When a force is constant, work is easy to calculate with W = Fd, but many real forces change as an object moves. Springs, magnets, rockets, and muscles can all apply forces that vary with position. In these cases, work is found by adding up many tiny pieces of force times distance.
On a Force vs Distance graph, this total work is the area under the curve.
Understanding Physics: Work Done by a Variable Force
The calculation comes from treating a changing force as a series of short intervals. Over one very short interval, the force changes so little that it can be treated as constant. The work for that interval is then the force value multiplied by the small movement.
Adding every interval gives a closer estimate of the total. Using narrower intervals improves the result. On a graph, the intervals appear as thin rectangles.
Some rectangles may sit slightly above the curve while others sit below it. This explains why a simple rectangle estimate is useful but not always exact.
The horizontal axis must show position or distance, not time. A force versus time graph can tell a different story. Its area is related to impulse, which changes momentum.
A force versus distance graph tracks energy transfer because each part of the graph is linked to an amount of movement. The units confirm this idea. Force is measured in newtons and distance is measured in metres.
A newton metre is one joule, the unit of energy. If a force acts at an angle to the movement, only the part of the force pointing along the movement contributes to the work in that direction.
Signs matter throughout the calculation. A force pointing in the same direction as the movement transfers energy into the object. A resistive force, such as friction or air resistance, points against the movement and transfers energy out of the object.
Its contribution must be counted as negative. A spring gives a useful example. While a person stretches a spring, the person applies a force in the direction of the stretch.
The spring pulls back in the opposite direction. The person supplies energy that becomes elastic potential energy in the spring.
When released, the spring can return that stored energy by pushing or pulling an object. Real springs may lose some energy as heating, especially if they are repeatedly stretched.
Students often meet variable-force work in practical graphs and data tables. A force sensor can measure the pull needed to extend a spring at many positions. The points can be joined with a smooth curve, or estimated with straight sections.
For each section, a rectangle or trapezium gives an approximate contribution to the total work. Check the axis scales carefully, since a graph can look steep or shallow because of its scale. Notice where the curve crosses the horizontal axis, since the contributions on opposite sides can cancel.
Finally, keep the direction of motion clear. Moving an object back to its starting position can give zero overall work from one force, even though energy was transferred during separate parts of the journey.
Key Facts
- Work done by a variable force along one dimension is W = ∫ F(x) dx.
- On a Force vs Distance graph, work equals the signed area under the F(x) curve.
- For constant force, W = Fd is the rectangular area under a horizontal force graph.
- For a linear force from 0 to Fmax over distance x, W = 1/2 Fmax x.
- For an ideal spring, F = kx and the work to stretch or compress it is W = 1/2 kx^2.
- Positive area means the force adds energy to the object, while negative area means the force removes energy.
Vocabulary
- Work
- Work is the energy transferred by a force acting through a displacement.
- Variable force
- A variable force is a force whose magnitude or direction changes as position changes.
- Integral
- An integral adds many tiny contributions to find a total amount, such as total work from changing force.
- Force vs Distance graph
- A Force vs Distance graph shows how the force on an object changes as the object moves along a path.
- Spring constant
- The spring constant k measures how stiff a spring is and appears in Hooke's law, F = kx.
Common Mistakes to Avoid
- Using W = Fd for every problem, which is wrong when the force changes with position. Use the area under the graph or W = ∫ F(x) dx instead.
- Forgetting that area below the distance axis is negative, which gives the wrong sign for work. A force opposite the direction of motion does negative work.
- Using the final force of a spring as if it acted for the whole distance, which doubles the correct answer. For a spring starting at x = 0, use W = 1/2 kx^2.
- Mixing units on the graph, which makes the work value incorrect. Force must be in newtons and distance must be in meters so the area is in joules.
Practice Questions
- 1 A force increases linearly from 0 N to 20 N as an object moves from x = 0 m to x = 5 m. Find the work done.
- 2 A spring with k = 300 N/m is stretched from x = 0 m to x = 0.20 m. How much work is required?
- 3 A force vs distance graph has equal positive area from 0 m to 4 m and negative area from 4 m to 8 m. Explain what the net work is and what that means for the object's kinetic energy.