Power (P = W/t)

Power is the rate at which work is done, $P = W/t$. It answers a different question from work: not "how much energy did this take?" but "how fast was it delivered?" Two apprentices carrying identical toolboxes up the same stairs do exactly the same work against gravity, and the one who takes the stairs two at a time develops more power. One watt is one joule per second, which makes the watt a small unit — a person working steadily manages perhaps 75 W, and a fit cyclist holds around 250 W for an hour.

Take 20 kg of tools hauled 12 m up a ladder. The work is $mgh = 20 \times 9.80665 \times 12 \approx 2354$ J regardless of how it is done. Take 40 s over it and $P = 2354/40 \approx 59$ W. Rush it in 15 s and the same job demands 157 W. The energy bill is identical; only the rate has changed, and it is the rate that decides whether a motor is big enough.

James Watt coined horsepower in the 1780s as a sales tool. He needed to tell mill owners how many horses one of his engines would replace, measured a horse turning a mill wheel, and settled on 33 000 foot-pounds per minute — about 745.7 W. It was a marketing unit before it was an engineering one, and it has outlived the argument it was built to win. The same rate appears in two other forms on this site: $P = Fv$ when the work is a force moving something along, and $P = \tau\omega$ when it is a torque turning a shaft.

The commonest error is treating power as though it were energy. A kilowatt-hour is not a unit of power — it is a power multiplied by a time, so it is an energy, equal to 3.6 MJ. A 100 W bulb does not consume "100 watts per hour"; it consumes 100 watts, which over an hour amounts to 0.1 kWh. The phrase "watts per hour" is almost always a symptom that the two ideas have been mixed. Second, watch which horsepower a figure is quoted in: mechanical horsepower is 745.7 W, but metric horsepower — PS, cv, ch — is 735.5 W, and European engine ratings are usually the latter, a 1.4% difference that quietly walks into converted specifications. Third, $t$ must be the time over which the work was actually done, not the length of the shift; a hoist that lifts for 20 s and then sits idle for 10 minutes has a duty cycle, and averaging over the whole ten minutes describes the energy consumption honestly but badly understates the motor the job needs.