Energy in SHM (KE ↔ PE)

Energy in SHM oscillates between KE and PE but total stays constant.

KE: ½mv² = ½m·ω²·(A² − x²). Maximum (½mω²A²) at x = 0, zero at x = ±A.

PE: ½kx² = ½m·ω²·x². Maximum (½mω²A²) at x = ±A, zero at x = 0.

Total: E = ½kA² = ½mω²A² — depends on amplitude squared.

PE and KE oscillate at TWICE the SHM frequency (sin² and cos² have period T/2).

Time-average of KE over a period = time-average of PE = ½ × Total energy.

Doubling amplitude QUADRUPLES total energy.

Energy conservation: ½kx² + ½mv² = ½kA² always.