Work: force times displacement
Work is done only when a force moves a body along the force: W = F × s. It can be positive, negative or zero, depending on the angle.
Work in science
In daily talk, reading or standing with a load counts as work. In science, work is done only when a force moves a body in the direction of the force. Work = force × displacement in the direction of the force: W = F × s. Its SI unit is the joule (J), also written N m. One joule is the work done when a force of 1 N moves a body through 1 m along the force.
When is no work done?
- No movement: pushing a wall that does not move. Here s = 0, so W = 0.
- Motion at right angles to the force: a porter walks on a level platform with a box on his head. He does no work against gravity, because gravity acts down but the box moves sideways.
- A satellite in a circular orbit: the earth's pull is always at right angles to its motion. So the pull does no work.
- Steady speed on a level road: the engine's push just balances friction. The net force is zero, so the net work done on the car is zero.
Positive, negative and zero work
| Angle between force and displacement | Work | Example |
|---|---|---|
| 0° (same direction) | Positive, W = F s | Lifting a bucket: your pull and the motion are both upward |
| Between 0° and 90° | Positive, but less than F s | Pulling a toy cart by a string held at an angle |
| 90° (at right angles) | Zero | Gravity on a bag carried along a level road |
| 180° (opposite) | Negative, W = −F s | Friction on a sliding box: friction backward, motion forward |
Work when the force is at an angle
Often the force and the motion are not in the same direction. Then only the part of the displacement along the force counts. If θ is the angle between them, this part is s cos θ. So W = F s cos θ. For θ = 0°, cos θ = 1 and W = F s. For θ = 90°, cos θ = 0 and W = 0. For θ = 180°, cos θ = −1, so W is negative. You will mostly need cos 0° = 1, cos 60° = ½, cos 90° = 0 and cos 180° = −1.
Worked example: lifting a bag of rice
A woman lifts a 10 kg bag of rice from the floor onto a shelf 1.5 m high. Find the work she does against gravity. (g = 9.8 m s⁻²)
- Force needed = weight of the bag = mg = 10 kg × 9.8 m s⁻² = 98 N, upward.
- Displacement = 1.5 m, upward, along the force.
- W = F × s = 98 N × 1.5 m = 147 J.
147 J of positive work
Worked example: pulling at an angle
A child pulls a toy cart 4 m along the floor with a force of 10 N. The string makes an angle of 60° with the floor. Find the work done. (cos 60° = 0.5)
- W = F s cos θ.
- W = 10 N × 4 m × 0.5 = 20 J.
- If the string were level (θ = 0°), the work would be 10 N × 4 m = 40 J.
20 J. Part of the pull acts upward and does no work along the floor
A coolie carrying luggage along a level platform does no work against gravity in the science sense. He still gets tired, because his muscles use energy to hold the load up.
2 marks: “When is work said to be done? Give one example where a force acts but no work is done.” Work is done when a force moves a body in the direction of the force: W = F × s. Example: pushing a wall that does not move does no work, because s = 0.
2 marks: “A force of 20 N moves a box 3 m along the force. Find the work done. What would it be if the force acted at 90° to the motion?” W = F × s = 20 × 3 = 60 J. At 90°, cos 90° = 0, so the work would be zero.
3 marks: “Explain positive, negative and zero work, with one example each.” Positive (θ = 0°): lifting a bucket; force and motion are both upward. Negative (θ = 180°): friction on a sliding box, opposite to the motion. Zero (θ = 90°): gravity on a bag carried along a level road.
No move, no work; across the force, zero; along it, plus; against it, minus.