Core principles

• Energy is conserved: total energy in an isolated system stays constant, although it may be transferred or transformed between forms.

• Work done by a force is a transfer of energy.

• Work done by the resultant force on a system equals the change in the system’s energy.

• In exam questions, identify the system, the initial and final energy stores, and whether non-conservative forces such as friction or air resistance are present.

• If non-conservative forces act, the change in total mechanical energy equals the work done by those forces.

This roller-coaster diagram shows gravitational potential energy changing into kinetic energy and back again as the car moves. It is useful for visualising mechanical energy conservation when friction is neglected. It also helps explain why speed is greatest at the lowest point. Source

Work done by a force

• For a constant force, $W = Fs\cos\theta$.

• $F$ is the force, $s$ is the displacement, and $\theta$ is the angle between the force and the displacement.

• Only the component of force parallel to the displacement does work.

• If $\theta = 0^\circ$, then maximum positive work is done: $W = Fs$.

• If $\theta = 90^\circ$, then no work is done.

• If $\theta > 90^\circ$, the work is negative: the force removes energy from the system.

• Unit of work: joule (J), where $1\text{ J} = 1\text{ N m}$.

• Common exam trap: a force can act on an object without doing work if there is no displacement or no force component along the displacement.

This diagram shows a force–distance graph, useful because the area under the graph represents work done. It helps students connect the idea of force acting through a displacement to energy transfer. It is especially good for explaining why changing force still leads to calculable work from graph area. Source

Mechanical energy

• Mechanical energy is the sum of kinetic energy, gravitational potential energy, and elastic potential energy.

• Kinetic energy: $E_k = \frac{1}{2}mv^2 = \frac{p^2}{2m}$.

• Near Earth’s surface, change in gravitational potential energy: $\Delta E_p = mg\Delta h$.

• Elastic potential energy in a spring: $E_H = \frac{1}{2}k(\Delta x)^2$.

• If frictional/resistive forces are absent, total mechanical energy is conserved.

• Typical conservation equation: $E_{k,i} + E_{p,i} + E_{H,i} = E_{k,f} + E_{p,f} + E_{H,f}$.

• If friction is present, include the work done by non-conservative forces separately.

• In many exam problems, equate loss of gravitational potential energy to gain in kinetic energy when friction is negligible.

• Be careful with signs in $\Delta h$: moving upward increases gravitational potential energy; moving downward decreases it.

Power and efficiency

• Power is the rate of doing work or the rate of energy transfer.

• $P = \frac{\Delta W}{\Delta t}$.

• For motion at speed $v$ with a force in the direction of motion: $P = Fv$.

• Unit of power: watt (W), where $1\text{ W} = 1\text{ J s}^{-1}$.

• A machine can do the same work in less time and therefore produce greater power.

• Efficiency measures how much useful output is obtained from an input.

• $\eta = \frac{E_{\text{output}}}{E_{\text{input}}} = \frac{P_{\text{output}}}{P_{\text{input}}}$.

• Efficiency is often expressed as a percentage: $\eta \times 100$.

• Efficiency is always less than or equal to 1 (or 100%) in real systems.

• Lost energy is usually transferred to the surroundings as thermal energy or sound.

This diagram compares two ways of moving the same object to the same height. It shows that the work done can be the same while the power differs because the time taken is different. It is a clear visual for the distinction between work and power. Source

Sankey diagrams and energy density

• Sankey diagrams show energy transfers and energy losses in a process.

• The width of each arrow is proportional to the amount of energy transferred.

• Useful output arrows show useful energy transfer; side arrows usually show wasted energy.

• In exam questions, use a Sankey diagram to discuss efficiency and identify where energy is dissipated.

• Energy density means the energy stored per unit mass or sometimes per unit volume of a fuel source.

• A fuel with higher energy density stores more energy for the same mass, which is important in comparing fuel sources.

• When comparing fuels, consider both energy density and efficiency of the device/system using the fuel.

This image shows the basic structure of a Sankey diagram, where arrow width represents the size of an energy flow. It is useful for understanding useful output, wasted energy, and efficiency in devices. Students can adapt this visual when drawing or interpreting energy-transfer diagrams in exams. Source