4.6.2 Area Under a Force-Displacement Graph

• The work done can also be found from a force-displacement graph
• If the force is not constant and is plotted against the displacement of the object:
  • The work done is equal to the area under the force-displacement graph

• This is because:

Work done = Force × Displacement

• The work done is therefore equivalent whether there is:
  • A small force over a long displacement 
  • A large force over a small displacement

• The graph may need to be split up into sections. The total area is the sum of the areas of each section

The area underneath the force-displacement graph is the work done

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

0.5 × 80 × (250 – 100) = 6000 J

Step 3: Calculate the area of section B

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

80 × 100 = 8000 J

Step 4: Calculate the total work done

    The total work done is the sum of both areas

Work done = 6000 + 8000 = 14 000 J

• The force on an object may not always be constant, this is known as a variable force
• This is more representative of a force in real life
• If a force is constant, then the following equations can be used:

W = Fs

P = Fv

• If a force is varying, the above equations cannot be used, instead, work done must be found from the area under the force-displacement graph
• If a varying force increases, then an object’s acceleration increases and vice versa

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

12 × 150 = 1800 J

Step 4: Calculate the area of section B

• • Section B is a trapezium where the area is

Step 5: Calculate the total work done

• • The total work done is the sum of both areas

Work done = 1800 + 1365 = 3165 J