Last updated: 2026-05-18

Work and power

In physics, work is the energy transferred to or from an object when a force acts on it through a displacement. It is the product of the force, the distance moved, and the cosine of the angle between them: W = F × d × cos(θ). Power is the rate at which that work is done, P = W ÷ t. Both quantities are calculated here from the force, displacement, angle, and elapsed time, with results in joules (J), kilojoules (kJ), kilowatt-hours (kWh), watts (W), kilowatts (kW), and horsepower (hp).

The work formula

$$W = F \times d \times \cos(\theta)$$

Symbol	Quantity	SI unit
W	Work	J (joules)
F	Force	N (newtons)
d	Displacement	m (metres)
θ	Angle between F and d	degrees or radians

Work is a scalar — it has magnitude but no direction. One joule equals one newton-metre (1 J = 1 N·m).

The role of the angle

The angle θ controls how much of the force contributes to motion:

θ	cos(θ)	Work	Example
0°	1	Maximum	Pushing a cart forward
45°	0.707	Partial	Dragging a suitcase with a handle at 45°
90°	0	Zero	Carrying a bag while walking horizontally
180°	−1	Negative	Friction opposing motion

A perpendicular force (θ = 90°) does zero work — the force doesn't contribute to or take from the object's kinetic energy. This is why gravity does no work on a satellite in a circular orbit: gravity points toward the centre, motion is tangential (perpendicular).

Power

$$P = \frac{W}{t}$$

Power is the rate of doing work — how quickly energy is transferred.

Unit	Equivalent
1 W	1 J/s
1 kW	1,000 W
1 hp	≈ 745.7 W

A 100 W light bulb does 100 J of work (mostly as heat and light) every second. A car engine producing 150 hp produces about 111.9 kW — enough to accelerate a 1,500 kg car from 0 to 100 km/h in under 10 seconds under ideal conditions.

Worked example

Negative work

When the force opposes displacement (θ = 180°), work is negative. This describes:

• Friction: always opposes motion, removing kinetic energy from the system.
• Braking: the brake force opposes vehicle motion; negative work slows the vehicle.
• Gravity on an ascending object: gravity acts downward while the object moves upward.

The work-energy theorem states: net work done on an object equals its change in kinetic energy:

$$W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

Frequently Asked Questions (FAQ)

What is the formula for work in physics?

Work is W = F × d × cos(θ), where F is the applied force, d is the displacement, and θ is the angle between the force vector and the direction of motion. When force and displacement are parallel (θ = 0°), the formula reduces to W = F × d. Work is measured in joules (J) — one joule equals one newton applied over one metre.

What does the angle in the work formula represent?

The angle θ is between the force vector and the displacement vector. θ = 0°: force and motion are parallel — maximum work done. θ = 90°: force is perpendicular to motion (e.g. carrying a bag horizontally while walking) — zero work done by that force. θ = 180°: force opposes motion (e.g. friction, braking) — negative work done, energy is removed from the system.

How is power related to work and time?

Power is the rate of doing work: P = W ÷ t. A motor that does 1,000 J of work in 10 seconds has a power of 100 W. A more powerful motor does the same work in less time. Power is measured in watts (W), kilowatts (kW), or horsepower (hp; 1 hp ≈ 745.7 W).

What is the difference between work and energy?

Work and energy have the same unit (joules) and are closely related: doing work on an object transfers energy to it. The work-energy theorem states that the net work done on an object equals its change in kinetic energy (W_net = ΔKE). Energy is the capacity to do work; work is the process of transferring that energy through a force applied over a distance.