Chapter 1: Review

The chapter problem, solved

The field centre stream brings 30 kg of water to the wheel each second, and the water falls 12 m. Take $g = 9.8\,\text{N/kg}$ .

1. The energy the water gives up each second.

\begin{aligned} E_p &= m g h \\ &= 30 \times 9.8 \times 12 \\ &= 3528\,\text{J} \end{aligned}

The water gives up 3528 J each second, so its gravitational potential store empties at a rate of 3528 J/s.

2. The greatest power the wheel could produce. Power is the rate of energy transfer, so if every joule reached the generator,

P = 3528\,\text{W} \approx 3.5\,\text{kW}

Real machines cannot reach this value, because some energy is always dissipated.

3. The efficiency of the generator. The useful output is 2.1 kW, which is 2100 W.

\begin{aligned} \text{efficiency} &= \frac{2100}{3528} \\ &= 0.60 \\ &= 60\% \end{aligned}

4. The energy the kettle needs.

\begin{aligned} \Delta E &= m c \Delta\theta \\ &= 1.5 \times 4200 \times 80 \\ &= 504000\,\text{J} \\ &= 5.0 \times 10^5\,\text{J} \end{aligned}

5. The time the kettle takes on the generator's output.

\begin{aligned} t &= \frac{E}{P} \\ &= \frac{504000}{2100} \\ &= 240\,\text{s} \\ &= 4.0\,\text{minutes} \end{aligned}

6. If the stream runs low. Suppose a dry spell halves the flow to 15 kg per second. The energy given up each second also halves, so the useful output power falls as well, from 2100 W to about 1060 W. The kettle then takes about 480 s, or 8.0 minutes, to transfer the same 504000 J.

7. Is the stream a sound energy resource? The stream is renewable: rain refills it, so it will not run out. It produces no carbon dioxide while it generates. Its weakness is reliability: the flow depends on the weather, so in a dry spell it may not deliver enough power. A centre that depended on the stream alone would need a way to store energy or a backup generator. This final judgement is a reasoned answer. It is not the kind of claim a computer can check, so compare your reasoning with the evidence above.

Mixed practice — choose the method first

For each question, say which equation or idea you will use before you calculate. Keep your units with your numbers.

Say which equation you will use, then calculate the energy gained by the gravitational potential store when a pupil of mass 70 kg climbs to a platform 5.0 m above the ground. Take $g = 9.8\,\text{N/kg}$ .
A ball of mass 0.40 kg moves at 25 m/s. Calculate the energy in its kinetic store.
A 3.0 kg block is heated from 10 °C to 60 °C. Its specific heat capacity is 500 J/kg°C. Calculate the energy transferred to the block.
A pump does 9000 J of work in 30 s. Calculate its power.
A device is 60% efficient and its useful output is 900 J. Calculate the total energy supplied.
A ball gives up 40 J from its gravitational potential store and gains 34 J in its kinetic store. Calculate the energy dissipated.
A lamp takes in 40 J of energy and transfers 8 J as useful light. Calculate the efficiency as a percentage.
A spring with spring constant 100 N/m is stretched by 0.30 m. Calculate the energy stored in its elastic potential store.
In the chapter problem a kettle heats 1.5 kg of water through 80 °C. The specific heat capacity of water is 4200 J/kg°C. Calculate the energy transferred to the water.
The kettle needs 504000 J and the generator supplies 2100 W. Calculate the time the kettle takes, in seconds and in minutes.
A cyclist brakes to a stop, and the brakes become warm. Say which store decreases, which store increases, and which equation links them.
In the chapter problem the stream is a renewable resource. Evaluate whether the centre should rely on the stream alone for its electricity. Give one reason for and one reason against.

Answers and concise solutions

1. Energy gained by the gravitational potential store

Answer: Use $E_p = mgh = 70 \times 9.8 \times 5.0 = 3430\,\text{J}$ , so $3400\,\text{J}$ to 2 significant figures.

2. Energy in the kinetic store

Answer: $E_k = \tfrac{1}{2}mv^2 = 0.5 \times 0.40 \times 25^2 = 125\,\text{J}$ , so $130\,\text{J}$ to 2 significant figures.

3. Energy transferred to the block

Answer: $\Delta E = mc\Delta\theta = 3.0 \times 500 \times 50 = 75000\,\text{J}$ , so $7.5 \times 10^4\,\text{J}$ to 2 significant figures.

4. Power of the pump

Answer: $P = \dfrac{E}{t} = \dfrac{9000}{30} = 300\,\text{W} = 3.0 \times 10^2\,\text{W}$ to 2 significant figures.

5. Total energy supplied

Answer: $0.60 = \dfrac{900}{\text{total input}}$ , so total input $= \dfrac{900}{0.60} = 1500\,\text{J}$ .

6. Energy dissipated

Answer: $\text{dissipated} = 40 - 34 = 6\,\text{J} = 6.0\,\text{J}$ to 2 significant figures. The energy is transferred to the thermal store of the air and the ball.

7. Efficiency of the lamp

Answer: $\text{efficiency} = \dfrac{8}{40} = 0.20 = 20\%$ .

8. Energy in the elastic potential store

Answer: $E_e = \tfrac{1}{2}ke^2 = 0.5 \times 100 \times 0.30^2 = 4.5\,\text{J}$ .

9. Energy transferred to the water

Answer: $\Delta E = mc\Delta\theta = 1.5 \times 4200 \times 80 = 504000\,\text{J}$ , so $5.0 \times 10^5\,\text{J}$ to 2 significant figures.

10. Time for the kettle

Answer: $t = \dfrac{E}{P} = \dfrac{504000}{2100} = 240\,\text{s} = 4.0\,\text{minutes}$ .

11. The stores and the equation

The kinetic store of the cyclist and bicycle decreases as they slow down.
The thermal store of the brakes and the air increases, because friction transfers energy there.
The energy transferred is found from the kinetic energy equation $E_k = \tfrac{1}{2}mv^2$ , using the speed before braking.

12. Should the centre rely on the stream alone?

The stream is renewable, because rain refills it, and it produces no carbon dioxide while it generates.
The flow depends on the weather, so in a dry spell the stream may not supply enough power. The centre would need stored energy or a backup generator.

What you can now do

name the energy stores and describe transfers between them;
calculate gravitational potential, kinetic and elastic potential energy, and the energy transferred by heating;
calculate power, the work done by a force, and the time or energy from a power;
use conservation of energy in a closed system;
explain dissipation and describe how to reduce unwanted transfer;
calculate efficiency as a decimal or a percentage;
compare renewable and non-renewable energy resources and explain trends in their use.

Your relevance reflection

Return to the prompt from the chapter opener: where in your own week do you pay for energy, or try to use less of it? Write one more example now that you have finished the chapter. Has the chapter changed what you notice about a bill, a hot drink or a journey?