§ 5.1 Energy conservation
Key ideas
- Work done W = Fs cos θ, where θ is the angle between the force and the displacement; work transfers energy.
- Energy is conserved: never created or destroyed, only transferred between stores or converted in form.
- Efficiency = useful output ÷ total input, as a ratio or a percentage.
Equations
W = F s cos θwork = force × displacement × cos(angle between them)J
efficiency = useful / totaluseful energy or power out ÷ total inno unit
Fig. 1 · Only the component of F along the motion does work: W = Fs cos θ; the perpendicular component does none at all.
Watch out: a force perpendicular to the motion does zero work, however large it is: cos 90° = 0. The weight of a satellite in circular orbit does no work on it.
§ 5.2 Gravitational PE and kinetic energy
Key ideas
- Kinetic energy Ek = ½mv², derived from W = Fs with F = ma and v² = u² + 2as.
- Gravitational potential energy change ΔEp = mgΔh, from the work mg does over a height change Δh.
- For a falling object with negligible resistance, Ep transfers to Ek joule for joule: the total stays constant.
- Power is the rate of doing work, P = W/t; for a constant force at steady speed, P = Fv.
Equations
Ek = ½ m v²kinetic energy of a moving massJ
ΔEp = m g Δhchange in gravitational potential energyJ
P = W / tpower = rate of doing workW
P = F vpower = force × velocity at constant speedW
Fig. 2 · The energy trade during a fall: Ep drains at exactly the rate Ek fills, so the two lines mirror each other beneath the constant total.
Watch out: the v in Ek = ½mv² is squared, so doubling speed quadruples kinetic energy. Equating mgΔh to ½mv² needs resistance to be negligible: say so when you use it.