Potential energy and conservation of energy

Potential energy is energy stored because of where a body is, or how it is arranged: a book on a high shelf, a stretched spring. Near the Earth, a body raised through h gains V = mgh.

Where the zero of height is put is our choice, so V itself can be positive, negative or zero. But the change in V between two points is the same whatever we choose.

Potential energy belongs only to conservative forces. Compare the two kinds:

A conservative force always points the way the potential energy decreases: F = –dV/dx. For V = ax² – bx, F = –2ax + b, which is zero at x = b/2a: that point is an equilibrium. At a minimum of V the equilibrium is stable (a ball in a bowl); at a maximum it is unstable (a ball on a hilltop); where V is flat it is neutral.

A spring pulls back with F = –kx, so it stores V = ½kx² when stretched or squashed by x. A block on a spring has its greatest speed at x = 0 and stops for an instant at the ends ±x_m. A body can only be where V ≤ E; where V = E it turns back.

Conservation of mechanical energy: if only conservative forces do work, K + V stays the same. Energy only changes form, from potential to kinetic and back.

If friction or air resistance also acts, it takes energy away as heat: E_f – E_i = W_nc, the work of the non-conservative forces. Example: a 5 kg ball falls 20 m (g = 10 m s⁻²) and lands at 10 m s⁻¹. It started with 5 × 10 × 20 = 1000 J and ended with ½ × 5 × 10² = 250 J, so air resistance did –750 J of work.

A bob whirled in a vertical circle of radius R slows as it rises. To get round the top with the string still tight, it needs at least √(gR) at the top, which means √(5gR) at the bottom; the string pulls hardest at the bottom. With less than √(2gR) at the bottom it only swings to and fro; in between, the string goes slack part-way up and the bob leaves the circle.