Area Under a Force-Displacement Graph (AQA A Level Physics): Revision Note

The graph shows how a force varies over a displacement of 80 m.

Calculate the work done.

[3]

Answer:

Step 1: Split the graph into sections

• The work done is the area under the graph

• The total area can be found by splitting the graph into sections A and B

Step 2: Calculate the area of section A

• Section A is a right-angled triangle where the area is 0.5 × base × height

$\mathrm{Area}\mathrm{of}\mathrm{section}\mathrm{A}=0.5\times 80\times (250–100)=6000\mathrm{J}$ [1 mark]

Step 3: Calculate the area of section B

• Section B is a rectangle where the area is base × height

$\mathrm{Area}\mathrm{of}\mathrm{section}\mathrm{B}=80\times 100=8000\mathrm{J}$ [1 mark]

Step 4: Calculate the total work done

• The total work done is the sum of both areas

$\mathrm{Work}\mathrm{done}=6000+8000=14000\mathrm{J}$ [1 mark]

Always check the units on the axes when calculating values from a graph. Sometimes the force will be given in kN or the displacement in km. These must be converted into SI units to calculate the work done in J.

A person is pulling a suitcase through an airport with a rough surface. They apply a force of 150 N over a distance of 12 m. Afterwards, the person gets progressively tired and the applied force is linearly reduced to 60 N. The total distance through which the suitcase has moved is 25 m.

Calculate the work done by the force applied by the person over 25 m.

[3]

Answer:

Step 1: Sketch a force-displacement graph and split it into sections

Step 2: Split the graph into sections

• The work done is the area under the graph

• The total area can be found by splitting the graph into sections A and B

Step 3: Calculate the area of section A

• Section A is a rectangle where the area is base × height

$\mathrm{Area}\mathrm{of}\mathrm{section}\mathrm{A}=12\times 150=1800\mathrm{J}$ [1 mark]

Step 4: Calculate the area of section B

• Section B is a trapezium, which can be split into a right-angled triangle and a rectangle

$A_{triangle}+A_{rectangle}=\left(\frac{1}{2}\times \mathrm{base}\times \mathrm{height}\right)+\left(\mathrm{base}\times \mathrm{height}\right)$

$\mathrm{Area}\mathrm{of}\mathrm{section}\mathrm{B}=0.5\left(25-12\right)\left(150-60\right)+\left(25-12\right)\left(60-0\right)$

$\mathrm{Area}\mathrm{of}\mathrm{section}\mathrm{B}=\left(0.5\times 13\times 90\right)+\left(13\times 60\right)$

$\mathrm{Area}\mathrm{of}\mathrm{section}\mathrm{B}=585+780=1365\mathrm{J}$ [1 mark]

Step 5: Calculate the total work done

• The total work done is the sum of both areas

$\mathrm{Work}\mathrm{done}=1800+1365=3165\mathrm{J}$ [1 mark]

When you sketch a graph, it doesn't have to be to scale. However, make sure you label the key points from the question on the x- and y-axes, so you can calculate the areas underneath the graph.