Work by a Variable Force
The work done by a force is the area under its force–position graph.
About this simulation
Choose a force law — constant, spring-like F = kx, or quadratic — and the shaded area under the F–x curve reads out as the work done while a block advances along the displacement. It makes the integral concrete: a spring stretched twice as far costs four times the work.
How to use it
- Pick a Force law: F = c, F = kx, F = ax² or Ramp, and see the shape of the force–displacement graph change.
- Watch the block slide along the track below the graph; the shaded area fills up to the red marker and its value is written as W in joules.
- Move the Displacement slider and compare the W written in the shaded area, once the block reaches the end, with the formula at the top right, which gives the same area exactly.
- Use the Force coefficient slider to scale the force; with F = kx, double the displacement and check that the work becomes four times as large.
- Divide the final W by the displacement d to find the average force: the height of a rectangle with the same area under the graph.
What to notice
- Work done by any force along a straight line equals the area between the F–x curve and the x-axis, whether the curve is flat, sloping or bending.
- For a constant force the area is a rectangle, F × d; for a spring-like force it is a triangle, ½kx².
- The block's kinetic energy label always equals the area swept so far, which is the work–energy theorem when this force is the only one doing work and the block starts from rest.
- In the Ramp law the force grows for the first half of the displacement and then stays constant, so the area is a triangle plus a rectangle.
The formulas
- W = ∫ F dx = area under the F–x graph
- Work W done by a force F that varies with position x, added up over the displacement.
- W = ½kx²
- Work done by a force F = kx over a displacement x, where k is the force constant in N/m.
- W = ax³ ⁄ 3
- Work done by a force F = ax² over a displacement x.
Questions
How do you find the work done by a variable force?
Split the displacement into tiny steps, multiply the force at each step by the step length, and add them up. In the limit this is the integral of F dx, which is the area under the force–displacement graph. In the simulation the shaded region grows as the block moves, and its area in joules is the work.
What is the work done in stretching a spring?
The force needed rises in proportion to the extension, F = kx, so the F–x graph is a straight line through the origin. The area under it is a triangle, giving W = ½kx². Because x is squared, stretching twice as far takes four times the work. Choose F = kx and change Displacement to check.
What does the area under a force–displacement graph represent?
It represents the work done by the force. Area above the x-axis is positive work, done in the direction of motion. The units confirm it: newtons on one axis times metres on the other gives newton-metres, which are joules. If the force acted against the motion, the area would count as negative work.
What is the work-energy theorem?
The work done by the net force on a body equals the change in its kinetic energy, W = ΔKE. If a body starts from rest and one force does all the work, its kinetic energy at the end equals the area under that force's F–x graph. The KE label on the block in the simulation tracks the shaded area for this reason.
