Know the principle of the conservation of energy and apply this principle to simple examples including the interpretation of simple flow diagrams

1.7 Energy, Work and Power

Learning objective

Know the principle of the conservation of energy and apply it to simple situations, including the interpretation and drawing of simple flow diagrams.

Key vocabulary (exam‑style definitions)

TermDefinition
EnergyAbility to do work or to produce heat.
Energy storeForm in which energy may be stored: kinetic, gravitational PE, elastic PE, thermal, electrical, chemical or nuclear.
WorkTransfer of energy when a force moves an object through a distance.
Mechanical: \(W = Fd\) (J = N·m).
Electrical: \(W = VIt = Pt\).
PowerRate of energy transfer.
\(P = \dfrac{E}{t}\)  or  \(P = \dfrac{W}{t}\) (W = J s⁻¹).
SystemPortion of the universe chosen for analysis.
Isolated systemSystem that does not exchange energy with its surroundings.
EfficiencyRatio of useful energy output to total energy input, expressed as a percentage:
\(\displaystyle \eta = \frac{E{\text{useful}}}{E{\text{input}}}\times100\%.\)

Energy stores, symbols, typical equations & everyday examples

Energy storeSymbolTypical equationEveryday example
Kinetic\(E_k\)\(E_k = \dfrac12 mv^{2}\)Moving car, wind‑turbine blades
Gravitational potential\(E_g\)\(E_g = mgh\)Roller‑coaster at the top of a hill
Elastic (spring) potential\(E_e\)\(E_e = \dfrac12 kx^{2}\)Compressed spring in a toy car
Thermal (internal) energy\(E_{th}\)\(E_{th}= mc\Delta T\)Heater warming a room
Electrical\(E_{el}\)\(E_{el}= VIt = Pt\)Battery powering a torch
Chemical\(E_{ch}\)Energy released in a reaction (e.g. combustion)Fuel in a car engine
Nuclear\(E_{nu}\)Energy released per fission/fusion eventPower from a nuclear reactor

Principle of conservation of energy

  • In an isolated system the total energy remains constant.

  • Mathematically: \(\displaystyle \sum E{\text{initial}} = \sum E{\text{final}}\)

  • If energy crosses the system boundary: \(\displaystyle \Delta E{\text{system}} = E{\text{in}} - E_{\text{out}}\)

  • All energies must be expressed in joules (J); power in watts (W = J s⁻¹).

Simple flow‑diagram template

Use the box‑arrow format below. Write the energy store symbols on the arrows and, where the question asks, include the amount of energy (J) on each arrow.

[Energy store 1] ──► [Energy store 2] ──► … ──► [Energy store n]
(symbol)          (symbol)               (symbol)
(J)                (J)                     (J)

Example (ball falling):

[Eg] ──► [Ek]
(J)      (J)

Worked examples

  1. Pendulum (gravitational ↔ kinetic)

    A 0.5 kg bob is released from a height of 0.20 m above its lowest point. Find its speed at the lowest point, neglecting air resistance.

    1. Initial gravitational energy: \(E_g = mgh = 0.5\times9.8\times0.20 = 0.98\ \text{J}\).

    2. At the lowest point all this energy is kinetic: \(E_k = \tfrac12 mv^{2}\).

    3. Set \(Eg = Ek\) → \(0.98 = \tfrac12(0.5)v^{2}\) → \(v^{2}=3.92\) → \(v = 1.98\ \text{m s}^{-1}\).

  2. Spring‑loaded toy car (elastic → kinetic)

    A toy car of mass 0.20 kg is launched by a spring compressed 0.05 m. Spring constant \(k = 800\ \text{N m}^{-1}\). Find the speed when the spring returns to its natural length, neglecting friction.

    1. Elastic energy stored: \(E_e = \tfrac12 kx^{2}= \tfrac12\times800\times(0.05)^{2}=1.0\ \text{J}\).

    2. All becomes kinetic: \(E_k = \tfrac12 mv^{2}\).

    3. Set \(Ee = Ek\) → \(1.0 = \tfrac12(0.20)v^{2}\) → \(v^{2}=10\) → \(v = 3.16\ \text{m s}^{-1}\).

  3. Electric heater (electrical → thermal)

    A 1500 W electric heater runs for 2 minutes. Calculate the thermal energy produced.

    1. Power \(P = 1500\ \text{W}=1500\ \text{J s}^{-1}\).

    2. Time \(t = 2\ \text{min}=120\ \text{s}\).

    3. Thermal energy \(E_{th}=Pt = 1500\times120 = 1.8\times10^{5}\ \text{J}\).

  4. Roller‑coaster (gravitational → kinetic → thermal)

    A coaster car of mass 500 kg starts from rest at the top of a 30 m hill. At the bottom its speed is 20 m s⁻¹. Determine the energy lost as thermal energy due to friction.

    1. Initial gravitational energy: \(E_{g,i}=mgh = 500\times9.8\times30 = 1.47\times10^{5}\ \text{J}\).

    2. Kinetic energy at the bottom: \(E_{k,f}= \tfrac12 mv^{2}= \tfrac12\times500\times20^{2}=1.00\times10^{5}\ \text{J}\).

    3. Thermal loss: \(E{th}=E{g,i}-E_{k,f}=1.47\times10^{5}-1.00\times10^{5}=4.7\times10^{4}\ \text{J}\).

Energy resources – brief overview

  • Fossil fuels (coal, oil, natural gas) – widely used, high energy density, but emit CO₂ and other pollutants.

  • Bio‑fuels (wood, ethanol, biogas) – renewable if sustainably managed; lower carbon intensity but can compete with food production.

  • Hydroelectric – converts water flow into electricity via turbines; very efficient, but large dams can disrupt ecosystems.

  • Geothermal – uses heat from the Earth’s interior; reliable base‑load power, limited to regions with suitable geology.

  • Nuclear – high energy density, low greenhouse‑gas emissions; concerns about radioactive waste and safety.

  • Solar (photovoltaic & thermal) – abundant and clean; intermittent and dependent on daylight.

  • Wind – clean and renewable; intermittent and site‑specific.

Practice – interpreting & drawing flow diagrams

Answer the questions below. Where required, sketch the diagram on a separate sheet and label each arrow with the appropriate energy store symbol and, if asked, the amount of energy (J).

  1. In a simple electric circuit a battery supplies electrical energy to a resistor which heats up.

    • Write the flow diagram using the template.

    • State the energy stores involved (include symbols).

  2. A roller‑coaster car at the top of a hill has 500 J of gravitational potential energy. After descending it has 300 J of kinetic energy; the remainder is lost as thermal energy due to friction.

    • Draw the flow diagram showing all three transfers.

    • Indicate the amount of energy (in joules) on each arrow.

  3. Explain why the total energy in the pendulum example (Example 1) remains constant even though the forms of energy change.

  4. A chemical reaction in a portable stove releases 2.5 MJ of chemical energy, which is used to heat water. If the water’s temperature rises by 30 °C and its mass is 5 kg, calculate:

    1. The thermal energy gained by the water.

    2. The efficiency of the stove (use \(c_{\text{water}} = 4180\ \text{J kg}^{-1}\text{K}^{-1}\)).

Reminder – units & significant figures

  • Always write the unit after each numerical answer (e.g., \(v = 1.98\ \text{m s}^{-1}\)).

  • Use SI units throughout: mass (kg), distance (m), time (s), energy (J), power (W).

  • Give the appropriate number of significant figures (usually 2–3 sf for IGCSE).

Summary

  • The total energy of an isolated system never changes – it is conserved.

  • Energy may be stored as kinetic, gravitational PE, elastic PE, thermal, electrical, chemical or nuclear.

  • Work transfers energy (mechanical: \(W=Fd\); electrical: \(W=VIt\)).

  • Power is the rate of energy transfer (\(P=E/t\) or \(P=W/t\)).

  • Efficiency links useful output to total input: \(\eta = \dfrac{E{\text{useful}}}{E{\text{input}}}\times100\%.\)

  • Flow diagrams provide a clear visual of energy transfers; in the exam you may be asked to draw them, interpret them, and include the energy values on the arrows.

  • Quantitative problems are solved by equating total initial energy with total final energy, remembering to include every relevant energy store and to state units explicitly.