(c) Work and power
- Syllabus
- 2024
- Topic
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- Level
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Learning objectives
4.11Work done equationKnow and use work done = force × distance moved in the force direction, W = Fd.4.12Work and energy transferKnow that work done is equal to energy transferred4.13Gravitational potential energyKnow and use gravitational potential energy = mass × gravitational field strength × height, GPE = mgh.4.14Kinetic energy equationKnow and use kinetic energy = ½ × mass × speed², KE = ½mv².4.15Energy conservation linksUnderstand how conservation of energy produces a link between gravitational potential energy, kinetic energy and work4.16PowerDescribe power as the rate of transfer of energy or the rate of doing work4.17Power equationUse power = work done (energy transferred) ÷ time taken, P = W/t.
Calculate work done by a force
A force does work when it moves an object through a distance in the direction of that force. The work done measures the energy transferred mechanically by the force.
W=F\times d
W is work done in joules (J), F is the force in newtons (N), and d is the distance moved in the force direction in metres (m). Rearrangements are F=W/d and d=W/F.
Example: a rope transfers 41 000 J of energy while pulling a truck 15 m in the rope's direction. F=W/d=41000/15=2733… N, so the force is 2.7×103 N to two significant figures.
Use the distance moved in the force direction, not automatically the total path length. Convert centimetres to metres before substituting, and keep work in joules.
Connect work done to energy transferred
Work done is equal to energy transferred. If a force does 50 J of work, it transfers 50 J of energy between stores.
W=\text{energy transferred}
| Force and motion | Store change caused by the work |
|---|---|
| a person lifts an object | the object's gravitational store increases |
| a force accelerates an object | the object's kinetic store increases |
| friction slows an object | its kinetic store decreases while thermal stores increase |
| a force compresses a spring | the spring's elastic store increases |
Work describes the transfer process, not a separate energy store. Both work done and energy transferred are measured in joules because they are equal amounts in the same energy account.
An applied force does no work on an object if it causes no displacement in its direction. A force can be present without transferring energy mechanically.
Calculate changes in gravitational potential energy
Raising an object increases its gravitational potential energy because work is done against the gravitational force. The change depends on mass, gravitational field strength and vertical height change.
\Delta \mathrm{GPE}=mgh
m is mass in kilograms (kg), g is gravitational field strength in newtons per kilogram (N/kg), and h is the vertical height change in metres (m). The energy change is in joules (J).
Example: a 14 g ball rises by 29 cm where g=10 N/kg. Convert first: m=0.014 kg and h=0.29 m. Then ΔGPE=0.014×10×0.29=0.0406 J, about 4.1×10−2 J.
Use the vertical height change, not the distance travelled along a slope. Convert grams to kilograms and centimetres to metres before calculating.
Calculate kinetic energy and speed
Kinetic energy is the energy in the store of a moving object. It depends on the object's mass and on the square of its speed.
\mathrm{KE}=\frac{1}{2}mv^2
m is mass in kilograms (kg), v is speed in metres per second (m/s), and kinetic energy is measured in joules (J). To find speed, use v=2KE/m.
Example: a 0.014 kg ball has 0.051 J in its kinetic store. v=(2×0.051)/0.014=7.29=2.7 m/s.
Square the speed, not the mass. When rearranging for speed, take the square root at the end. At the same mass, doubling speed makes kinetic energy four times as large.
Link GPE, KE and work through conservation
Conservation of energy links gravitational potential energy, kinetic energy and work: energy leaving one store must appear in another store or be transferred to the surroundings.
| Situation | Conserved energy account |
|---|---|
| object falls with negligible resistance | GPE lost = KE gained |
| person pushes a falling hammer downwards | KE gained = GPE lost + work done by the person |
| brakes stop a vehicle | KE lost = work done by the braking force, transferred mainly to thermal stores |
| object moves up a rough ramp | input work = GPE gained + energy transferred to thermal stores by friction |
Example: a 0.0055 kg marble descends 0.21 m, so GPE lost is 0.0055×10×0.21=0.01155 J. At 0.76 m/s its KE is 0.5×0.0055×0.762=0.00159 J. The difference, about 0.010 J, has been transferred to other stores by friction and resistance.
Do not set GPE lost equal to KE gained when friction, air resistance or external work matters. Add every transfer to the energy account before equating totals.
Understand power as a rate
Power is the rate of energy transfer or the rate of doing work. It describes how quickly an energy transfer happens, not the total amount transferred.
One watt means one joule transferred each second: 1W=1J/s. A 500 W device transfers 500 J each second while operating at that power.
| Comparison | Power conclusion |
|---|---|
| same energy transferred in less time | greater power |
| more energy transferred in the same time | greater power |
| same power used for twice as long | twice as much energy transferred |
Two people may do the same work climbing the same stairs. The person who completes the climb in less time has the greater power, even though both transfer the same amount of energy to their gravitational stores.
Power and energy are different quantities: watts measure a rate, while joules measure an amount of energy or work. A high-power device is not necessarily more efficient.
Use the power equation
Power equals the work done, or energy transferred, divided by the time taken.
P=\frac{W}{t}=\frac{E}{t}
P is power in watts (W), W is work done and E is energy transferred in joules (J), and t is time in seconds (s). Rearrangements are E=Pt and t=E/P.
Example: a heater transfers 39 kJ in 290 s. Convert 39 kJ to 39000 J, then P=39000/290=134.5… W, which is 1.3×102 W (130 W) to two significant figures.
Convert minutes to seconds and kilojoules to joules before using watts. Match the rearrangement to the unknown instead of multiplying energy by time.