Work and Power

Work done — the energy transferred when a force moves its point of application:

W = Fs cosθ

where θ is the angle between the force and the displacement. Measured in joules: one joule is the work done when a force of one newton moves through one metre in its own direction.

Why the cosine is there — only the component of the force along the direction of motion does work. A force perpendicular to the motion does none at all, because cos90° = 0.

Example: carrying a bag horizontally across a room does no work on the bag against gravity. The weight acts downwards and the motion is horizontal, so the angle is 90°. You certainly get tired, but that energy goes into your muscles, not into the bag.

Work can be negative — when the force opposes the motion, cosθ is negative and the work done is negative, meaning energy is taken out of the body. Friction and drag do negative work.

Power — the rate of doing work, or the rate of energy transfer:

P = W / t

Measured in watts, where 1 W = 1 J s⁻¹.

Power for a force moving at constant velocity — substituting W = Fs gives

P = F v

Example: a car travelling at constant speed does work against resistive forces of 600 N at 30 m s⁻¹. Its output power is 600 × 30 = 18 kW. At constant velocity the driving force equals the resistive force, which is what makes this calculation possible.

Efficiency —

efficiency = useful output energy ÷ total input energy

expressed as a decimal or a percentage. It can never exceed 1, or 100%, because that would require creating energy. Any claim above 100% means a transfer has been miscounted.