Work, Energy and Power

Work done

Work done is the energy transferred when a force moves an object:

W = F s cos θ

where F = force, s = displacement, and θ = angle between the force and the displacement. If the force is along the motion, W = Fs. Work is measured in joules (J). No work is done if there is no movement, or if the force is perpendicular to motion.

Conservation of energy

Energy cannot be created or destroyed, only transferred. In mechanics the main stores are:

kinetic energy:  Ek = ½mv²
gravitational potential energy:  Ep = mgh
elastic potential energy:  E = ½kx²  (or ½Fx)

In the absence of resistive forces, energy transfers between these stores with the total conserved (e.g. a falling object: Ep → Ek).

Power

Power is the rate of energy transfer (or rate of doing work), in watts (W = J/s):

P = W ÷ t = E ÷ t
P = Fv        (force × velocity, for a constant force)

Efficiency

efficiency = useful energy (or power) output ÷ total input

Always less than 100% because some energy is dissipated (usually as heat). Multiply by 100 for a percentage.

Hooke's law and springs

A spring obeys Hooke's law up to the limit of proportionality:

F = kx      (k = spring constant, x = extension)

The elastic potential energy stored = ½Fx = ½kx² = the area under a force–extension graph.

Worked example

A 1200 kg car climbs a hill, rising 50 m in 20 s. Find the useful power output (g = 9.81).

• Work done against gravity = mgh = 1200 × 9.81 × 50 = 588 600 J.
• Power = W ÷ t = 588 600 ÷ 20 = 29 430 W ≈ 29.4 kW. ✓

Common mistakes

• Forgetting the cos θ when force and displacement aren't aligned.
• Confusing power (per second) with total energy/work.
• Forgetting elastic PE is the area under the force–extension graph (½Fx), not Fx.

Exam tips

• Use W = Fs cos θ and check the angle carefully.
• Apply conservation of energy (Ep ↔ Ek) for falling/projectile problems (ignoring resistance).
• Use P = Fv for moving objects at constant velocity.

Key facts to remember

• Work = Fs cos θ (joules); energy stores Ek = ½mv², Ep = mgh, elastic = ½kx².
• Power = energy/time = Fv (watts); efficiency = useful ÷ total (< 100%).
• Hooke's law F = kx; elastic PE = area under the force–extension graph = ½Fx.