Power

Definition

Power

The rate at which energy is transferred.

Power is the rate at which energy is transferred, which means how much energy is transferred each second.
When a force does work, the energy transferred equals the work done, so power is also the rate at which work is done.
Power is calculated with $donetime\text{power}=\dfrac{\text{work done}}{\text{time}}$ , an equation you must recall.
In symbols the equation is $P=WtP=\dfrac{W}{t}$ $P = \frac{W}{t}$ .
1. $P$ is the power in watts, $W\text{W}$ .
2. $W$ is the work done, or energy transferred, in joules, $J\text{J}$ .
3. $t$ is the time taken in seconds, $s\text{s}$ .
The same equation can be written as $P=EtP=\dfrac{E}{t}$ , where $E$ is the energy transferred in joules.
Rearranging gives $W = P t$ to find the work done and $t=WPt=\dfrac{W}{P}$ to find the time.

The watt

Definition

Watt

The unit of power, equal to an energy transfer of one joule per second.

Power is measured in watts, with the symbol $W\text{W}$ .
One watt is a rate of energy transfer of one joule per second, so $J/s1\ \text{W}=1\ \text{J/s}$ .
Larger powers are given in kilowatts and megawatts.
1. $W1\ \text{kW}=1000\ \text{W}$ , which is roughly the power of a strong cyclist sprinting for a few seconds.
2. $W1\ \text{MW}=1\,000\,000\ \text{W}$ , which is the scale of the power of a train engine or a large wind turbine.
Times must be in seconds before you substitute, so a time of $2$ minutes becomes $s120\ \text{s}$ .

Comparing power

A more powerful machine transfers energy more quickly than a less powerful one.
Two things decide the power: the energy transferred and the time taken.
1. If two machines transfer the same energy, the one that takes less time has the greater power.
2. If two machines run for the same time, the one that transfers more energy has the greater power.
Two pupils of equal weight who climb the same staircase do the same work against their weight, but the pupil who runs up does it in less time and so develops a greater power.
A crane lifting a load to the top of a building does the same work whether it lifts slowly or quickly, but the faster crane needs a more powerful motor.
Power describes how quickly energy is transferred, not how much, so a low-power machine running for a long time can transfer more energy than a high-power machine running briefly.

Example

Power of a lift motor

A lift motor does $J54\,000\ \text{J}$ of work raising a lift between floors in $s18\ \text{s}$ .

P=\dfrac{W}{t}

P=\dfrac{54\,000\ \text{J}}{18\ \text{s}}=3000\ \text{W}

The power of the lift motor is $W3000\ \text{W}$ , which is $kW3.0\ \text{kW}$ .

Example

Power of a pupil running upstairs

A pupil with a weight of $N600\ \text{N}$ runs up a staircase with a vertical height of $m3.0\ \text{m}$ in $s4.0\ \text{s}$ .
Find the work done against the pupil's weight first.

W=Fs=600\ \text{N}\times3.0\ \text{m}=1800\ \text{J}

P=\dfrac{W}{t}=\dfrac{1800\ \text{J}}{4.0\ \text{s}}=450\ \text{W}

The pupil develops a power of $W450\ \text{W}$ while running up the stairs.

Example

Time for a crane to do work

A $kW2.5\ \text{kW}$ crane motor does $kJ75\ \text{kJ}$ of work lifting a load.
Convert to base units, so $WP=2500\ \text{W}$ and $JW=75\,000\ \text{J}$ .

t=\dfrac{W}{P}=\dfrac{75\,000\ \text{J}}{2500\ \text{W}}=30\ \text{s}

The crane takes $s30\ \text{s}$ to lift the load.

Common Mistake

Do not define power as "the amount of energy", because power is the rate of energy transfer.
Do not confuse the unit watt, $W\text{W}$ , with the symbol for work done, $W$ , which is measured in joules.
Do not substitute a time in minutes or hours, because the equation needs the time in seconds.

Exam technique

Power questions

Write $P=WtP=\dfrac{W}{t}$ from memory, because you are expected to recall it.
Use the word rate in every definition of power, for example "the rate at which energy is transferred".
When you compare two machines, mention both the energy transferred and the time taken before you conclude which is more powerful.
In a two-step question, find the work done with $W = F s$ first, then divide by the time.

Self review

Define power.
Write the equation that links power, work done and time.
How many joules per second is $kW1\ \text{kW}$ ?
Calculate the power of a motor that does $J1200\ \text{J}$ of work in $s8.0\ \text{s}$ .
Two cranes lift identical loads to the same height, so how can you tell which is more powerful?

Power

Definition

Power

The rate at which energy is transferred.

Power is the rate at which energy is transferred, which means how much energy is transferred each second.
When a force does work, the energy transferred equals the work done, so power is also the rate at which work is done.
Power is calculated with $donetime\text{power}=\dfrac{\text{work done}}{\text{time}}$ , an equation you must recall.
In symbols the equation is $P=WtP=\dfrac{W}{t}$ $P = \frac{W}{t}$ .
1. $P$ is the power in watts, $W\text{W}$ .
2. $W$ is the work done, or energy transferred, in joules, $J\text{J}$ .
3. $t$ is the time taken in seconds, $s\text{s}$ .
The same equation can be written as $P=EtP=\dfrac{E}{t}$ , where $E$ is the energy transferred in joules.
Rearranging gives $W = P t$ to find the work done and $t=WPt=\dfrac{W}{P}$ to find the time.

The watt

Definition

Watt

The unit of power, equal to an energy transfer of one joule per second.

Power is measured in watts, with the symbol $W\text{W}$ .
One watt is a rate of energy transfer of one joule per second, so $J/s1\ \text{W}=1\ \text{J/s}$ .
Larger powers are given in kilowatts and megawatts.
1. $W1\ \text{kW}=1000\ \text{W}$ , which is roughly the power of a strong cyclist sprinting for a few seconds.
2. $W1\ \text{MW}=1\,000\,000\ \text{W}$ , which is the scale of the power of a train engine or a large wind turbine.
Times must be in seconds before you substitute, so a time of $2$ minutes becomes $s120\ \text{s}$ .

Comparing power

A more powerful machine transfers energy more quickly than a less powerful one.
Two things decide the power: the energy transferred and the time taken.
1. If two machines transfer the same energy, the one that takes less time has the greater power.
2. If two machines run for the same time, the one that transfers more energy has the greater power.
Two pupils of equal weight who climb the same staircase do the same work against their weight, but the pupil who runs up does it in less time and so develops a greater power.
A crane lifting a load to the top of a building does the same work whether it lifts slowly or quickly, but the faster crane needs a more powerful motor.
Power describes how quickly energy is transferred, not how much, so a low-power machine running for a long time can transfer more energy than a high-power machine running briefly.

Example

Power of a lift motor

A lift motor does $J54\,000\ \text{J}$ of work raising a lift between floors in $s18\ \text{s}$ .

P=\dfrac{W}{t}

P=\dfrac{54\,000\ \text{J}}{18\ \text{s}}=3000\ \text{W}

The power of the lift motor is $W3000\ \text{W}$ , which is $kW3.0\ \text{kW}$ .

Example

Power of a pupil running upstairs

A pupil with a weight of $N600\ \text{N}$ runs up a staircase with a vertical height of $m3.0\ \text{m}$ in $s4.0\ \text{s}$ .
Find the work done against the pupil's weight first.

W=Fs=600\ \text{N}\times3.0\ \text{m}=1800\ \text{J}

P=\dfrac{W}{t}=\dfrac{1800\ \text{J}}{4.0\ \text{s}}=450\ \text{W}

The pupil develops a power of $W450\ \text{W}$ while running up the stairs.

Example

Time for a crane to do work

A $kW2.5\ \text{kW}$ crane motor does $kJ75\ \text{kJ}$ of work lifting a load.
Convert to base units, so $WP=2500\ \text{W}$ and $JW=75\,000\ \text{J}$ .

t=\dfrac{W}{P}=\dfrac{75\,000\ \text{J}}{2500\ \text{W}}=30\ \text{s}

The crane takes $s30\ \text{s}$ to lift the load.

Common Mistake

Do not define power as "the amount of energy", because power is the rate of energy transfer.
Do not confuse the unit watt, $W\text{W}$ , with the symbol for work done, $W$ , which is measured in joules.
Do not substitute a time in minutes or hours, because the equation needs the time in seconds.

Exam technique

Power questions

Write $P=WtP=\dfrac{W}{t}$ from memory, because you are expected to recall it.
Use the word rate in every definition of power, for example "the rate at which energy is transferred".
When you compare two machines, mention both the energy transferred and the time taken before you conclude which is more powerful.
In a two-step question, find the work done with $W = F s$ first, then divide by the time.

Self review

Define power.
Write the equation that links power, work done and time.
How many joules per second is $kW1\ \text{kW}$ ?
Calculate the power of a motor that does $J1200\ \text{J}$ of work in $s8.0\ \text{s}$ .
Two cranes lift identical loads to the same height, so how can you tell which is more powerful?

2.2.15 Power as the rate of energy transfer

Power

The watt

Comparing power

Power of a lift motor

Power of a pupil running upstairs

Time for a crane to do work

Power questions

Review 2.2.15 Power as the rate of energy transfer by teaching Genie

2.2.15 Power as the rate of energy transfer

Power

The watt

Comparing power

Power of a lift motor

Power of a pupil running upstairs

Time for a crane to do work

Power questions

Review 2.2.15 Power as the rate of energy transfer by teaching Genie

2.2.15 Power as the rate of energy transfer

Power

The watt

Comparing power

Power of a lift motor

Power of a pupil running upstairs

Time for a crane to do work

Power questions

Review 2.2.15 Power as the rate of energy transfer by teaching Genie

2.2.15 Power as the rate of energy transfer Revision Guide

2.2.15 Power as the rate of energy transfer

Power

The watt

Comparing power

Power of a lift motor

Power of a pupil running upstairs

Time for a crane to do work

Power questions

Review 2.2.15 Power as the rate of energy transfer by teaching Genie

2.2.15 Power as the rate of energy transfer Revision Guide