force.
• Explains that work done equals energy transferred, so the area also represents the energy transferred to or from the object.
• For a constant force, calculates area as a rectangle: area = force × displacement.
• For a variable force, finds area by counting squares, using the trapezium rule or integrating the force–displacement relationship.
• Interprets area below the displacement axis as negative work done, indicating energy transferred away from the object.
• Applies the result to a spring obeying Hooke's law, where the triangular area gives elastic potential energy ½kx².

Common mistakes and how to put them right

• Confusing the area under a force–displacement graph with the gradient.

  Put it right by recalling that area = work done and gradient = force constant for a spring.

• Using the maximum force instead of the average force for a variable force.

  Put it right: For a linear variation the average force is half the maximum, but for a non-linear graph the area must be found by squares or integration.

• Ignoring the sign of the area when the force acts opposite to the displacement.

  Put it right: Area below the axis represents negative work done, not a magnitude to be made positive.

• Forgetting to convert units such as cm to m or N to kN before calculating area, which gives an incorrect value for work done.

Exam tips

• Always label the axes and state the unit of area, for example N m or J, before or after calculating the area.
• When a graph is not a simple shape, count squares within the curve and multiply by the area represented by one square, showing your working clearly.
• If the force–displacement graph is a straight line through the origin, use the triangle area formula ½ × base × height rather than counting squares.
• Check whether the question asks for work done on the object or by the object, and use the sign of the area to support your answer.