Power: the rate of doing work

Two ways of climbing

Two students of the same mass climb the same stairs. One walks; the other runs. Both do the same work against gravity, $mgh$. But the runner does it faster, and is out of breath. We need a quantity for how fast work is done.

Power

Power is the rate of doing work: the work done in one second.

with $W$ in joules, $t$ in seconds and $P$ in watts ($\text{W}$). One watt is one joule each second: $1\ \text{W} = 1\ \text{J/s}$.

Bigger units: $1\ \text{kW} = 1000\ \text{W}$ and $1\ \text{MW} = 1\,000\,000\ \text{W}$. An LED bulb uses a few watts, a fan about $60\ \text{W}$, a water heater about $2000\ \text{W}$. An old unit, the horsepower, is about $746\ \text{W}$.

Power and energy

Power is also the rate at which energy is changed: $P = \dfrac{\text{energy}}{\text{time}}$. A $60\ \text{W}$ lamp changes $60\ \text{J}$ of electrical energy into light and heat every second. Power tells you how fast, not how much in total.

When a machine works faster at some moments than at others, its average power is the total work divided by the total time.

You should find that half the time gives twice the power, while the work stays the same. A heavier climber, or taller stairs, needs more work and so more power for the same time.

Watch out. The time must be in seconds when the work is in joules. Change minutes to seconds first.

A worked example

A student of $48\ \text{kg}$ climbs stairs $3.5\ \text{m}$ high in $8\ \text{s}$. Then a pump lifts $250\ \text{kg}$ of water $10\ \text{m}$ in $1$ minute. Find each power.

1. The student's work. $W = 48\ \text{kg} \times 9.8\ \text{m/s}^2 \times 3.5\ \text{m} = 1646.4\ \text{J}$.
2. The student's power. $P = \dfrac{1646.4\ \text{J}}{8\ \text{s}} = 205.8\ \text{W}$.
3. The pump. $W = 250\ \text{kg} \times 9.8\ \text{m/s}^2 \times 10\ \text{m} = 24\,500\ \text{J}$ in $60\ \text{s}$, so $P = 408\ \text{W}$.

Practise with new numbers

Try it in Python

The program finds the power of the same climb done at three speeds.

Remember these

• Power is the rate of doing work: $P = \dfrac{W}{t}$, in watts.
• $1\ \text{W} = 1\ \text{J/s}$, $1\ \text{kW} = 1000\ \text{W}$.
• More power means the same work in less time, not more work.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A machine does $4200\ \text{J}$ of work in $15\ \text{s}$. What is its power?

2. How much energy does a $75\ \text{W}$ lamp use in $1$ minute?

3. A $5\ \text{W}$ toy motor runs for an hour, and a $500\ \text{W}$ mixer for $10\ \text{s}$. Which does more work?

4. Two cranes lift the same load to the same height. Crane A takes half the time of crane B. Compare their work and their power.