Work, energy and power

A-Level Physics 9702 · Topic 5 revision notes · AS

One conservation law doing all the work: work done, efficiency and power (P = Fv), and gravitational potential and kinetic energy.

8 definitions to learn word for word, 8 equations in their own column, 6 diagrams and graphs. Read it, cover it, test yourself.

Definitions (mark-scheme wording)EquationsExam traps

5.1aWork done and energy conservation

Work done: the product of the force and the displacement in the direction of the force.
Joule: the work done when a force of 1 N moves its point of application 1 m in the direction of the force.
Principle of conservation of energy: energy cannot be created or destroyed; it can only be transferred from one form to another (the total energy of a closed system is constant).

Work done = energy transferred.
Force at angle θ to the motion: only F cos θ does work.
Force perpendicular to motion does zero work (cos 90° = 0), e.g. a centripetal force.
Force-displacement graph: area under the graph = work done.
Gas expanding at constant pressure does work W = pΔV on its surroundings.
Work done against friction or drag is transferred as thermal energy.

Only the component of F along the displacement does work: W = Fs cos θ.

Force-displacement graph: the area under it is the work done (here a stretched spring).

A large force that does not move its point of application (or acts at 90°) does no work.
θ is the angle between the force and the displacement, not the horizontal by default.

Efficiency: the ratio of useful energy (or power) output to total energy (or power) input.
Power: the rate of doing work (the work done or energy transferred per unit time).
Watt: a rate of energy transfer of one joule per second.

Efficiency is a ratio or percentage; it is always less than 1 (100%) for a real machine.
Wasted energy usually ends up as thermal energy in the surroundings.
Moving at constant speed v against a resistive force F: power needed P = Fv.
At top speed, driving force = resistive forces, so P = F_resist v_max.
Derivation: P = W/t = F s/t = Fv.

Sankey diagram for an electric motor: arrow widths are to scale; efficiency = 60 J / 100 J = 60%.

Car at top speed: driving force equals total resistance, so P = F v_max.

Efficiency greater than 100% means input and output were swapped.
P = Fv uses the force in the direction of motion and the speed at that instant.

Kinetic energy: the energy a body has because of its motion.
Gravitational potential energy: the energy a mass has because of its position in a gravitational field.

Derive E_k: W = Fs, F = ma, v² = 2as (from rest), so W = m(v²/2) = ½mv².
Derive ΔE_p: lifting at constant speed needs F = mg over Δh, so W = mgΔh.
ΔE_p = mgΔh only in a uniform field (near the Earth's surface).
No resistance: loss in E_p = gain in E_k, so v = √(2gΔh), independent of mass.
With resistance: E_p lost = E_k gained + work done against friction.
Doubling the speed quadruples the kinetic energy.

Falling without air resistance: E_p falls as E_k rises; the total stays constant.

Pendulum or roller coaster: E_p at the top becomes E_k at the bottom.

Equating mgΔh and ½mv² is only valid when resistive forces are negligible: say so.
Forgetting to square v, or using mass in grams.