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IB DP Physics · Theme A Space, time and motion · A.3 · ~7 min read

A.3 Work, energy and power

Work as energy transfer, kinetic, gravitational and elastic potential energy, conservation of mechanical energy, Sankey diagrams, power P = Fv, efficiency and the energy density of fuels.

Work doneKinetic & potential energyConservation of energySankey diagramsPowerEfficiencyEnergy density
RSby Riyaz Sir
mentorriyaz.in
No tag = SL and HLHL ONLY HL students onlyDATA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Energy is conserved. It is never made or destroyed, only transferred from one store to another. Work is energy transferred by a force: W = Fs cos θ.
  • Mechanical energy = kinetic + gravitational potential + elastic potential. With no friction or drag it stays constant, so speeds and heights follow from Ep → Ek.
  • Power is the rate of doing work, P = ΔW/Δt, and for a steady force P = Fv. Efficiency = useful out ÷ total in, by energy or by power.
  • A.3 is the same at SL and HL.

By the end you can

  • Calculate work done by a force at an angle, and from the area under a force–displacement graph.
  • Use Ek = ½mv2, ΔEp = mgΔh and EH = ½kx2 with conservation of energy.
  • Draw and interpret Sankey diagrams.
  • Calculate power, including P = Fv, and efficiency.
  • Compare fuels using energy density.

1Work

FθF cos θdisplacement sW = Fs cos θonly the component alongthe motion does work
Only the component of the force along the displacement transfers energy.
W = Fs cos θ DATA BOOKLET
θ is the angle between F and the displacement; W in joules
  • θ = 90°: no work (the centripetal force, the normal force on a level floor). θ = 180°: negative work, energy is taken out (friction, braking).
  • The work done by the resultant force equals the change in kinetic energy (the work–energy principle).
Worked example

A crate is pulled 5.0 m along the floor by a 20 N rope at 30° to the horizontal. How much work does the rope do?

  1. W = Fs cos θ = 20 × 5.0 × cos 30° M1
  2. W = 87 J A1
sFxFW = Fsconstant forceW = ½Fx= ½kx²spring: F = kxWork done = area under a force–displacement graph
For any force, work is the area under the F–s graph. For a spring the area is a triangle.

2Energy stores

Ek = ½mv2 = p22m DATA BOOKLET
kinetic energy
ΔEp = mgΔh DATA BOOKLET
gravitational potential energy near Earth's surface (uniform g)
EH = ½k(Δx)2 DATA BOOKLET
elastic potential energy of a spring obeying Hooke's law

Conservation of mechanical energy

ABCDABCDblue = Ep, orange = Ek: each bar has the same total length (no friction)Mechanical energy is conserved, so the trolley can never rise above its starting height.
As the trolley falls, Ep becomes Ek; as it climbs, Ek turns back into Ep. The total stays the same.
Worked example

A pendulum bob is released from 0.45 m above its lowest point. Find its speed at the bottom (no air resistance).

  1. Ep lost = Ek gained: mgh = ½mv2 M1
  2. v = √(2gh) = √(2 × 9.8 × 0.45) M1
  3. v = 3.0 m s−1, whatever the mass A1
Worked example

A spring (k = 200 N m−1) compressed by 10 cm launches a 50 g ball horizontally. Find the launch speed.

  1. EH = ½ × 200 × 0.102 = 1.0 J M1
  2. ½ × 0.050 × v2 = 1.0 M1
  3. v = 6.3 m s−1 A1
Worked example

A 70 kg skier slides 40 m down a slope, dropping 12 m in height, and reaches 12 m s−1. Find the average friction force.

  1. Ep lost = 70 × 9.8 × 12 = 8232 J M1
  2. Ek gained = ½ × 70 × 122 = 5040 J M1
  3. work against friction = 3192 J = F × 40 M1
  4. F = 80 N A1
Common mistakeUsing h as the length of the slope. ΔEp = mgΔh uses the vertical height change only.
Exam tipEnergy methods ignore direction and the shape of the path. If a question asks for a speed after a height change, try energy before suvat: it works on curved paths too.

3Power and efficiency

P = ΔWΔt = Fv DATA BOOKLET
power in watts (J s−1)
η = useful work outtotal work in = useful power outtotal power in DATA BOOKLET
efficiency, a ratio or a percentage
Worked example

A car's engine delivers 60 kW of useful power at a steady 25 m s−1. Find the resistive force.

  1. at constant speed, driving force = resistive force R1
  2. F = P/v = 60 000/25 M1
  3. F = 2400 N A1
Worked example

A motor lifts a 50 kg load 3.0 m in 4.0 s and draws 600 W from the supply. Find its efficiency.

  1. useful work = mgh = 50 × 9.8 × 3.0 = 1470 J M1
  2. input energy = 600 × 4.0 = 2400 J M1
  3. η = 1470/2400 = 61 % A1
100 J chemicalenergy in the fuel25 J useful kinetic energy75 J wasted:thermal energy, soundSankey diagram: width ∝ energyefficiency = 25/100 = 25 %
The arrows show where the energy goes. Their widths are drawn to scale.
Common mistakeDrawing a Sankey diagram whose output arrows do not add up to the input. Energy is conserved: the widths must add up.

4Energy density of fuels

Energy density — energy released per unit mass of fuel, in J kg−1. (Some questions use energy per unit volume, J m−3; check the units.)

wood16 MJ kg⁻¹coal30 MJ kg⁻¹petrol46 MJ kg⁻¹natural gas55 MJ kg⁻¹hydrogen142 MJ kg⁻¹Approximate specific energies of fuels. Uranium in a reactor: about 8 × 10⁷ MJ kg⁻¹.
Fossil fuels store tens of MJ per kilogram. Nuclear fuel stores about a million times more.
  • Choosing a fuel is about more than energy density: cost, availability, pollution and CO2, storage and safety all matter. Hydrogen has a high energy per kilogram but a low energy per litre, so it is hard to store.
Worked example

A power station burns coal (30 MJ kg−1) at 38 % efficiency to give 500 MW of electrical power. How much coal does it burn per second?

  1. input power = 500/0.38 = 1316 MW M1
  2. mass per second = 1316 × 106 / 30 × 106 M1
  3. ≈ 44 kg s−1 A1
Linking questionEnergy conservation reappears everywhere: in collisions (A.2), thermal physics (B.1), circuits (B.5) and orbits (D.1). Which of these problems could you solve with forces alone?
Idea for your IA / explorationMeasure the efficiency of a small electric motor lifting a mass: record V and I with a joulemeter or multimeters, time the lift, and investigate how efficiency changes with the load.

Test yourself — cover the answers first

Quick check

  1. When does a force do no work on a moving object?
  2. What does the area under a force–displacement graph represent?
  3. Write two expressions for kinetic energy.
  4. A ball is dropped from 5.0 m. Its speed just before landing, ignoring air resistance?
  5. State the unit of power in base units.
  6. A device takes in 200 J and wastes 150 J. Its efficiency?

Exam-style

  1. A 1500 kg car accelerates from 10 m s−1 to 25 m s−1 in 8.0 s on a level road. Find the average useful power.
  2. A 2.0 kg block is pushed 3.0 m up a frictionless 25° slope by a 20 N force parallel to the slope, starting from rest. Find its final speed.
  3. A wind turbine has a useful output of 2.0 MW and an efficiency of 40 %. How much energy does the wind deliver to it in one hour?
Show answers
  1. when it is perpendicular to the displacement
  2. the work done
  3. ½mv2 and p2/2m
  4. v = √(2 × 9.8 × 5.0) = 9.9 m s−1
  5. kg m2 s−3 (W = J s−1)
  6. 50/200 = 25 %
  7. ΔEk = ½ × 1500 × (252 − 102) = 393 750 J
    P = 393 750 / 8.0
    ≈ 49 kW
  8. work by push = 20 × 3.0 = 60 J
    ΔEp = 2.0 × 9.8 × 3.0 sin 25° = 24.8 J
    Ek = 35.2 J = ½(2.0)v2
    v = 5.9 m s−1
  9. input power = 2.0/0.40 = 5.0 MW
    E = 5.0 × 106 × 3600
    = 1.8 × 1010 J
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Next chapterA.4 Rigid body mechanics →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.