Work, Energy and Power

Topic-by-topic explanations

1. Introduction

Work is energy transferred by a force through displacement. Work and energy are scalar quantities measured in joules; the sign of work indicates whether the force transfers energy into or out of the selected body.

2. Notions of work and kinetic energy: The work-energy theorem

The work-energy theorem connects the net work of all forces with the change in kinetic energy: W_net=K_f−K_i. It provides a direct link between forces and speed without needing the time history.

3. Work

For a constant force, W=Fs cosθ, where θ is the angle between the force and displacement. Work is positive along displacement, negative opposite it and zero if force is perpendicular or displacement is zero.

4. Kinetic energy

Kinetic energy is K=½mv². It is non-negative in classical mechanics and grows with the square of speed, so doubling speed quadruples kinetic energy.

5. Work done by a variable force

For a force that varies with position, work is the signed area under an F_x-versus-x graph or W=∫F_x dx. Areas below the displacement axis represent negative work.

6. The work-energy theorem for a variable force

The work-energy theorem also holds for variable forces: total work by all forces equals ΔK. When force is given as a function of position, integrate or calculate graph area to find work.

7. The concept of potential energy

Potential energy is associated with configuration for a conservative force. Near Earth's surface U_g=mgh; for a conservative force, W_cons=−ΔU.

8. The conservation of mechanical energy

If only conservative forces do work, mechanical energy K+U remains constant. If friction or another non-conservative force does work, Δ(K+U)=W_non-conservative.

9. The potential energy of a spring

An ideal spring stretched or compressed by x stores U_s=½kx², valid while the spring obeys Hooke's law. Doubling extension makes stored energy four times as large.

10. Power

Average power is P=W/Δt; instantaneous power is P=F·v. The SI unit is watt (J s⁻¹). Efficiency is useful output energy or power divided by input, multiplied by 100%.

11. Collisions

Momentum is conserved in an isolated collision. Elastic collisions also conserve total kinetic energy; inelastic collisions do not. In a perfectly inelastic collision, bodies move together after collision.

Formula sheet

Worked example

Common mistakes and exam checklist

Use net work from all forces. Momentum conservation does not imply kinetic-energy conservation. Work depends on the angle between force and displacement.

• Identify all forces before finding net work.
• Use energy conservation only when the assumptions apply.
• Show working, units and signs; check that the answer is reasonable.