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SHM energy fractions at half amplitude

Short answer

At x = a/2, PE/E = (a/2)²/a² = 1/4, so KE/E = 3/4. Energy is half KE and half PE when ½kx² = ½E → x² = a²/2, so x = ±a/√2 ≈ ±0.707a.

In simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is KE and what fraction is PE? At what displacement is the energy half KE and half PE?

For SHM with amplitude a and displacement x:

Total energy  E = ½ k a²
PE = ½ k x²
KE = ½ k (a² − x²)

Energy fractions at x = a/2

PE/E = x²/a² = (a/2)²/a² = 1/4
KE/E = 1 − PE/E = 1 − 1/4 = 3/4

Answer: KE = 3/4 (75%) of the total energy and PE = 1/4 (25%).

Displacement at which KE = PE

Equal halves means PE = E/2:

½ k x² = ½ (½ k a²)
x² = a²/2
x = ± a/√2 = ± 0.707 a

Answer: the energy is half KE and half PE at x = ± a/√2 ≈ ± 0.707 a.

Check: at x = a/√2, KE = ½k(a² − a²/2) = ¼ka², PE = ¼ka²; each is ½E.

✓ Answer checked by Claude Opus 5.5 (high effort)

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