Result

Energy of an oscillator

(W1.2.1) W1.2 · Energy in an Oscillation →

E = \frac{1}{2} k A^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}

Statement

Total energy, fixed by the amplitude, equals kinetic plus potential energy at any instant.

Symbols and units

$E$	total mechanical energy	J
$k$	spring/restoring constant	N/m
$A$	amplitude	m
$m$	mass	kg
$v$	speed at that instant	m/s
$x$	displacement from equilibrium	m

When it applies

Ideal SHM with no energy loss to friction or resistance.

Don't use when

Assumes no energy loss to friction, drag, or resistance — a damped oscillator's energy decays and this formula no longer gives the total at later times.

Derivation

At maximum displacement (x = A), velocity is zero, so all the energy is potential — this fixes the total:

E = \frac{1}{2} k A^{2}

At any other instant, the same total is split between kinetic and potential energy:

E = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}

Worked examples

Speed at the centre of the swing

At x = 0, all energy is kinetic, so setting the two expressions equal gives the maximum speed:

\frac{1}{2} m v_{m a x}^{2} = \frac{1}{2} k A^{2} \Rightarrow v_{m a x} = A \frac{k}{m} = A ω

A frequently needed relation, derived directly from energy conservation rather than memorized separately.

Watch out for

Speed is maximum at x = 0 (all energy kinetic) and zero at x = ±A (all energy potential) — the two extremes are easy to swap by mistake.