Energy, work and power. Attnom Academy

S4.9.1 Chapter 1

Calculating energy stores

A car drives at 10 m/s and then at 20 m/s. What happens to its kinetic energy?

Your guess

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

Your guess

A It stays the same, because the car is the same car.
You picked A. Read on. At the end, the chapter comes back to your guess.
B It doubles.
You picked B. Read on. At the end, the chapter comes back to your guess.
C It becomes four times as big.
You picked C. Read on. At the end, the chapter comes back to your guess.
D It depends only on the mass, not on the speed.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate kinetic energy and gravitational potential energy
Use energy conservation to find the speed of a falling object

The car is the same car, but the faster drive holds more energy. How much more? Energy is counted in joules, written J, so we can work it out exactly. A raised thing holds energy too, and we can count that as well.

Kinetic energy is the energy in the kinetic store of a moving thing. It depends on two things: the mass m in kg and the speed v in m/s.

kinetic energy = ½ × mass × speed². In symbols, Ek = ½ × m × v². The speed is squared. The mass is not.

A 2 kg ball. Double the speed, from 1 m/s to 2 m/s, and the energy is four times as big: 1 J becomes 4 J.

Look at the bars. Speeds of 1, 2, 3 and 4 m/s give energies of 1, 4, 9 and 16 J. The energy grows much faster than the speed.

Worked example

A car has a mass of 1 500 kg and moves at 20 m/s. How much kinetic energy does it have?

Ek = ½ × m × v².
v² = 20 × 20 = 400.
Ek = ½ × 1 500 × 400 = 300 000 J = 300 kJ.

Answer: 300 kJ

Three mistakes are common here. First, squaring the mass instead of the speed. Second, leaving out the half. Third, using the wrong units. Mass must be in kg, not g. Speed must be in m/s, not km/h.

To change km/h into m/s, divide by 3.6. A speed of 72 km/h is 72 ÷ 3.6 = 20 m/s.

Gravitational potential energy is the energy in the gravitational potential store of a raised thing. It depends on the mass m, the height h in m, and the gravitational field strength g. The height is measured above the place you choose as zero, such as the floor.

Near the surface of the Earth, g is about 9.8 N/kg. It changes a little from place to place. On the Moon it is about 1.6 N/kg, so the same lift stores less energy. Some books use 10 N/kg to shorten sums. This book uses 9.8 N/kg.

gravitational potential energy = mass × g × height. In symbols, Ep = m × g × h.

The numbers for the bag. Each one has its unit.

Worked example

A bag of 3 kg is lifted 4 m. How much energy does it gain in its gravitational potential store? Use g = 9.8 N/kg.

Ep = m × g × h.
m × g = 3 × 9.8 = 29.4.
Ep = 29.4 × 4 = 117.6 J.

Answer: 117.6 J

Energy is never made or destroyed. When a ball falls, gravitational potential energy changes into kinetic energy. This is the conservation of energy. With no air resistance, the two always add up to the same total.

Draw it first

A falling ball keeps its total

1 At 5 m the ball is still. Ek is 0 J. Ep is 98 J, all of the energy.
2 At 2.5 m half the height is gone. Ep is 49 J and Ek is 49 J.
3 At 0 m, on the ground, Ep is 0 J and Ek is 98 J. The total is still 98 J.

Energy moves from one store to the other. None of it is lost on the way.

Now use this to find a speed. Just before the ball lands, all the energy is kinetic. So ½ × m × v² = m × g × h. The mass m is on both sides, so it cancels. That leaves v² = 2 × g × h. Then v is the square root of 2 × g × h.

The mass has gone. The speed of a fall does not depend on the mass. This holds when air resistance is too small to matter. In air, a real ball reaches a little less than this speed.

A 0.5 kg ball dropped from 10 m. All 49 J ends up as kinetic energy.

Worked example

A ball is dropped from a height of 10 m. Ignore air resistance. How fast is it moving just before it lands? Use g = 9.8 N/kg.

All the gravitational potential energy becomes kinetic energy: ½ × m × v² = m × g × h.
The mass cancels. So v² = 2 × g × h. Here 2 × g = 19.6.
v² = 19.6 × 10 = 196.
v = √196 = 14 m/s.

Answer: 14 m/s

A wrong idea many people have

A heavy object falls faster, so it lands with more speed.

Why it feels right: A heavy stone does hit the ground harder than a feather, and a feather floats down slowly.

The test that breaks it: Drop a small stone and a large stone together from the same height onto soft soil. You hear one thud. In v² = 2 × g × h the mass cancels, so the speed cannot depend on it.

What is true: With no air resistance, everything dropped from the same height lands at the same speed. A heavy object has more kinetic energy at that speed, so it hits harder, but it is not faster. A feather is slowed by the air.

Try it

Do heavy things fall faster?

You need: a small stone, a larger stone, soft ground, such as soil or sand, a friend to listen with you.

Be safe: Drop only onto soft ground and keep your feet well back. Never drop things from a window, a roof or any high place.

Stand on soft ground. Hold the two stones side by side at the height of your chest.
Open both hands at the same moment.
Listen to the landing. Is it one sound or two?
Repeat it three times. Try a potato in place of one stone.

What you will see: You hear one thud each time. Both stones land together, so they speed up in the same way. The larger stone has more kinetic energy when it lands, which is why it digs a deeper hole in the soil.

If you cannot try it, this is what happens.

Key words

kinetic energy: the energy in the kinetic store of a moving thing, Ek = ½ × m × v².
gravitational potential energy: the energy in the gravitational potential store of a raised thing, Ep = m × g × h.
gravitational field strength: the force of gravity on each kg of mass, written g. Near the Earth it is about 9.8 N/kg.
conservation of energy: energy is never made or destroyed, only moved from one store to another.

Back to your guess

The answer is C: It becomes four times as big.

The speed is squared in the sum. Twice the speed means 2 × 2 = 4 times the kinetic energy.

Keep it

Copy this into your exercise book.

Ek = ½ × m × v². Ep = m × g × h, with g = 9.8 N/kg. A falling thing turns Ep into Ek. Ignoring air, v² = 2 × g × h, and the mass cancels.

Draw: A ball at three heights, with a bar beside each. Every bar is the same length, split into Ep and Ek.

S4.9.2 Chapter 2

Springs and elastic energy

Within its limit, a spring stretches 2 cm under a 1 N weight. How far will a 3 N weight stretch it?

Your guess

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

Your guess

A 3 cm
You picked A. Read on. At the end, the chapter comes back to your guess.
B 5 cm
You picked B. Read on. At the end, the chapter comes back to your guess.
C 9 cm
You picked C. Read on. At the end, the chapter comes back to your guess.
D 6 cm
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use Hooke's law to calculate the extension of a spring
Calculate the energy stored in a stretched spring

A weight hung on a spring makes it longer. Keep your guess in mind as you read, and see how far a spring stretches and how much energy it keeps.

A spring has a natural length. That is its length with nothing pulling on it. Hang a load on it and it grows longer. The extension is how much longer: extension = length now − natural length.

See it three ways

As a picture

In words

A spring of natural length 10 cm is 14 cm long with a 4 N load. Its extension is 14 − 10 = 4 cm. Twice the load gives twice the extension.

In symbols

e = length now − natural length. F = k × e.

Robert Hooke found that the extension of a spring is proportional to the force on it. Double the force and the extension doubles. This is Hooke's law.

force = spring constant × extension. In symbols, F = k × e. The force F is in N. The extension e is in m. The spring constant k is in N/m. A stiff spring has a large k.

Spring S has a natural length of 10 cm. Each 2 N of load adds 2 cm.

Mind the units. The sum needs the extension in metres. A stretch of 4 cm is 4 ÷ 100 = 0.04 m. Forgetting to change cm to m is the commonest slip in this chapter.

Worked example

Spring S has a natural length of 10 cm. A load of 4 N makes it 14 cm long. Find its spring constant k in N/m.

The extension is 14 − 10 = 4 cm = 0.04 m.
k = F ÷ e = 4 ÷ 0.04 = 100 N/m.

Answer: 100 N/m

Worked example

A spring has k = 250 N/m. A load of 15 N hangs on it. The spring stays within its limit. What is its extension in cm?

F = k × e, so e = F ÷ k.
e = 15 ÷ 250 = 0.06 m.
e = 0.06 × 100 = 6 cm.

Answer: 6 cm

Hooke's law has a limit. If you pull a spring too far, the extension grows faster than the force. The line on a graph of force against extension stops being straight. The point where it bends is the limit of proportionality.

The line is straight up to the limit, then it bends. Past the limit, F = k × e does not work.

Past the limit of proportionality a spring may also stay stretched when you let go. It does not return to its natural length. So keep loads small when you test a spring. The point past which a spring stays stretched is the elastic limit. It is at or a little past the limit of proportionality.

Stretching a spring puts energy into its elastic store. The force is 0 at the start and F at the end. The energy stored is the area under the line on the graph. That area is a triangle: ½ × base × height = ½ × e × F.

At 6 cm the force is 6 N. The hatched triangle under the line from 0 to 6 cm has an area equal to the stored energy.

Put F = k × e into ½ × e × F. The energy stored is ½ × k × e². This is the elastic potential energy, written Ee. It works only up to the limit of proportionality, where the line is straight.

Note the square. The extension is squared. Double the extension and the energy is four times as big.

Worked example

Spring S has k = 100 N/m. It is stretched by 6 cm, which is within its limit. How much energy is stored in it?

Ee = ½ × k × e².
e = 6 cm = 0.06 m.
e² = 0.06 × 0.06 = 0.0036.
Ee = ½ × 100 × 0.0036 = 0.18 J.

Answer: 0.18 J

A wrong idea many people have

Stretch a spring twice as far and it stores twice the energy.

Why it feels right: Twice the stretch needs twice the force, so it seems the energy should double as well.

The test that breaks it: Take k = 100 N/m. A stretch of 2 cm stores ½ × 100 × 0.02 × 0.02 = 0.02 J. A stretch of 4 cm stores ½ × 100 × 0.04 × 0.04 = 0.08 J. That is four times as much.

What is true: Both the force and the distance double, so the energy goes up four times. The extension is squared in ½ × k × e².

A wrong idea many people have

F = k × e holds for any stretch of a spring, however far.

Why it feels right: A formula in a book looks like a law of nature that always holds.

The test that breaks it: Look at the graph. The line is straight only up to the limit of proportionality. After that it bends, so the force is no longer in proportion to the extension.

What is true: F = k × e and Ee = ½ × k × e² work only up to the limit of proportionality. Beyond it a spring may be damaged.

Try it

Test a spring with water

You need: a spring from an old pen, string, a small plastic cup, water, a teaspoon, a ruler with mm marks, a pencil laid across two cups, to hang it from.

Be safe: Never let go of a stretched spring: it can snap back into your face. Do not stretch the spring far. Keep your face away from it.

Hang the spring from the pencil. Tie the cup to its lower end with string.
With the empty cup hanging, measure the length of the spring in mm. Call it the starting length. The cup already stretches the spring a little, so the extensions you find are for the water alone.
Add 10 level teaspoons of water, about 50 mL. This is a load of about 0.5 N. Measure the length again.
Add 50 mL twice more, up to 150 mL, and measure each time.
For each load, work out the extension: length now minus the starting length.

What you will see: The extension doubles when the water doubles, and triples when the water triples. The stretch is proportional to the load. Your numbers will differ with your spring, but the pattern is the same. With much more water the spring may stretch faster than the load grows. That is the limit of proportionality.

If you cannot try it, this is what happens.

Key words

natural length: the length of a spring with nothing pulling on it.
extension: how much longer a spring is than its natural length. Change it to m before using F = k × e.
Hooke's law: the extension of a spring is proportional to the force, F = k × e, up to the limit of proportionality.
spring constant: the force for each metre of extension, k in N/m. A stiff spring has a large k.
limit of proportionality: the point where the force extension line stops being straight.
elastic potential energy: the energy in a stretched spring, Ee = ½ × k × e², up to the limit of proportionality.

Back to your guess

The answer is D: 6 cm

Within the limit of the spring, the stretch is proportional to the force. Three times the force gives three times the stretch: 3 × 2 = 6 cm.

Keep it

Copy this into your exercise book.

Extension e = length now − natural length. F = k × e, with e in m, up to the limit. Ee = ½ × k × e². Double e, four times the energy.

Draw: A spring with its natural length and extension marked, and a force extension line that bends after the limit.

S4.9.3 Chapter 3

Work done

A force of 30 N moves a trolley 5 m in the direction of the force. How much work is done?

Your guess

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

Your guess

A 6 J
You picked A. Read on. At the end, the chapter comes back to your guess.
B 25 J
You picked B. Read on. At the end, the chapter comes back to your guess.
C 150 J
You picked C. Read on. At the end, the chapter comes back to your guess.
D 35 J
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate work done as force times distance moved
Explain that work done equals energy transferred

In everyday speech, work is anything tiring. In science, work done has one exact meaning. Work is done when a force moves something.

work done = force × distance moved in the direction of the force. In symbols, W = F × s. W is in joules (J). F is in newtons (N). s is in metres (m).

One joule is the work done when a force of 1 N moves something 1 m. So 1 J = 1 N × 1 m.

A force of 50 N pushes a box 4 m. The dashed box is where it ends up.

Worked example

A force of 50 N pushes a box 4 m along the floor, in the direction of the force. How much work is done on the box?

W = F × s.
W = 50 × 4 = 200 J.

Answer: 200 J

Work done is the same as energy transferred. When a force does 200 J of work, 200 J of energy moves from one store to another. That is why both are measured in joules.

Pushing a box at a steady speed over a rough floor. The work you do ends up as thermal energy.

Where does the energy go? If you push a box at a steady speed, it is not speeding up, so its kinetic energy does not change. The work you do against friction warms the box and the floor a little.

Now lift something. To lift at a steady speed, your force must equal the weight. The weight in newtons is mass × g, with g = 9.8 N/kg. The distance is the height you lift it.

A 4 kg bag lifted 2.5 m. The lifting force is 39.2 N, the weight of the bag.

Worked example

A bag of 4 kg is lifted 2.5 m at a steady speed. How much work does the lifting force do? Use g = 9.8 N/kg.

The lifting force equals the weight, and W = F × s.
F = m × g = 4 × 9.8 = 39.2 N.
W = 39.2 × 2.5 = 98 J.

Answer: 98 J

Compare with Chapter 1. The bag gains m × g × h = 4 × 9.8 × 2.5 = 98 J of gravitational potential energy. That is the same number. The work done by the lifting force is the energy transferred into the gravitational potential store.

Take care with direction. Only the distance moved in the direction of the force counts. Suppose you carry a bag along a level path. The force holding the bag up points up, and the bag moves sideways. Up and sideways are at right angles.

The force is up. The bag moves sideways. The distance moved up is 0 m, so the work done by this force is 0 J.

The holding force does no work on the bag, because the bag does not move up or down. Your legs and arms still use energy. That energy goes into your own body and into the air, not into the bag.

A graph shows work too. Plot force against the distance moved. For a steady force the line is flat. The work done is the area under the line, a rectangle of height F and width s.

A steady 10 N over 4 m. The rectangle under the line is 10 × 4 = 40, so 40 J of work is done.

Three mistakes to avoid. Change cm to m before you multiply. A mass in kg is not a force, so turn it into a weight with m × g. And count only the distance moved in the direction of the force.

Worked example

A cart of mass 4 kg starts from rest on a smooth level floor. A force of 8 N pushes it 9 m. Ignore friction. How fast is it moving at the end?

W = F × s = 8 × 9 = 72 J.
All of it goes to the kinetic store, so Ek = 72 J.
Ek = ½ × m × v², so v² = 2 × Ek ÷ m = 2 × 72 ÷ 4 = 36.
v = √36 = 6 m/s.

Answer: 6 m/s

The cart starts from rest. The 8 N force acts over all 9 m.

A wrong idea many people have

If you hold a heavy box still, you are doing work on it.

Why it feels right: Holding a heavy box makes your arms ache, and aching feels like work.

The test that breaks it: Work done = force × distance moved. A box held still moves 0 m. So W = F × 0 = 0 J.

What is true: You do no work on a box that does not move. Your muscles still use chemical energy to stay tense, and that is why you tire. The energy goes into your body, not into the box.

Try it

Work done lifting a bag of rice

You need: a bag of rice or flour with its mass printed in kg, a table or shelf, string, a ruler or tape measure.

Be safe: Lift only a bag you can carry comfortably. Bend your knees, keep your back straight and keep your feet clear. Use a steady table.

Read the mass printed on the bag, in kg. Work out its weight: mass × 9.8.
Hold a string from the floor to the top of the table. Measure the string in m. This is the height.
Lift the bag from the floor to the table, slowly and at a steady speed.
Work out the work done: weight × height.
Now carry the bag across the room at the same height. How much work do you do on the bag in the up direction?

What you will see: Take a 2 kg bag and a table 0.75 m high. The weight is 19.6 N. The work is 19.6 × 0.75 = 14.7 J. Carrying the bag across the room adds 0 J, because it does not go up or down.

If you cannot try it, this is what happens.

Key words

work done: the energy transferred by a force, found as force × distance moved in the direction of the force.
joule: the unit of work and energy, written J. One joule is the work done by a force of 1 N moving something 1 m.

Back to your guess

The answer is C: 150 J

Work done is force × distance moved. 30 × 5 = 150 J. Adding or dividing the two numbers does not give an energy.

Keep it

Copy this into your exercise book.

W = F × s, with s moved in the direction of the force. Work done = energy transferred, in J. Force at right angles, or no movement: no work. Lifting force = m × g.

Draw: A box, a force arrow and a dimension line for the distance moved, with W = F × s beside it.

S4.9.4 Chapter 4

Power

A motor transfers 200 J in 8 s. What is its power?

Your guess

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

Your guess

A 1 600 W
You picked A. Read on. At the end, the chapter comes back to your guess.
B 192 W
You picked B. Read on. At the end, the chapter comes back to your guess.
C 0.04 W
You picked C. Read on. At the end, the chapter comes back to your guess.
D 25 W
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate power in watts from energy and time
Measure your own power by timing yourself climbing stairs

The same person carries a bag up the same stairs twice, once walking and once running. The energy transferred is the same, but the run is faster. Power measures how fast energy is transferred.

power = energy transferred ÷ time taken. In symbols, P = E ÷ t. Energy E is in joules (J). Time t is in seconds (s). Power P is in watts (W).

One watt is one joule each second. So a 5 W device transfers 5 J every second. A kilowatt (kW) is 1 000 W. Work done is energy transferred, so P = W ÷ t as well.

The two units of power you need.

Worked example

A motor transfers 480 J in 12 s. What is its power?

P = E ÷ t.
P = 480 ÷ 12 = 40 W.

Answer: 40 W

Here is how power compares across things you know. These are typical sizes. Real ones differ from one make to another.

About how much power each uses. Heating things takes much more power than lighting.

A graph shows power as steepness. Plot the energy transferred against time. A machine with a power of 40 W adds 40 J every second. A 10 W machine adds only 10 J every second, so its line is less steep.

The steeper line is the machine with more power. In 10 s, A transfers 400 J and B transfers 100 J.

Before you use P = E ÷ t, change the units. Time must be in seconds: 1 minute is 60 s. Power must be in watts: 1 kW is 1 000 W. Skip this step and the answer is wrong by 60 or by 1 000.

You can turn the sum around. energy transferred = power × time, so E = P × t. And time = energy ÷ power.

Worked example

An electric kettle has a power of 2 kW. It runs for 3 minutes. How much energy does it transfer?

E = P × t.
2 kW = 2 000 W, and 3 minutes = 3 × 60 = 180 s.
E = 2 000 × 180 = 360 000 J = 360 kJ.

Answer: 360 kJ

Now use your own body. When you climb stairs, you lift your own mass. The useful energy you transfer is your gain in gravitational potential energy, m × g × h. Divide it by the time and you have your useful power. Your body uses more, because much is wasted as heat. So this is the least power you made.

A person of 50 kg climbs 18 steps of 18 cm in 12 s.

Worked example

A person of 50 kg climbs 18 steps, each 18 cm high, in 12 s. What is her power? Use g = 9.8 N/kg.

The useful energy she transfers is her gain in gravitational potential energy, m × g × h.
The height is 18 × 0.18 = 3.24 m.
E = 50 × 9.8 × 3.24 = 1 587.6 J.
P = E ÷ t = 1 587.6 ÷ 12 = 132.3 W.

Answer: 132.3 W

A wrong idea many people have

A more powerful machine always does more work.

Why it feels right: In daily talk, a powerful engine means a strong one that gets more done.

The test that breaks it: A 100 W lamp on for 1 s transfers 100 J. A 10 W lamp on for 100 s transfers 10 × 100 = 1 000 J. The weaker lamp has transferred ten times as much energy.

What is true: Power is the rate: energy per second. A powerful machine does more work in the same time. A weaker one can do more work if it runs for longer. Energy = power × time.

Do not mix up the units. A joule is an amount of energy. A watt is a rate: joules each second. A lamp does not use 40 W. It uses energy at a rate of 40 W, which is 40 J every second.

Try it

Measure your power on the stairs

You need: a flight of stairs, a tape measure or ruler, a watch or phone timer, a friend to time you, your mass in kg, from a scale.

Be safe: Use stairs with a rail and clear, dry steps. Walk fast but do not run. Stop at once if you feel pain or dizzy. If you have been told not to exercise hard, do not try this.

Count the steps. Measure the height of one step in cm. Work out the total height in m.
Find your mass in kg. If you have no scale, use 50 kg to practise the sum.
Stand at the bottom. Your friend starts the timer when you start. Walk up briskly, one step at a time, holding the rail. Walk back down slowly.
Your friend stops the timer when you reach the top. Write the time in s.
Work out the energy: mass × 9.8 × height. Divide by the time. That is your power in W.

What you will see: For the example below, a person of 60 kg climbs 16 steps of 17 cm in 10 s. The height is 2.72 m. The energy is 60 × 9.8 × 2.72 ≈ 1 599 J. The power is about 160 W. Try it again more slowly. The energy is the same and the time is longer, so your power is smaller.

If you cannot try it, this is what happens.

Key words

power: the energy transferred each second, P = E ÷ t.
watt: the unit of power, written W. One watt is one joule each second.
kilowatt: a unit of power, written kW. One kilowatt is 1 000 watts.

Back to your guess

The answer is D: 25 W

Power is energy ÷ time: 200 J ÷ 8 s = 25 W. Multiplying or subtracting the numbers does not give a rate.

Keep it

Copy this into your exercise book.

P = E ÷ t. Watts: 1 W is 1 J each second. Seconds, not minutes. Watts, not kW: 1 kW is 1 000 W. E = P × t. Stairs: P = m × g × h ÷ t.

Draw: A flight of stairs with its height marked, and P = E ÷ t beside it.

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

A 20%
You picked A. Read on. At the end, the chapter comes back to your guess.
B 80%
You picked B. Read on. At the end, the chapter comes back to your guess.
C 500%
You picked C. Read on. At the end, the chapter comes back to your guess.
D 120%
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 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.

efficiency = useful output ÷ total input.
efficiency = 6 ÷ 100 = 0.06.
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 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?

wasted energy = input − useful output = 400 − 300 = 100 J.
efficiency = 300 ÷ 400 = 0.75.
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 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?

useful output power = efficiency × input power.
useful output power = 0.8 × 300 = 240 W.
wasted power = 300 − 240 = 60 W.
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.

Mark a height of 1 m on the wall, from the floor.
Hold the bottom of the ball level with the mark. Drop it, do not throw it.
Watch the top of the bounce and mark how high the bottom of the ball rises. Ask a friend to help.
Divide the bounce height by the drop height. This is the efficiency of the bounce.
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.

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%.

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.

S4.9.6 Chapter 6

Specific heat capacity

The same energy is given to 1 kg of water and to 1 kg of aluminium. Which one gets hotter?

Your guess

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

Your guess

A The water, because it is a liquid and can move heat around.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The water, because water holds heat well, so it gets hotter.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The aluminium, because it takes less energy to warm it each degree.
You picked C. Read on. At the end, the chapter comes back to your guess.
D Both warm by the same amount, because they have the same mass.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use specific heat capacity to calculate the energy for a temperature rise
Explain why coasts have milder weather than places far inland

Put a 200 g metal pan and 200 g of water in the same sunshine. The pan is hot to touch long before the water warms. The masses are the same, so the difference comes from the materials. Different materials need different amounts of energy to warm up.

The specific heat capacity of a material is the energy needed to warm 1 kg of it by 1 °C. It is written c. Its unit is joules for each kg for each °C, or J/kg °C.

Water has a specific heat capacity of 4 200 J/kg °C. So it takes 4 200 J to warm 1 kg of water by 1 °C. A high number means a lot of energy for each degree. So the temperature rises slowly.

Approximate values. Water is far higher than the metals.

The energy needed depends on three things. The mass m in kg, the specific heat capacity c, and the change in temperature ΔT in °C. Here ΔT is the final temperature minus the start temperature.

change in thermal energy = mass × specific heat capacity × change in temperature. In symbols, ΔE = m × c × ΔT.

Worked example

How much energy is needed to warm 2 kg of water from 20 °C to 70 °C? The specific heat capacity of water is 4 200 J/kg °C.

ΔE = m × c × ΔT.
ΔT = 70 − 20 = 50 °C.
ΔE = 2 × 4 200 × 50 = 420 000 J = 420 kJ.

Answer: 420 kJ

Mind the units. The mass must be in kg. Divide grams by 1 000. And ΔT is the change, not the final temperature. A rise from 20 °C to 70 °C is a change of 50 °C, not 70 °C.

Worked example

A block of aluminium has a mass of 500 g. It is heated from 15 °C to 55 °C. Its specific heat capacity is 900 J/kg °C. How much energy does it take in?

m = 500 g = 0.5 kg.
ΔT = 55 − 15 = 40 °C.
ΔE = 0.5 × 900 × 40 = 18 000 J = 18 kJ.

Answer: 18 kJ

The formula can be turned around. c = ΔE ÷ (m × ΔT) gives the specific heat capacity. And ΔT = ΔE ÷ (m × c) gives the temperature rise. A heater of power P running for a time t supplies ΔE = P × t.

A 100 W heater in 0.5 kg of water, before and after 210 s.

Worked example

A 100 W heater warms 0.5 kg of water for 210 s. All its energy goes into the water. By how many °C does the temperature rise?

The energy supplied is ΔE = P × t = 100 × 210 = 21 000 J.
ΔE = m × c × ΔT, so ΔT = ΔE ÷ (m × c).
m × c = 0.5 × 4 200 = 2 100 J/°C.
ΔT = 21 000 ÷ 2 100 = 10 °C.

Answer: 10 °C

This test is done in a laboratory with a low voltage heater. It is not done at home, because mains electricity and water are dangerous together. In a real test some heat escapes to the air, so the water warms a little less than the sum says.

Now to the coast. Water has a specific heat capacity of 4 200 J/kg °C. Dry soil and rock have about 800 J/kg °C. So the same sunshine warms the land about five times as much as the same mass of sea water.

At night the same thing happens in reverse. The land loses heat and cools fast. The sea needs to lose a lot of energy to cool by 1 °C, so it cools slowly. The sea also mixes, which spreads its heat through deep water.

A made-up day to show the pattern. The temperature scale starts at 12 °C, not 0. The land swings from 14 °C to 30 °C. The sea only moves between 19 °C and 21 °C.

Winds that blow from the sea carry this steadiness onto the coast. So a coast is usually cooler than inland when it is hot, and warmer than inland when it is cold. Far inland there is no sea to hold the temperature steady, so the range is bigger.

This holds where the wind comes from the sea and where no cold sea current runs along the coast. A coast can be different in a place where the wind blows from the land.

A wrong idea many people have

A material with a high specific heat capacity heats up faster.

Why it feels right: A high number sounds like a quick or strong thing.

The test that breaks it: Give 4 200 J to 1 kg of water: it warms by 1 °C. Give 4 200 J to 1 kg of lead, with c = 130: it warms by 4 200 ÷ 130, about 32 °C.

What is true: A high specific heat capacity means a lot of energy is needed for each degree. So the material heats up slowly, and cools slowly too.

Try it

Soil and water in the sun

You need: two identical dark cups or tins, dry soil, cold water, a sunny spot, a thermometer, if you have one.

Be safe: Never look at the Sun. Touch only with a finger. Do not use hot water, and do not leave the water where a child or an animal may drink it.

Fill one cup to the top with dry soil. Fill the other to the top with water. Use the same cup size.
Put both cups in the same sunny place. Do not move them.
After an hour, feel the top of the soil and the water with a finger. If you have a thermometer, measure both.
Move the cups into the shade in the evening and feel them again an hour later.
Write down which cup warmed more and which cooled faster.

What you will see: The soil feels warmer after the hour in the sun. Late in the day it also cools faster. The soil heats up and cools down more quickly than the water because it has a lower specific heat capacity. This is a rough test, because the Sun also warms the soil and the water in different ways.

If you cannot try it, this is what happens.

Key words

specific heat capacity: the energy needed to raise the temperature of 1 kg of a material by 1 °C, in J/kg °C.
change in temperature: the final temperature minus the start temperature, written ΔT.

Back to your guess

The answer is C: The aluminium, because it takes less energy to warm it each degree.

Aluminium has a much lower specific heat capacity than water. The same energy warms 1 kg of aluminium by a much larger number of degrees than 1 kg of water.

Keep it

Copy this into your exercise book.

ΔE = m × c × ΔT. Mass in kg. ΔT is the change. Water: c = 4 200 J/kg °C. High c: slow to warm and slow to cool. The sea keeps coasts mild.

Draw: A container of water with a heater and a thermometer, and ΔE = m × c × ΔT beside it.

S4.9.7 Chapter 7

Energy resources

Which resource gives electricity at any hour, and gives out no carbon dioxide while it runs?

Your guess

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

Your guess

A Coal
You picked A. Read on. At the end, the chapter comes back to your guess.
B Solar power
You picked B. Read on. At the end, the chapter comes back to your guess.
C Wind power
You picked C. Read on. At the end, the chapter comes back to your guess.
D Nuclear power
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Compare fossil, nuclear, solar, wind, hydro and biofuel resources
Judge which resources suit a sunny, windy or hilly place

Every home, factory and train needs energy from somewhere. An energy resource is a source we can get energy from. Some resources will run out. Others will not. Each has good points and bad points.

A renewable resource is one that is used up no faster than it is replaced, or that never runs out. A non-renewable resource is one that will run out, because it is used far faster than it forms.

Fossil fuels are coal, oil and natural gas. They formed from dead plants and animals over millions of years. They are non-renewable. Burning them gives out carbon dioxide, which adds to the warming of the Earth.

Nuclear fuel is uranium, which is mined. There is a limited amount, so it is non-renewable. A reactor gives out no carbon dioxide while it runs. It makes radioactive waste, which must be stored safely for a very long time.

A fossil fuel power station. A generator then turns the turbine's movement into electricity. A nuclear station is the same, but its heat comes from the uranium.

The other four resources are renewable. Solar panels turn sunlight into electricity. They give out no carbon dioxide and use no fuel. They work only in daylight, and less well under cloud.

A sunny place with clear skies is a good place for solar panels.

Wind turbines use moving air to turn a generator. They also use no fuel and give out no carbon dioxide while they run. They work only when the wind blows. In a very strong wind they are switched off to protect them.

A place with steady wind is a good place for a wind turbine.

Hydroelectric power uses falling water. A dam holds water in a lake high up, called a reservoir. The water runs down a pipe and turns a turbine. It gives out no carbon dioxide while it runs. The stored water can be used at any time.

A hilly place with plenty of rain is a good place for hydroelectric power.

Water high up holds energy in its gravitational potential store. Chapter 1 gave the sum for it.

Worked example

A dam lets 1 000 kg of water fall 50 m every second. How much gravitational potential energy does the water lose each second, and what power is that? Use g = 9.8 N/kg.

Ep = m × g × h.
Ep = 1 000 × 9.8 × 50 = 490 000 J.
This happens each second, so the power is 490 000 J ÷ 1 s = 490 000 W = 490 kW.

Answer: 490 kW

A real station gets a little less than this, because the pipe, the turbine and the generator waste some energy. The sum gives the most that is possible.

A biofuel is a fuel made from plants or from animal waste, such as wood, plant oil or biogas. It is renewable, because new plants can be grown. Burning it gives out carbon dioxide.

The plants took in carbon dioxide as they grew. So the carbon dioxide from burning can roughly balance over time. But growing, drying and moving the fuel use energy too. And land used for fuel crops cannot grow food.

Now put the six side by side. Always say what a table cannot: each one has good points and bad points, and the best choice depends on the place.

CO₂ means carbon dioxide given out while running. Fossil means fossil fuels. Biofuel gives out CO₂ when burnt, but growing plants took it in.

Carbon dioxide is not the only harm. Burning coal, oil and wood also makes smoke that harms the lungs. A nuclear station costs a great deal to build and to take apart. An accident is rare, but it can close land for many years.

Wind turbines are noisy close up and can harm birds. Today, batteries and pumped water storage can keep solar and wind energy for the night or a calm day, at extra cost.

Other harms besides carbon dioxide and waste. Every resource has some.

Solar and wind depend on the weather. Hydro works while the lake holds water. Fossil fuel and biofuel stations can be started when needed. A nuclear station runs day and night but is slow to start or stop.

Solar panels give electrical power. The energy from a panel is E = P × t, with the time in seconds, as in Chapter 4.

Worked example

A solar panel delivers 200 W in full sunshine. How much energy does it deliver in 5 minutes?

E = P × t.
5 minutes = 5 × 60 = 300 s.
E = 200 × 300 = 60 000 J = 60 kJ.

Answer: 60 kJ

To judge a place, ask three questions. Is it sunny for most of the year? Is the wind steady? Are there hills, rivers and rain? The answers point to solar, wind or hydroelectric power.

A wrong idea many people have

Renewable energy has no bad points.

Why it feels right: Sunshine, wind and rain cost nothing, and none of these runs out.

The test that breaks it: Think of each one. Solar panels give no power at night. Wind turbines stand still in a calm. A hydroelectric dam floods a valley and changes the life of the river.

What is true: Renewable resources do not run out, but each has costs. They may depend on the weather, take up land or harm wildlife. Choose by looking at the place and the need.

A wrong idea many people have

Nuclear power gives out no carbon dioxide while it runs, so it must be renewable.

Why it feels right: Clean and renewable both sound like the same good thing.

The test that breaks it: Ask where the fuel comes from. Uranium is mined from the ground, and the supply is limited.

What is true: Renewable means the resource is replaced or never runs out. Nuclear fuel is not replaced, so nuclear power is non-renewable, even though a reactor gives out no carbon dioxide while it runs.

Try it

A sun and wind diary

You need: a stick or a pole, a strip of light cloth or a plastic bag torn into a ribbon, string, paper and a pencil.

Be safe: Never look at the Sun to see if it is out. Look at your shadow instead. Keep plastic bags away from small children.

Tie the strip of cloth to the top of the stick. Push the stick into the ground in an open place, away from walls and trees.
Make a table with three columns: day, sun and wind.
Three times a day, look at the sky and at the cloth. Write yes or no for sun. Judge it by whether you have a sharp shadow.
Write none, light or strong for the wind. Hanging down is none. Waving is light. Streaming out flat is strong.
After a week, count the days with sun and the days with strong wind.

What you will see: Many days with sun and a sharp shadow mean solar panels would work well in your place. Many days with the cloth streaming out mean a wind turbine would turn well. If the cloth hangs down most of the week, wind power would give little. One week is only a start. Engineers measure for a year, high up where the blades would turn.

If you cannot try it, this is what happens.

Key words

renewable: a resource that is replaced as fast as it is used, or that never runs out.
non-renewable: a resource that will run out, because it forms far more slowly than it is used.
fossil fuels: coal, oil and natural gas, formed from dead plants and animals over millions of years.
biofuel: a fuel made from plants or animal waste, such as wood or biogas.
reservoir: a lake held behind a dam, where water is stored high up for a hydroelectric station.

Back to your guess

The answer is D: Nuclear power

Nuclear power gives steady electricity day and night, and a reactor gives out no carbon dioxide while it runs. Coal gives out carbon dioxide. Solar and wind depend on the weather.

Keep it

Copy this into your exercise book.

Renewable: replaced or never runs out. Fossil fuels and uranium are not. Solar needs sun, wind needs wind, hydro needs hills and water. Every resource has a cost. Choose by the place.

Draw: Three small pictures: a sun over panels, a turbine in wind, a lake on a hill with a pipe.

S4.9.8 Chapter 8

Energy at home

A 1 kW iron is used for 2 hours. How much energy does it use?

Your guess

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

Your guess

A 0.5 kWh
You picked A. Read on. At the end, the chapter comes back to your guess.
B 1 kWh
You picked B. Read on. At the end, the chapter comes back to your guess.
C 3 kWh
You picked C. Read on. At the end, the chapter comes back to your guess.
D 2 kWh
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate the cost of running an appliance from its power and time
Compare the yearly cost of a filament lamp and an LED lamp

A home pays for energy. The joule is far too small for the bill. A lamp that is on for one hour already uses tens of thousands of joules. So the bill counts energy in a bigger unit.

A kilowatt hour (kWh) is the energy used by a 1 kW appliance running for 1 hour. 1 kWh = 1 kW × 1 h = 1 000 W × 3 600 s = 3 600 000 J, which is 3.6 MJ.

energy in kWh = power in kW × time in hours. The cost is the energy in kWh times the price for each kWh.

Prices differ from place to place. In this chapter the price is 10 coins for each kWh. Many bills call one kWh a "unit", so a price for each unit is a price for each kWh.

Use kW and h together, and the answer comes out in kWh.

Worked example

A 2 kW electric heater runs for 3 hours. The price is 10 coins for each kWh. How much energy does it use, and what does it cost?

energy = power × time.
E = 2 × 3 = 6 kWh.
cost = 6 × 10 = 60 coins.

Answer: 6 kWh, which costs 60 coins

A meter on the wall counts the kWh used. Read it at the start and at the end. The difference is the energy used. The last dial, shaded here, shows tenths of a kWh.

The meter before and after the heater ran. 2 310.5 − 2 304.5 = 6 kWh.

Most appliances are rated in watts. Change watts to kilowatts by dividing by 1 000. Change minutes to hours by dividing by 60. Forget either change and the answer is wrong by a large factor.

Worked example

A 500 W iron is used for 90 minutes. How much energy does it use, in kWh?

energy in kWh = power in kW × time in h.
500 W = 0.5 kW, and 90 minutes = 90 ÷ 60 = 1.5 h.
E = 0.5 × 1.5 = 0.75 kWh.

Answer: 0.75 kWh

Do not mix up the units. A kilowatt is a rate of using energy. A kilowatt hour is an amount of energy. A 2 kW heater has a power of 2 kW. It uses 2 kWh in each hour.

More power does not always mean more cost. The cost depends on the power and the time. A strong appliance used for a short time can cost less than a weak one left on for hours.

The kettle runs for 6 minutes and the television for 5 hours. The kettle has twenty times the power, yet uses less energy, because it runs for a short time.

Devices left on standby still use power all day. A device on standby at 5 W uses 0.005 kW × 24 h = 0.12 kWh in a day. Newer devices often use less than 1 W on standby. A few of them still add up over a year.

Now compare two lamps that give the same light. A filament lamp uses 50 W. An LED lamp uses 10 W. This matches Chapter 5. A typical filament lamp turns about 6% of its energy into light. A typical LED lamp turns about 30%. So both give about 3 W of light. Real lamps differ. A filament lamp turns only a few percent of its energy into light. An LED lamp turns about a third.

Worked example

A filament lamp uses 50 W and an LED lamp uses 10 W. They give the same light. Each is on for 1 000 hours in a year. The price is 10 coins for each kWh. How much more does the filament lamp cost to run for the year?

energy in kWh = power in kW × time in h.
Filament: 0.05 kW × 1 000 h = 50 kWh, which costs 50 × 10 = 500 coins.
LED: 0.01 kW × 1 000 h = 10 kWh, which costs 10 × 10 = 100 coins.
The filament lamp costs 500 − 100 = 400 coins more.

Answer: 400 coins

Both lamps give the same light for 1 000 hours. The filament lamp uses five times the energy.

An LED lamp costs more to buy. So a fair comparison also asks how long the yearly saving takes to pay back the higher price. The saving here is large, so the LED lamp pays back its extra price in a short time.

A wrong idea many people have

A kilowatt hour is a unit of power, kilowatts for each hour.

Why it feels right: Power units such as watts already mean "for each second", so "for each hour" sounds like a power too.

The test that breaks it: Multiply: kW × h. A 1 kW iron running for 2 hours gives 1 × 2 = 2 kWh. That is a power times a time, which is an energy, like a W times a s gives joules.

What is true: A kilowatt hour is a unit of energy: the energy used by 1 kW running for 1 hour. It equals 3.6 MJ.

Try it

What does your home use in a day?

You need: paper and a pencil, three electrical appliances to look at, such as a kettle, a lamp and a television.

Be safe: Only read the labels. Do not open any appliance, and do not touch plugs or sockets with wet hands. Leave a hot kettle alone until it cools.

Look for the label on each appliance. It is on the back or the bottom. Write down its power in W or kW.
Change each power to kW. Divide a W by 1 000.
Estimate for how many hours a day each one is on. A kettle may be on for only 6 minutes, which is 0.1 h.
For each, work out the energy in kWh: power in kW × hours.
Which appliance uses the most energy in a day? Is it the one with the highest power?

What you will see: In the example, the kettle has by far the highest power but is on for the shortest time. The television uses the most energy in the day. The answer depends on the power and the time together.

If you cannot try it, this is what happens.

Key words

kilowatt hour: a unit of energy, written kWh. It is the energy used by a 1 kW appliance in 1 hour, which is 3.6 MJ.
power rating: the power an appliance uses when it runs, printed on its label in W or kW.

Back to your guess

The answer is D: 2 kWh

Energy in kWh is power in kW times time in hours: 1 kW × 2 h = 2 kWh. Adding the numbers gives 3, and dividing gives 0.5.

Keep it

Copy this into your exercise book.

E (kWh) = P (kW) × t (h). W to kW: ÷ 1 000. Minutes to hours: ÷ 60. Cost = kWh × price. 1 kWh = 3.6 MJ. LED lamp: same light, less power, lower cost.

Draw: A meter before and after, with the difference in kWh, and E = P × t beside it.

A falling ball keeps its total

Do heavy things fall faster?

Test a spring with water

Work done lifting a bag of rice

Measure your power on the stairs

The efficiency of a bounce

Soil and water in the sun

A sun and wind diary

What does your home use in a day?

Key words, A to Z