Work, Power and Energy — The Complete Physics Guide
Work, power, and energy are three of the most fundamental concepts in physics. Work quantifies the energy transferred when a force causes displacement. Power measures how quickly that work is done. Energy is the capacity to do work — it takes many forms (kinetic, potential, thermal, electrical, chemical) and is conserved in all physical processes. Together, these three concepts underpin mechanics, thermodynamics, and every engineering discipline.
Work — Energy Transferred by a Force
Work W is done when a force causes displacement in the direction of the force: W = F·d·cosθ, where F is force (N), d is displacement (m), and θ is the angle between them. When force and displacement are parallel (θ = 0°), cosθ = 1 and W = Fd. When perpendicular (θ = 90°), cosθ = 0 and no work is done — a centripetal force, for example, does no work because it is always perpendicular to velocity.
Work is a scalar (no direction), measured in joules (J = N·m). Negative work occurs when the force opposes the displacement — for example, friction always does negative work (removes energy from the system). The work-energy theorem states: net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv² − ½mu².
For a variable force, work is the integral of force with respect to displacement: W = ∫F(x)dx — the area under the force-displacement graph. For a spring: W = ∫₀ˣ kx' dx' = ½kx² (the elastic potential energy). For constant force, this simplifies to W = Fd as expected.
Power — The Rate of Doing Work
Power P is the rate of energy transfer: P = W/t = ΔE/t. For a constant force and constant velocity: P = Fv — power equals force times velocity. SI unit: the watt (W = J/s). One watt is one joule per second.
The watt is named after James Watt, who improved steam engine efficiency and introduced the concept of "horsepower" as a marketing tool to sell his engines (comparing their output to horse labour). One horsepower ≈ 746 watts. A typical car engine produces 75–200 kW (100–270 horsepower). A human sprinter reaches about 2 kW peak power; a Tour de France cyclist sustains about 400 W for hours.
For rotating systems: P = τω — power equals torque times angular velocity. This connects linear and rotational power calculations and is used to specify motor and engine performance. A motor delivering 100 N·m at 1,500 rpm (ω = 157 rad/s) produces P = 100 × 157 = 15,700 W = 15.7 kW.
Worked Example 1 — Work Against Gravity
Problem: A 70 kg person climbs a 12 m staircase in 15 seconds. Calculate the work done against gravity and the average power output.
Work = mgh = 70 × 9.81 × 12 = 8,241 J
Power = W/t = 8,241/15 = 549 W ≈ 0.74 horsepower. This is only the useful output — actual metabolic power is 3–4× higher due to muscle and body inefficiency.
Worked Example 2 — Car Engine Power
Problem: A car exerts a driving force of 3,200 N at a constant speed of 28 m/s (about 100 km/h). Find the engine power output and the power wasted overcoming air resistance if the engine efficiency is 30%.
Useful power = Fv = 3200 × 28 = 89,600 W = 89.6 kW
At 30% efficiency, total fuel power = 89,600/0.30 = 298,667 W. Power wasted as heat = 298,667 − 89,600 = 209,067 W — about 70% of fuel energy becomes heat. This is why car engines need cooling systems.
Energy Conservation and Efficiency
Energy is conserved in all physical processes — it can neither be created nor destroyed, only converted from one form to another. This is the first law of thermodynamics. In mechanical systems: total mechanical energy (KE + PE) is conserved in the absence of dissipative forces (friction, air resistance). When friction is present, some mechanical energy converts to thermal energy.
Efficiency η = useful output energy / total input energy × 100%. No real system has 100% efficiency — there are always losses to heat (second law of thermodynamics). Internal combustion engines: ~25–35% efficient. Electric motors: ~85–95% efficient. Power stations: 35–45% (fossil fuel), 30–40% (nuclear), ~100% (gravitational hydro). This is why electrification of transport (using efficient motors) and generation mix (moving toward renewables) both matter for total energy system efficiency.
The joule is named after James Prescott Joule, who in the 1840s precisely measured the mechanical equivalent of heat — demonstrating that 4.184 J of mechanical work raises 1 gram of water by 1°C. This established the unity of mechanical and thermal energy, one of the foundations of thermodynamics and a cornerstone of the conservation of energy principle.