In Work work was force times distance. That holds when the force points along the motion. When it points at an angle, only part of it, its component along the motion, does work.
Key idea
The work done by a constant force is $W = Fd\cos\theta$, where $\theta$ is the angle between the force and the displacement. Work is in joules ($1\ \text{J} = 1\ \text{N}\cdot\text{m}$) and can be positive, negative or zero.
Why it works
Work measures how much a force adds to (or takes from) an object's energy of motion. A force helps the motion only through its part along the motion, $F\cos\theta$. If the force points backward ($\theta = 180^\circ$, like friction), it takes energy away: negative work. If it is perpendicular ($\theta = 90^\circ$), it neither helps nor hinders: zero work, however hard it pushes.
| Angle between F and d | cos θ | Work |
|---|---|---|
| 0° (along the motion) | 1 | positive, Fd |
| between 0° and 90° | between 0 and 1 | positive, less than Fd |
| 90° (perpendicular) | 0 | zero |
| 180° (against the motion) | −1 | negative, −Fd |
Worked examples
Worked example: A pull at an angle
A rope pulls a sled $15\ \text{m}$ across level snow with $70\ \text{N}$ at $30^\circ$ above the horizontal. How much work does the rope do?
- $W = Fd\cos\theta = (70)(15)\cos 30^\circ$.
- $W \approx 909\ \text{J}$.
Answer: about $909\ \text{J}$
Worked example: Work by friction
Friction of $40\ \text{N}$ acts on a box as it slides $6\ \text{m}$. How much work does friction do?
- Friction points opposite to the motion: $\theta = 180^\circ$.
- $W = (40)(6)(-1) = -240\ \text{J}$.
Answer: $-240\ \text{J}$
Worked example: Work by gravity
A $5\ \text{kg}$ box is lifted $2\ \text{m}$. How much work does gravity do on it?
- Gravity, $49\ \text{N}$, points down; the box moves up: $\theta = 180^\circ$.
- $W_g = -(49)(2) = -98\ \text{J}$.
Answer: $-98\ \text{J}$
Common mistake
Thinking every force on a moving object does work. The normal force on a car driving along a level road is perpendicular to the motion, so it does zero work; so does the string's tension on a ball moving in a circle. Check: find the angle between each force and the displacement before you multiply.