Work: force times distance
Pushing a lawn mower 25 m with 75 N at 35° below the horizontal does 1,540 J of work, about a third of a food Calorie.
Work in physics is narrower than work in life
Holding a heavy bag still feels like hard work. In physics, you do no mechanical work on the stationary bag. Work has a strict meaning: a force does work on an object only when the object moves, and only by the part of the force that points along the motion. When work is done, energy is transferred. That link is the reason the idea matters.
For a constant force the rule is W = F d cos θ. F is the size of the force, d is the distance moved, and θ is the angle between the force and the direction of motion. F cos θ is the part of the force that points along the motion. Work is measured in joules, and one joule is one newton pushing through one metre: 1 J = 1 N·m.
When the work is zero
OpenStax uses a briefcase to sort out the cases. A person standing still and holding a briefcase does no work on it, because d = 0. Your arm muscles still tire, because inside them fibres work against each other, but no energy goes into the briefcase. A person walking across a level floor at a steady speed while carrying the case also does no work on it. The lifting force points up and the motion is sideways, so θ = 90° and cos 90° = 0.
Carry the same case up the stairs and work is done, because the lifting force now has a part along the motion. That energy stays stored in the briefcase and Earth as gravitational potential energy. Lower the case gently and the force from your hand still points up while the motion points down. Then θ = 180°, cos 180° = −1 and the work is negative: your hand takes energy out of the briefcase. Friction is the most common example of negative work. It always opposes sliding, so it removes energy from moving objects and turns it into heat.
| Situation | Angle θ | Work |
|---|---|---|
| Push along the motion | 0° | F d (largest) |
| Push at an angle | between 0° and 90° | F d cos θ (smaller) |
| Force at right angles to motion | 90° | 0 |
| Force against the motion | 180° | −F d (negative) |
| Holding still | any | 0, because d = 0 |
A worked example: the lawn mower
A gardener pushes a lawn mower with a constant force of 75.0 N along the handle, which points 35° below the horizontal, and moves it 25.0 m across level ground. The work done by the gardener's push is W = 75.0 × 25.0 × cos 35° = 1,536 J, about 1.54 × 10³ J. That sounds like a lot, but in food energy it is 1,536 ÷ 4,186 = 0.37 kcal. OpenStax notes that even on a busy day less than 10 percent of the energy we eat goes into work; the rest becomes body heat or is stored.
In the lab below, set the force to 50 N and the distance to 10 m. Along the floor (0°) the push does 500 J. Raise the angle to 60° and only half the force points forward, because cos 60° = 0.5, so the work drops to 250 J. At 90° the work is zero however hard you pull.
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Gravity does work too. Drop a 0.5 kg ball from a height of 2 m. Gravity pulls down with m g = 0.5 × 9.8 = 4.9 N, the ball moves 2 m downward in the same direction, so the work done by gravity is 4.9 × 2 = 9.8 J. Throw the same ball straight up 2 m and gravity does −9.8 J on the way up, taking energy away as the ball slows. Over the round trip back to your hand, gravity's total work is zero. That is a first hint of potential energy, which unit 4 takes up.
| Work | W = F d cos θ, energy moved by a force |
| No work when… | Nothing moves (d = 0) or the force is at 90° to the motion |
| Negative work | Force against the motion, for example friction |
| 1 joule | 1 newton through 1 metre |
- W = F d cos θ; only the part of the force along the motion counts.
- No motion or a force at 90° means zero work.
- A force against the motion does negative work and takes energy out.
- Work and energy share the unit joule, 1 J = 1 N·m.
- You push a box 10 m along the floor with 50 N in the direction of motion. Work in joules?
- Same push, but the force is at 60° to the floor. Work in joules?
- Lawn mower: 75.0 N at 35° for 25.0 m. Work in joules? (nearest whole number)
- You hold a 10 kg bag still for one minute. Work done on the bag?
- Zero
- 98 J
- 588 J
- 5,880 J
- Carrying a bag across a level room at constant speed, the lifting force does:
- No work, the force is at 90° to the motion
- Positive work
- Negative work
- Work equal to the weight times the distance
- Friction on a sliding box does:
- Negative work
- Positive work
- No work
- Work only when the box stops
- How many food Calories (kcal) is 4,186 J of work?
- A force of 20 N moves a cart 3 m in the force's direction. Work in joules?
- Which unit is the same as a joule?
- N·m
- N/m
- W/s
- kg·m/s
- You lower a 2 kg box 1.5 m at constant speed. What is the work done by your hand, in joules? (g = 9.8, one decimal, include the sign)
Sources for this course (8)
- OpenStax College Physics 2e (openstax.org/books/college-physics-2e): 7.1 Work: the scientific definition (W = F d cos θ, briefcase cases, 1 J = 1 N·m, 1 J lifts a 100 g apple about 1 m, lawn mower 75.0 N at 35° for 25.0 m = 1.54 × 10³ J, 1 kcal = 4186 J, under 10% of food energy goes into work)
- 7.2 Kinetic energy and the work-energy theorem (KE = ½ m v², 30.0 kg package at 0.500 m/s = 3.75 J, 120 N push with 5.00 N friction over 0.800 m = 92.0 J net, final speed 2.53 m/s, a car at 100 km/h has four times the KE it has at 50 km/h)
- 7.3 Gravitational potential energy (ΔPE = m g h, reference level, path independence, 0.500 kg cuckoo clock weight raised 1.00 m = 4.90 J, 60.0 kg person landing stiffly from 3.00 m, roller coaster 20.0 m hill: 19.8 m/s, and 20.4 m/s with a 5.00 m/s start, mass cancels)
- 7.4 Conservative forces and potential energy (spring PE = ½ k x², 0.100 kg toy car, k = 250 N/m, 4.00 cm, 0.180 m rise)
- 7.5 Nonconservative forces (friction turns mechanical energy into thermal energy)
- 7.6 Conservation of energy (statement of the law, forms of energy, Table 7.1: 1000 kg car at 90 km/h 3.1 × 10⁵ J, tennis ball at 100 km/h 22 J, mosquito 1.3 × 10⁻⁶ J, daily food intake 1.2 × 10⁷ J, 1 gallon gasoline 1.2 × 10⁸ J, 1 g fat 3.9 × 10⁴ J; efficiency, Table 7.2: cycling and climbing 20%, swimming at the surface 2%, gasoline engine 30%, diesel 35%, coal power plant 42%, electric motor 98%, solar cell 10%)
- 7.7 Power (P = W ÷ t, 1 W = 1 J/s, 1 hp = 746 W, 60.0 kg woman up 3.00 m stairs in 3.50 s reaching 2.00 m/s = 538 W, Table 7.3: Sun 4 × 10²⁶ W, car 8 × 10⁴ W, person at rest 100 W, heart at rest 8 W, clothes dryer 4 × 10³ W, a 60 W incandescent bulb gives 5 W of light, power plants turn 35 to 40% of fuel energy into electricity, E = P t, 0.200 kW computer 6.00 h a day for 30.0 days = 36.0 kWh, $4.32 at $0.120 per kWh)
- g = 9.8 m/s² as in OpenStax examples (NIST standard gravity 9.80665 m/s², physics.nist.gov/cgi-bin/cuu/Value?gn).
Checked: 2026-10-17