Energy in Simple Harmonic Motion

Energy is never destroyed in SHM — it perpetually converts between kinetic and potential form. The total stays constant.

The energy exchange

Watch KE and PE trade places as the block oscillates. Their sum — total energy E — never changes:

SpringMass amplitude 80 k 100 m 1
Show spring block wall energy KE PE E graph-x
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Green bar = KE. Orange bar = PE. Blue bar = total E (always full).

All PE at the extremes

Block at $x = +A$. Spring fully stretched. All energy is stored as potential energy. KE is zero — the block has momentarily stopped:

SpringMass amplitude 80 k 100 m 1
Snapshot at extreme
Show spring block wall displacement-arrow energy KE PE E
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$PE_{max} = \frac{1}{2}kA^2, \quad KE = 0$

All KE at the centre

Block passes through equilibrium. Spring at natural length — zero PE. All energy is kinetic. This is where the block moves fastest:

SpringMass amplitude 80 k 100 m 1
Snapshot at centre
Show spring block wall velocity-arrow energy KE PE E
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$KE_{max} = \frac{1}{2}kA^2, \quad PE = 0$

KE = PE at $x = A/\sqrt{2}$

There is a precise position where kinetic and potential energy are exactly equal — each is half the total:

SpringMass amplitude 80 k 100 m 1
Snapshot at equal-energy
Show spring block wall displacement-arrow energy KE PE E
Hide graph-x readouts velocity-arrow acceleration-arrow equilibrium

$KE = PE = \frac{E}{2} \quad \text{at} \quad x = \pm\frac{A}{\sqrt{2}}$

Energy vs displacement

The PE curve is a parabola. KE is the gap between PE and total energy E. The position marker (white line) shows where the block currently is:

SpringMass amplitude 80 k 100 m 1
Snapshot at equal-energy
Show spring block wall graph-energy KE PE E
Hide energy readouts displacement-arrow velocity-arrow acceleration-arrow equilibrium

The formulas

$KE = \frac{1}{2}m v^2 = \frac{1}{2}k(A^2 - x^2)$

$PE = \frac{1}{2}kx^2$

$\boxed{E = KE + PE = \frac{1}{2}kA^2 \quad \text{(constant)}}$

Total energy depends only on $k$ and amplitude. Not on position. Not on time. Doubling the amplitude quadruples the total energy ($E \propto A^2$).