Library / Foundations: Physics / FPH-07
Foundations: Physics
Work done by a force at an angle
Work from force, displacement and the angle between them.
- Ref
- FPH-07
- Rev
- v1
- Field
- Foundations: Physics
- Result
- W [J]
θ is the angle between force and displacement (or velocity). Assumes constant force and straight displacement. At θ = 0 this reduces to W = Fs.
Only the part of the force pointing along the movement does work.
Calculator
- Force component along motion43.3 N
- Work if θ = 0500 J
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A force at right angles to the motion does no work, such as the pull of a string on a swinging weight.
- Pulling at a shallow angle is much more efficient than at a steep one.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Displacement | 0.1 to 1,000 m | 4.33 | 43,300 | 43,300 | +10% | +10 % |
| F Force | 1 to 1,000 N | 8.66 | 8,660 | 8,652 | +10% | +10 % |
| θ Angle | 0 to 75 ° | 129.4 | 500 | 370.6 | +10% | −3.16 % |
Sweep of Angle
| Angle (°) | Work done (J) |
|---|---|
| 0 | 500 |
| 9.375 | 493.3 |
| 18.75 | 473.5 |
| 28.13 | 441 |
| 37.5 | 396.7 |
| 46.88 | 341.8 |
| 56.25 | 277.8 |
| 65.63 | 206.4 |
| 75 | 129.4 |
Sweep one variable
Variables and how each one contributes
Work done
What the equation gives you. Everything else feeds it.
Force
Applied force. Directly proportional.
Displacement
Distance moved. Directly proportional.
Angle
Angle between force and displacement. Cuts the useful part: at 60° only half the force does work.
Worked example with the values above
- cos 30° = 0.866.
- Component along the motion: 50 × 0.866 = 43.3 N.
- W = 43.3 × 10 = 433 J.
Assumptions and limits
- Constant force and straight-line displacement.
- Angle measured between force and motion.
Common mistakes
- Using the angle to the vertical.
- Calculator in radian mode.
Statistics and uncertainty
Angle errors have little effect at small angles (cos is flat near 0°) but matter more above 45°.
Where it is used
- Towing and hauling loads.
- Work on a ramp or sloping surface.
Maths you need
- Cosine: component along a direction
- Work: force times distance