3.2 · Energy

Energy: the accountant of physics

Nothing destroys energy — it is only moved between stores and partly dissipated into less useful forms. OxfordAQA's second topic builds the accounting: work and power, the three energy-store equations, conservation with Sankey diagrams, and the choices behind national energy resources.

OxfordAQA International GCSE Physics 9203, topic 3.2 (Version 5.1). Statements marked (P) are Paper-2-only.

Forces and energy

Work 功 is done when a force moves its point of application through a distance; work against friction transfers energy by heating (shuttle re-entry, meteorites, brakes). A stretched spring stores elastic potential energy $\frac{1}{2}ke^2$; a raised mass stores gravitational potential energy mgh; a moving mass carries kinetic energy ½mv² — doubling mass doubles it, doubling speed quadruples it (the road-safety link). Power 功率 is the rate of energy transfer.

Worked check. A 1500 kg car at 20 m/s brakes to rest: KE removed $= \frac{1}{2} \times 1500 \times 400 = 300\,000$ J — all transferred to heating the brakes. Lifting it 2 m at g = 10 N/kg stores $1500 \times 10 \times 2 = 30\,000$ J.

Worked example (elastic store to kinetic store). A catapult's elastic has $k = 100$ N/m and is stretched by 0.20 m. Stored energy $E_e = \tfrac12 \times 100 \times 0.20^2 = \tfrac12 \times 100 \times 0.040 = 2.0$ J (square the extension first). If all of it goes to a 0.050 kg stone: $\tfrac12 \times 0.050 \times v^2 = 2.0$, so $v^2 = 80$ and $v = 8.9$ m/s.

Worked example (falling). A ball is dropped from 5.0 m. Its GPE becomes KE: $mgh = \tfrac12 mv^2$, so $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5.0} = 10$ m/s, whatever its mass (ignoring air resistance).

Worked example (power). A crane lifts a 500 kg load through 12 m in 20 s. Energy transferred $= mgh = 500 \times 10 \times 12 = 60\,000$ J, so $P = 60\,000 \div 20 = 3000$ W $= 3.0$ kW.

Energy transfers and dissipation

When a system changes, energy is transferred — usefully, stored, or dissipated 耗散 ("wasted") into less useful stores; it is never created or destroyed. Friction and air resistance dissipate energy by heating. (P) A pendulum swaps kinetic and gravitational potential energy each swing.

Sankey diagrams show the redistribution: arrows split in proportion to the energy flows (draw them to scale — width ∝ energy).

Worked check. A 60 W lamp gives 6 W of light: efficiency 效率 $= 6/60 \times 100 = 10\%$.

Worked example (wasted energy). A motor takes 800 W and gives out 600 W of useful power. Efficiency $= 600/800 \times 100 = 75\%$. The other 200 W is dissipated, so in one minute $200 \times 60 = 12\,000$ J heats the motor and its surroundings. On a Sankey diagram the input arrow splits into a useful arrow three times as wide as the wasted one.

Energy resources

Fuels are chosen for ease of storage, energy content and safety; some transfer more of their energy usefully than others. National generation trades renewability against cost, reliability and the environmental cost of extraction, use and disposal.

Common mistakes

• Forgetting the square in $E_e = \tfrac12 ke^2$ or $E_k = \tfrac12 mv^2$. Square $e$ or $v$ first, then multiply.
• Using a spring's length instead of its extension, or centimetres instead of metres.
• Using mass instead of weight in $W = F \times d$ for a lift. The force needed is the weight, $mg$.
• Getting an efficiency above 100%. Useful output can never be more than the total input.
• Saying dissipated energy is destroyed. It still exists, spread out in the surroundings as thermal energy.
• Drawing a Sankey diagram whose output arrows do not add up to the input arrow.

Check yourself

1. Calculate the work done by a 50 N force moving 4 m.
2. Calculate the elastic energy in a spring (k = 200 N/m) stretched 0.15 m.
3. A 2 kg book is lifted 3 m at g = 10 N/kg. Calculate the GPE gained.
4. What happens to the kinetic energy of a car when it brakes?
5. A motor transfers 800 J in 20 s. Calculate its power.
6. State the efficiency equation in energy and power forms.
7. A lamp gives out 12 J of light for every 100 J in. Calculate the efficiency.
8. What does a Sankey diagram's arrow width show?
9. Give one advantage and one drawback of wind power.
10. What does "dissipated" mean?

Answers: 1 $W = 50 \times 4 = 200$ J; 2 $E_e = \frac{1}{2} \times 200 \times 0.0225 = 2.25$ J; 3 $E_p = 2 \times 10 \times 3 = 60$ J; 4 it is transferred to thermal energy of the brakes and surroundings by the friction force (the brakes heat up); 5 $P = 800/20 = 40$ W; 6 useful energy out ÷ total energy in (×100 %); equally useful power out ÷ total power in; 7 $12/100 \times 100 = 12\%$; 8 the energy flowing in that direction — width is proportional to energy; 9 renewable with no fuel or emissions; unreliable output with visual impact; 10 transferred to less useful stores (often waste heat), so it cannot readily do what we want.