The smoothest motion in physics — and the regular-motion counterpart to our chaotic double pendulum. Release a mass on a spring or a simple pendulum and watch it draw its own sine wave in real time, trading kinetic and potential energy back and forth. Turn on the ideal-sine overlay to see exactly where a wide-swinging pendulum stops being "simple." It all runs on your device.
The period doesn't care about the amplitude. An ideal oscillator takes exactly the same time to complete a big swing as a small one — a property called isochronism. A mass on a spring obeys T = 2π√(m/k) and a small-angle pendulum obeys T = 2π√(L/g), and neither formula contains the amplitude. Drag the mass out further and the measured period barely moves for the spring; push a pendulum to a wide angle and watch the real period grow past the ideal — the point where "simple" harmonic motion stops being simple.
How It Works
A restoring force pulling toward equilibrium, integrated step by step, drawing a sine wave as it goes.
1
A restoring force
Displace the mass and something pulls it back toward equilibrium. For a spring that force is −k·x, exactly proportional to the displacement; for a pendulum it is −(g/L)·sin θ. Because the pull grows with distance, the mass overshoots equilibrium, is pulled back again, and oscillates forever in the absence of friction.
2
It draws a sine wave
The simulator advances the state with fourth-order Runge–Kutta at a small fixed time step. As it runs, the bottom panel plots displacement against time — and for a proportional restoring force that plot is a pure sinusoid, x(t) = A·cos(ωt), with angular frequency ω = √(k/m) for the spring or √(g/L) for the pendulum.
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Ideal versus real
The dashed line is the ideal sine the formula predicts. For the spring the real motion sits exactly on it. For the pendulum it matches at small angles, but as the swing widens sin θ falls below θ, the restoring force weakens, and the real curve visibly lags the ideal — the small-angle approximation breaking down before your eyes.
What is simple harmonic motion?
Simple harmonic motion (SHM) is the back-and-forth motion of a system whose restoring force is proportional to how far it has been displaced from equilibrium and always points back toward it. A mass on an ideal spring is the classic example. The result is a smooth oscillation whose position traces a sine wave in time, with a period that does not depend on the amplitude.
What is the period of a mass on a spring?
For a mass m on a spring of stiffness k, the period is T = 2π√(m/k). Heavier masses and softer springs oscillate more slowly. The period is independent of the amplitude, so pulling the mass further from equilibrium does not change how long each swing takes.
Does amplitude change the period of an oscillator?
For ideal simple harmonic motion, no. A mass on an ideal spring keeps exactly the same period whether you release it from a small or a large displacement — a property called isochronism. A real pendulum only obeys this at small angles; at large swings its period grows noticeably longer than the ideal formula predicts.
Why is a pendulum only approximately simple harmonic?
A pendulum's restoring force is proportional to sin θ, not θ. For small angles sin θ ≈ θ, so the motion is very close to simple harmonic with period T = 2π√(L/g). As the swing angle grows, sin θ falls below θ, the restoring pull weakens, and the real period lengthens and departs from the ideal sine — which you can watch happen with the ideal-sine overlay above.