4.4 & 4.5 Conservation of Energy, Efficiency and Sankey Diagrams — Physics with Kate

.pwk-lesson [data-board]{display:block !important;} .pwk-lesson span[data-board], .pwk-lesson a[data-board]{display:inline-block !important;} /* the player's src is set by script, so with no JS show the "coming soon" panel and hide the empty player */ .pwk-lesson [data-video-state="ready"]{display:none !important;}

Your exam board

4.4 & 4.5

Edexcel International GCSE Physics · Section 4 Energy resources and energy transfer

Conservation of Energy, Efficiency and Sankey Diagrams

In this lesson you'll learn to: state the principle of conservation of energy, explain why wasted energy is dissipated rather than lost, draw and read a Sankey diagram drawn to scale, and use efficiency = (useful energy output ÷ total energy input) × 100% forwards and backwards.

▶ Prefer to watch? Scroll to the bottom for the video on this topic ▶ A video walkthrough is coming soon — it will appear at the bottom of this page

▶

Start here — build a Sankey diagram

Don't read anything yet. Pick an object, then drag the efficiency slider and watch the arrows. The input comes in on the left, the useful output carries straight on to the right, and the wasted energy turns down at the bottom. Every width is drawn to scale, and the joules underneath change with it.

Then press Different object →. A car, a filament lamp and an LED are the same three arrows with different labels — that is the whole trick to this topic.

Object:

Efficiency 25% Total energy input 2000 J

The calculation, live

efficiency = (useful energy output ÷ total energy input) × 100%

Every arrow is drawn to scale — its width is proportional to the joules it carries. Copy the black outline; the colours are only here to help you see the split.

▲ Drag efficiency to 100% and watch the wasted arrow vanish — that is the diagram of a device that cannot exist.

Now learn the two things the examiner wants

One sentence for conservation of energy, and one equation used three different ways. The notes below give you both, plus the three marks a Sankey diagram is worth.

Read the notes ↓

✎

Revision notes

4.4 The principle of conservation of energy

This is a one-mark definition that appears again and again, so learn the wording exactly.

Learn this definition

Energy cannot be created or destroyed. It can only be transferred from one store to another, stored, or dissipated. The total energy in a closed system always stays the same.

That means the energy you put in has to come out somewhere. Split what comes out into two piles and you get the sentence the whole of this topic rests on:

⭐ The one line that unlocks every question

total energy input = useful energy output + wasted energy

So if a question gives you any two of those three, you can always find the third by subtracting. No equation needed.

"Lost" energy is not lost

Wasted energy is dissipated — spread out into the thermal store of the surroundings, usually by friction, air resistance, sound, or the resistance of wires. It is spread so thinly that we cannot get it back and do anything useful with it, but it is still there. Saying energy is "lost" or "used up" contradicts 4.4 and loses marks.

✗ Don't write

"Some of the energy is lost as heat."
"The energy is used up by friction."

✓ Write instead

"Some energy is dissipated to the thermal store of the surroundings."
"Energy is transferred to the thermal store by friction."

Useful and wasted — it depends on the job

The same store can be useful in one device and wasted in another. Thermal energy is waste in a car engine, but it is the entire point of a kettle. Always ask: what is this device for?

Device	Energy input	Useful output	Wasted
Petrol car	chemical	kinetic	thermal, sound
Filament lamp	electrical	light	thermal
LED lamp	electrical	light	thermal (much less)
Electric kettle	electrical	thermal (in the water)	thermal (kettle, air), sound
Wind-up toy	elastic	kinetic	thermal
Gas power station	chemical	electrical	thermal

Sankey diagrams

A Sankey diagram is a picture of conservation of energy. The width of each arrow is proportional to the amount of energy it carries, so the diagram shows at a glance how much is useful and how much is wasted.

input — enters from the left useful — carries on to the right wasted — turns downwards

Because energy is conserved, the arrows leaving must add up to exactly the width of the arrow going in. If they don't, the diagram is wrong — and that is often how a question is marked.

💡 Reading a Sankey in the exam

If the diagram is drawn on a grid, count the squares. If a 40-square-wide input arrow splits into a 10-square useful arrow, the efficiency is 10 ÷ 40 = 0.25 = 25%. You can answer an efficiency question straight from the widths without being told any joules at all.

4.5 Efficiency

Efficiency tells you what fraction of the energy you put in came out doing the job you wanted.

efficiency = (useful energy output ÷ total energy input) × 100%

the version on the Edexcel iGCSE formula list

Symbol	Quantity	Unit
—	efficiency	no unit (a % or a decimal)
E	useful energy output	joule, J
E	total energy input	joule, J

There is a second version for when a question gives you power in watts instead of energy in joules. It works exactly the same way:

efficiency = (useful power output ÷ total power input) × 100%

use this one whenever the numbers are in watts

💡 Input or output? Both are right — here's why

Some books write "÷ total energy output" instead of "÷ total energy input". By conservation of energy those two are equal: everything that goes in must come back out somewhere. Edexcel's formula list says total energy input, so use that one and you can never be marked down.

Percentage or decimal?

Divide first: 40 ÷ 200 = 0.2. That is the efficiency as a decimal.
Multiply by 100 to get 20%. That is the same answer as a percentage.
Efficiency has no unit. Never write joules or watts after it.
Efficiency can never be more than 1 (or 100%). If yours is, you have divided the wrong way round — swap the numbers over.

Rearranging the equation

Three questions, one equation. Learn these three shapes:

Given input and efficiency → useful output = efficiency × total input
(remember to turn a percentage into a decimal first: 20% → 0.20)
Given useful output and efficiency → total input = useful output ÷ efficiency
Given input and useful output → wasted = total input − useful output

Why nothing is ever 100% efficient

Every real device has moving parts that rub, wires with resistance, or air pushing against it. Each of those transfers some energy to the thermal store of the surroundings, so there is always a wasted arrow on the Sankey diagram. You can make it thinner, but you can never remove it:

Lubrication — oil between moving parts reduces friction.
Streamlining — a smoother shape reduces air resistance.
Insulation — a material with a low thermal conductivity slows energy escaping.
Thicker, lower-resistance wires — less energy dissipated in the cables.

Reducing the wasted energy makes the useful arrow wider for the same input, so the efficiency goes up.

✍

How to draw a Sankey diagram for 3 marks

The workbook question asks for a labelled Sankey diagram and it is worth 3 marks. Those three marks are always for the same three things:

Work out the three numbers first. Total input, useful output, and wasted = input − useful. Do the arithmetic before you pick up a ruler.
Draw the input arrow from the left, to scale. Pick a simple scale — 1 cm per 50 J, or one grid square per 10 J — and make the arrow that wide.
Split it: useful straight on, wasted turning down. The two widths must add up to the input width. Use the same scale for all three.
Label every arrow with the store AND the number of joules. "kinetic 40 J", "thermal 160 J". A width with no label scores nothing.

⭐ The three marks, in the mark scheme's words

1. Arrows drawn to scale (widths in proportion to the energies).
2. Useful output going forwards, wasted energy turning downwards.
3. All arrows labelled with the correct store and value.

🔢

Worked examples

EXAMPLE 1A wind-up toy robot stores 200 J in the elastic store of its spring. It transfers this to its kinetic store with an efficiency of 20%. Calculate the energy transferred to the thermal store of the surroundings.

Useful output first — turn the % into a decimal useful output = 0.20 × 200 J = 40 J

Wasted = total in − useful out wasted = 200 − 40 = 160 J

energy transferred to the thermal store = 160 J Set the animation to the wind-up toy, 200 J, 20% and you get exactly this diagram.

20% efficient — the wasted arrow is four times the useful one.

EXAMPLE 2A filament lamp transfers 60 J of energy from the mains each second. Only 3 J of that is transferred as light. Calculate its efficiency.

Put the numbers into the equation efficiency = (3 ÷ 60) × 100%

Divide, then multiply by 100 = 0.05 × 100% = 5%

efficiency = 5% The unit is % when stating 5%, else no unit if given when a decimal is used 0.05. The other 57 J is dissipated to the thermal store — which is why old bulbs were too hot to touch.

5% efficient — almost the whole arrow turns down.

EXAMPLE 3An electric motor is 75% efficient and transfers 300 J to the kinetic store of a load. Calculate the total energy input, and the energy wasted.

Rearrange: total input = useful output ÷ efficiency total input = 300 ÷ 0.75 = 400 J

Then subtract for the wasted energy wasted = 400 − 300 = 100 J

total energy input = 400 J, energy wasted = 100 J Sense check: 300 J out of 400 J really is three quarters, so 75% is right.

75% efficient — three quarters of the width carries on.

EXAMPLE 4A kettle has a power input of 2000 W. It transfers energy to the water at a useful rate of 1800 W. Calculate the efficiency.

Watts, so use the power version — it works the same way efficiency = (1800 ÷ 2000) × 100%

Divide, then multiply by 100 = 0.9 × 100% = 90%

efficiency = 90% Never mix the two: joules with joules, watts with watts.

90% efficient — power splits exactly like energy.

EXAMPLE 5Reading a Sankey. A Sankey diagram for a hairdryer shows an input arrow 50 squares wide. The useful thermal output arrow is 36 squares and the kinetic (fan) arrow is 4 squares. How wide is the wasted (sound) arrow, and what is the efficiency?

Conservation of energy — the widths must balance wasted = 50 − 36 − 4 = 10 squares

Useful output is the heating AND the fan — both do the job efficiency = ((36 + 4) ÷ 50) × 100% = 0.8 × 100% = 80%

wasted arrow = 10 squares, efficiency = 80% You never needed a single joule — widths are proportional to energy, so the ratio is all that matters.

Widths in grid squares. Two useful arrows carry on, one wasted arrow turns down.

📘 Now do it in your workbook

The efficiency formula, a Sankey diagram to label, and the full exam-style wind-up robot question — with worked answers.

Edexcel iGCSE Physics Workbook · ENERGY · page 6

⬇ Free: page 6 (PDF) Get the full workbook on Amazon

Free sample = the page for this lesson. The full workbook covers the whole of Section 4 Energy resources and energy transfer with exam-style questions and worked answers.

▶

Prefer to watch? Here's the whole thing

Video walkthrough — coming soon

Conservation of energy, the efficiency equation used three ways, and a Sankey diagram drawn from scratch for the three marks — explained out loud.

Conservation of energy, the efficiency equation used three ways, and a Sankey diagram drawn from scratch for the three marks — the video is on its way.

🎬

Video coming soon

The walkthrough for this lesson is being filmed and will appear right here. Until then, the notes and the free workbook pages above cover every mark.

▲ In GHL you can also use the lesson's built-in video field instead of this embed.

Now put it into practice

Page 6 of the workbook is free. It has the wind-up robot question in full — the 4-mark calculation and the 3-mark Sankey diagram — with the worked answers, so you can see exactly where each mark comes from.

⬇ Free: page 6 (PDF) Full workbook on Amazon