Chapter 5 of 8

All 8 chapters
  1. 1Calculating energy stores, readunderstood
  2. 2Springs and elastic energy, readunderstood
  3. 3Work done, readunderstood
  4. 4Power, readunderstood
  5. 5Efficiency, readunderstood
  6. 6Specific heat capacity, readunderstood
  7. 7Energy resources, readunderstood
  8. 8Energy at home, readunderstood

S4.9.5 Chapter 5

Efficiency

A lamp takes in 100 J of electrical energy and gives out 20 J of light. What is its efficiency?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

  • You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Calculate efficiency as useful output over total input
  • Draw a Sankey diagram for a lamp or a motor

Most machines do not turn all the energy they are given into the energy you want. A lamp is meant to give light, but it also gets hot. What counts as useful depends on the job. Efficiency tells you how much of the energy is useful.

efficiency = useful energy output ÷ total energy input. The answer is a decimal between 0 and 1. Multiply by 100 to write it as a percentage.

The energy that is not useful is wasted energy. wasted energy = total input − useful output. Wasted energy is not destroyed. It spreads into the surroundings, mostly as heat.

A Sankey diagram shows this. It is an arrow that splits. The arrow going in is as wide as the energy put in. Each arrow going out is as wide as its energy. The useful arrow goes straight on. The wasted arrows bend away and are hatched.

A Sankey diagram of a filament lamp. A wide band labelled electrical 100 J comes in from the left. A thin band going straight on is labelled light 6 J. A wide hatched band bending down is labelled thermal 94 J.electrical 100 Jlight 6 Jthermal 94 J
A typical filament lamp. Of 100 J that go in, only about 6 J come out as light.

Worked example

A filament lamp takes in 100 J of electrical energy and gives out 6 J of light. Find its efficiency as a decimal and as a percentage.

  1. efficiency = useful output ÷ total input.
  2. efficiency = 6 ÷ 100 = 0.06.
  3. As a percentage, 0.06 × 100 = 6%.

Answer: 6%

An LED lamp is a different design. It is far better at turning electrical energy into light. Draw it with the same input band, and the light arrow is clearly wider.

A Sankey diagram of an LED lamp. A wide band labelled electrical 100 J comes in from the left. A band going straight on is labelled light 30 J. A hatched band bending down is labelled thermal 70 J.electrical 100 Jlight 30 Jthermal 70 J
A typical LED lamp, drawn on the same scale. Of 100 J, about 30 J come out as light.

Both lamps are drawn in the same way, and the widths are exact. The two output arrows always add up to the input arrow. If they did not, energy would have appeared from nowhere, or vanished.

Worked example

A motor takes in 400 J of electrical energy. It gives out 300 J of kinetic energy. How much energy is wasted, and what is the efficiency?

  1. wasted energy = input − useful output = 400 − 300 = 100 J.
  2. efficiency = 300 ÷ 400 = 0.75.
  3. As a percentage, 0.75 × 100 = 75%.

Answer: 100 J wasted, efficiency 75%

A motor can waste energy in more than one way. Its parts rub together and warm up. It also makes some sound. A Sankey diagram draws each waste as its own bent arrow. The smaller arrow bends inside the larger one, and each has its label below its tip.

A Sankey diagram of an electric motor. A wide band labelled electrical 100 J comes in from the left. A band going straight on is labelled kinetic 70 J. Two hatched bands bend down, one labelled thermal 20 J and one labelled sound 10 J.electrical 100 Jkinetic 70 Jsound 10 Jthermal 20 J
A motor. 70 + 20 + 10 = 100, so the arrows add up. The efficiency is 70 ÷ 100 = 0.7.

To draw a Sankey diagram on squared paper, follow these steps. Pick a scale, such as 1 square for each 10 J. Draw the input arrow that many squares wide. Split it into the output arrows, each as wide as its joules. Check that the output widths add up to the input width. Hatch the wasted arrows.

You can use power instead of energy. efficiency = useful output power ÷ total input power. So useful output power = efficiency × input power.

Worked example

A motor has an input power of 300 W and an efficiency of 0.8. How much energy does it waste in 3 s?

  1. useful output power = efficiency × input power.
  2. useful output power = 0.8 × 300 = 240 W.
  3. wasted power = 300 − 240 = 60 W.
  4. energy wasted = 60 × 3 = 180 J.

Answer: 180 J

Three mistakes. Putting the sum upside down gives an answer above 1, which is impossible. Forgetting to multiply by 100 leaves a decimal where a percentage was asked for. And mixing up watts and joules: use power with power, or energy with energy.

Machines can be made more efficient. Oil on moving parts reduces friction. A lagged pipe loses less heat. A streamlined shape loses less to the air. Each change cuts the wasted energy.

A wrong idea many people have

A machine with an efficiency of 120% is better than one with an efficiency of 80%.

Why it feels right: In school, 120 looks bigger and better than 80, and some adverts talk about "over 100%".

The test that breaks it: If 100 J go in and 120 J of useful energy come out, 20 J must have appeared from nowhere. Energy is never made, so this cannot happen.

What is true: Efficiency can never be more than 1, or 100%. Almost every machine wastes some energy. A result above 100% means the sum is upside down or something was missed.

A wrong idea many people have

Wasted energy is destroyed.

Why it feels right: The energy is gone from the lamp or the motor. It is no use to you, so it seems to have vanished.

The test that breaks it: Hold your hand near a lamp that has been on for a few minutes. It is warm. The wasted energy has gone into the thermal store of the lamp and the air.

What is true: Wasted energy is not destroyed. It spreads into the surroundings, mostly as thermal energy, where it is too spread out to be useful.

Try it

The efficiency of a bounce

You need: a ball that bounces, a wall or a door, a ruler, tape measure or string, a hard floor.

Be safe: Bounce the ball in a clear space, away from windows and from people.

  1. Mark a height of 1 m on the wall, from the floor.
  2. Hold the bottom of the ball level with the mark. Drop it, do not throw it.
  3. Watch the top of the bounce and mark how high the bottom of the ball rises. Ask a friend to help.
  4. Divide the bounce height by the drop height. This is the efficiency of the bounce.
  5. Try a soft surface, such as a folded blanket, and compare.

What you will see: The ball has gravitational potential energy m × g × h, so for one ball the energy is in proportion to the height. A ball that drops 1 m and bounces 0.6 m returns 0.6 of its energy, an efficiency of 0.6, or 60%. The rest is wasted as heat and sound. On a soft surface the bounce is lower, so the efficiency is lower.

If you cannot try it, this is what happens.
A table of bounce results from a drop of 1 m, with the columns Floor, Bounce and Efficiency. Tiles: bounce 0.6 m, efficiency 0.6. Carpet: bounce 0.3 m, efficiency 0.3.FloorBounceEfficiencytiles0.6 m0.6carpet0.3 m0.3

Key words

efficiency
useful energy output ÷ total energy input, as a decimal up to 1 or a percentage up to 100%.
wasted energy
energy that is not useful. It spreads into the surroundings, mostly as heat.
Sankey diagram
a picture of energy transfers where each arrow is as wide as its energy.

Back to your guess

The answer is A: 20%

Efficiency is useful energy out ÷ energy in: 20 ÷ 100 = 0.2, which is 20%. It can never be above 100%.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A lamp takes in 50 J of electrical energy and gives out 15 J of light. What is its efficiency?

    Tap your answer.

    1. Not this one. You divided the input by the output, which is upside down. An efficiency can never be above 1. Try another.

    2. Not this one. You took the output from the input. That gives the wasted energy, 35 J, not the efficiency. Try another.

    3. Yes, that is right. efficiency = useful output ÷ total input = 15 ÷ 50 = 0.3.

    4. Not this one. You divided the wasted energy, 35 J, by the input. Use the useful energy, 15 J. Try another.

    See the answer

    0.3

    efficiency = useful output ÷ total input = 15 ÷ 50 = 0.3.

    • If you chose A: You divided the input by the output, which is upside down. An efficiency can never be above 1.
    • If you chose B: You took the output from the input. That gives the wasted energy, 35 J, not the efficiency.
    • If you chose D: You divided the wasted energy, 35 J, by the input. Use the useful energy, 15 J.

  2. The diagram shows a radio. What is its efficiency as a percentage?

    A Sankey diagram of a radio. A wide band labelled electrical 80 J comes in from the left. A band going straight on is labelled sound 20 J. A hatched band bending down is labelled thermal 60 J.electrical 80 Jsound 20 Jthermal 60 J
    Tap your answer.

    1. Not this one. You used the wasted energy, 60 J, instead of the useful sound, 20 J. Try another.

    2. Not this one. You used the useful energy, 20 J, as the percentage. Divide it by the input, 80 J. Try another.

    3. Not this one. You divided the wasted energy by the useful energy. That is not an efficiency, and it can never be above 100%. Try another.

    4. Yes, that is right. The useful output is the sound, 20 J. efficiency = 20 ÷ 80 = 0.25, which is 25%.

    See the answer

    25%

    The useful output is the sound, 20 J. efficiency = 20 ÷ 80 = 0.25, which is 25%.

    • If you chose A: You used the wasted energy, 60 J, instead of the useful sound, 20 J.
    • If you chose B: You used the useful energy, 20 J, as the percentage. Divide it by the input, 80 J.
    • If you chose C: You divided the wasted energy by the useful energy. That is not an efficiency, and it can never be above 100%.

  3. A motor takes in 200 J of electrical energy and gives out 150 J of kinetic energy. You draw its Sankey diagram with 1 square for each 10 J. How many squares wide is the wasted arrow?

    A table of the energy of a motor with the columns Energy and Amount. The input is 200 J. The useful kinetic energy is 150 J.EnergyAmountinput200 Juseful kinetic150 J
    Tap your answer.

    1. Not this one. That is the useful arrow, 150 J. The wasted arrow is the rest of the input. Try another.

    2. Yes, that is right. The wasted energy is 200 − 150 = 50 J. At 1 square for each 10 J, the wasted arrow is 50 ÷ 10 = 5 squares wide. The useful arrow is 15 squares and the two add up to the 20 squares of the input.

    3. Not this one. That is the input arrow, 200 J. The wasted arrow is only the part that is not useful. Try another.

    4. Not this one. You found the 50 J that is wasted but forgot the scale. One square is 10 J, so divide by 10. Try another.

    See the answer

    5 squares

    The wasted energy is 200 − 150 = 50 J. At 1 square for each 10 J, the wasted arrow is 50 ÷ 10 = 5 squares wide. The useful arrow is 15 squares and the two add up to the 20 squares of the input.

    • If you chose A: That is the useful arrow, 150 J. The wasted arrow is the rest of the input.
    • If you chose C: That is the input arrow, 200 J. The wasted arrow is only the part that is not useful.
    • If you chose D: You found the 50 J that is wasted but forgot the scale. One square is 10 J, so divide by 10.

  4. A machine takes in 400 J of energy. A student says it gives out 500 J of useful energy. Could this be true?

    Tap your answer.

    1. Not this one. Energy is never made. It can only move from one store to another. Go to S3.8.3. Try another.

    2. Yes, that is right. Energy is never made. The useful output plus the wasted output equals the input, so the useful output is at most 400 J.

    3. Not this one. Even a very efficient machine gives out no more useful energy than it takes in. Efficiency never goes above 100%. Try another.

    4. Not this one. With no friction nothing is wasted as heat, and the output equals the input at most. It still cannot be more. Try another.

    See the answer

    No, because a machine cannot give out more energy than it takes in.

    Energy is never made. The useful output plus the wasted output equals the input, so the useful output is at most 400 J.

    • If you chose A: Energy is never made. It can only move from one store to another. Go to S3.8.3.
    • If you chose C: Even a very efficient machine gives out no more useful energy than it takes in. Efficiency never goes above 100%.
    • If you chose D: With no friction nothing is wasted as heat, and the output equals the input at most. It still cannot be more.

  5. A motor has an input power of 200 W and an efficiency of 0.6. What is its useful output power?

    Tap your answer.

    1. Yes, that is right. useful output power = efficiency × input power = 0.6 × 200 = 120 W.

    2. Not this one. You found the wasted power, 200 × 0.4 = 80 W. The question asks for the useful power. Try another.

    3. Not this one. You gave the whole input as useful power. That would mean no energy is wasted. Useful power = efficiency × input power = 0.6 × 200. Try another.

    4. Not this one. You divided by 0.6. The output is the efficiency times the input, and it cannot be more than the input. Try another.

    See the answer

    120 W

    useful output power = efficiency × input power = 0.6 × 200 = 120 W.

    • If you chose B: You found the wasted power, 200 × 0.4 = 80 W. The question asks for the useful power.
    • If you chose C: You gave the whole input as useful power. That would mean no energy is wasted. Useful power = efficiency × input power = 0.6 × 200.
    • If you chose D: You divided by 0.6. The output is the efficiency times the input, and it cannot be more than the input.

  6. Which statement about wasted energy is true?

    Tap your answer.

    1. Not this one. Energy is never destroyed. The total stays the same. Go to S3.8.3. Try another.

    2. Not this one. If it were stored somewhere useful, it would not be wasted. It spreads out, too thinly to use. Try another.

    3. Not this one. Spread out heat is too thin to do the job again. It does not change back by itself. Try another.

    4. Yes, that is right. Wasted energy moves into the thermal store of the surroundings. It is not lost, but it is no longer useful.

    See the answer

    It spreads into the surroundings, mostly as thermal energy.

    Wasted energy moves into the thermal store of the surroundings. It is not lost, but it is no longer useful.

    • If you chose A: Energy is never destroyed. The total stays the same. Go to S3.8.3.
    • If you chose B: If it were stored somewhere useful, it would not be wasted. It spreads out, too thinly to use.
    • If you chose C: Spread out heat is too thin to do the job again. It does not change back by itself.

  7. A room heater takes in 500 J of electrical energy and gives out 500 J of thermal energy to the room. The job of a heater is to warm the room. What is its efficiency for that job?

    Tap your answer.

    1. Not this one. For a heater, the heat is the job it is built to do. It is useful, not wasted. Try another.

    2. Not this one. Nothing here is lost. All 500 J ends up warming the room. Try another.

    3. Yes, that is right. The useful output is the thermal energy in the room, 500 J. efficiency = 500 ÷ 500 = 1, which is 100%. For a heater, the heat is the useful output. Only a device whose job is heating, such as an electric heater, can reach 100%.

    4. Not this one. Efficiency is the output divided by the input, 500 ÷ 500, and not the output alone. Try another.

    See the answer

    100%

    The useful output is the thermal energy in the room, 500 J. efficiency = 500 ÷ 500 = 1, which is 100%. For a heater, the heat is the useful output. Only a device whose job is heating, such as an electric heater, can reach 100%.

    • If you chose A: For a heater, the heat is the job it is built to do. It is useful, not wasted.
    • If you chose B: Nothing here is lost. All 500 J ends up warming the room.
    • If you chose D: Efficiency is the output divided by the input, 500 ÷ 500, and not the output alone.

  8. From an earlier chapter

    A machine transfers 150 J in 10 s. What is its power?

    Tap your answer.

    1. Not this one. You took the time from the energy. Power is energy divided by time. Go to S4.9.4. Try another.

    2. Yes, that is right. P = E ÷ t = 150 ÷ 10 = 15 W.

    3. Not this one. You multiplied the energy by the time. Power is energy divided by time. Go to S4.9.4. Try another.

    See the answer

    15 W

    P = E ÷ t = 150 ÷ 10 = 15 W.

    • If you chose A: You took the time from the energy. Power is energy divided by time. Go to S4.9.4.
    • If you chose C: You multiplied the energy by the time. Power is energy divided by time. Go to S4.9.4.

  9. A device takes in 250 J of energy and gives out 100 J of useful energy. What is its efficiency as a percentage?

    See the answer

    40%

    efficiency = 100 ÷ 250 = 0.4, and 0.4 × 100 = 40%.

  10. A motor takes in 600 J of energy and is 85% efficient. How much energy does it waste, in joules?

    See the answer

    90 J

    The useful energy is 85% of 600 J = 0.85 × 600 = 510 J. The wasted energy is 600 − 510 = 90 J.

  11. A motor has an input power of 900 W and an efficiency of 0.4. What is its useful output power, in watts?

    See the answer

    360 W

    useful output power = efficiency × input power = 0.4 × 900 = 360 W.

Keep it

Copy this into your exercise book.

Efficiency = useful output ÷ total input. At most 1, or 100%. Wasted = input − useful, and it spreads out as heat. A Sankey diagram: output arrows add up to the input arrow.

Draw: A Sankey diagram of a lamp: a wide arrow in, a thin light arrow straight on, a wide hatched arrow bending away.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (758 KB). It opens on any phone.