Work, Energy and Power
Work and energy stores
Work done by a force is W = Fd cos θ, where θ is the angle between the force and displacement. Work transfers energy between stores. Kinetic energy KE = ½mv²; gravitational potential energy ΔGPE = mgΔh; elastic potential energy in a spring = ½kx². The work–energy theorem states that the resultant work done on a body equals its change in kinetic energy.
Conservation and power
Energy is conserved but can dissipate to thermal stores through friction or air resistance. Power is the rate of energy transfer: P = W/t = Fv for a constant force. Efficiency = useful output energy ÷ total input energy × 100%.
Worked example
A 50 kg box is pushed 4.0 m up a 30° incline by a 400 N force parallel to the incline. Friction is 60 N. Work by the applied force = 400 × 4.0 = 1600 J. Work against friction = 60 × 4.0 = 240 J. Height gained = 4.0 sin 30° = 2.0 m, so GPE gain = 50 × 9.81 × 2.0 = 981 J. By conservation, KE gain = 1600 − 240 − 981 = 379 J. The box speeds up because the applied work exceeds the losses and GPE gain.
Traps
Using cos θ when the force is already parallel to the displacement; forgetting work done against friction; using the sloping distance instead of vertical height for GPE; confusing average power with instantaneous power; omitting the 100% factor in efficiency.