Oscillator energy and pendulums
A harmonic oscillator's energy is fixed and just sloshes between potential and kinetic — and a small-swing pendulum keeps time regardless of mass.
Energy sloshing
Total energy E = (1/2)kA^2 is set by stiffness and amplitude, and it trades form continuously: all potential (1/2)kx^2 at the turning points x = plus or minus A, all kinetic (1/2)mv^2 at equilibrium. Damping leaks this budget away into heat and sound, which is why real oscillators need a periodic push to keep going.
E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}mv_{\max}^2The pendulum
A pendulum's restoring force is gravity's tangential component, mg sin(theta), which is proportional to displacement only for small angles where sin(theta) is nearly theta in radians. That small-angle approximation gives T = 2 pi sqrt(L/g) — independent of both the bob's mass and the swing amplitude. Push a playground swing gently or hard and the period barely budges, until angles get large.
T = 2\pi\sqrt{\frac{L}{g}} \quad (\text{small angles})Worked example
Energy check: the 0.50 kg mass on the k = 200 N/m spring swinging with amplitude 0.10 m holds E = (1/2)(200)(0.10)^2 = 1.0 J. All 1.0 J is spring potential at the turning points, all 1.0 J is kinetic at the center, so v_max = sqrt(2E/m) = sqrt(2.0/0.50) = 2.0 m/s — matching A omega = (0.10)(20). Pendulum check: L = 1.0 m gives T = 2 pi sqrt(1.0/9.8) = 2.0 s, the classic seconds pendulum.
v_{\max} = \sqrt{\frac{2E}{m}} = \sqrt{\frac{2(1.0)}{0.50}} = 2.0\ \text{m/s}Flashcard
Total energy of a harmonic oscillator
Check your understanding
A 0.50 kg block on a spring with k = 200 N/m oscillates with amplitude 0.10 m. What is its maximum speed?