Simple Harmonic Motion Calculator

A mass on a spring. 0.5 kg swinging 10 cm at 2 Hz, starting at the top (φ = 0). ω = 2π × 2 = 12.566 rad/s. At t = 0.1 s the phase is 1.2566 rad, so x = 0.1 × cos 1.2566 = 3.090 cm, v = −0.1 × 12.566 × sin 1.2566 = −1.195 m/s (heading back through the centre) and a = −ω²x = −4.880 m/s². The spring constant is k = mω² = 78.96 N/m and the energy ½kA² = 0.3948 J.

Amplitude from a push. Released at x₀ = 5 cm with v₀ = 0.6 m/s at 1.5 Hz (ω = 9.425 rad/s): A = √(x₀² + (v₀ ÷ ω)²) = √(0.05² + 0.06366²) = 8.095 cm, and φ = atan2(−v₀ ÷ ω, x₀) = −51.85°.

The common mistake: assuming steady speed. A swing with a 2 s period and 5 cm amplitude starts at the end. Half-way to the centre is half of a quarter-period, 0.25 s, only if the speed were constant. In fact cos(ωt) = 0.5 at ωt = 60°, so t = T ÷ 6 = 0.333 s: the mass starts from rest at the end and spends longer near it.

Galileo Galilei noticed in the early 1600s that a pendulum’s small swings take the same time whatever their size, the defining property of simple harmonic motion. Christiaan Huygens turned that into the pendulum clock in 1656 and in Horologium Oscillatorium (1673) showed that only a cycloidal path makes the period exactly independent of amplitude.

Robert Hooke published the law that makes a spring oscillate harmonically in 1678, “ut tensio, sic vis” (as the extension, so the force), after hiding it in an anagram two years earlier. A force proportional to displacement and pointing back to the centre, F = −kx, is exactly what gives a = −ω²x and the sine-wave motion on this page, with ω = √(k ÷ m).

Real oscillators lose energy. The quality factor Q, now used for everything from bridges to quartz crystals, was introduced around 1914 by the engineer K. S. Johnson at Western Electric for radio coils. Critical damping, ζ = 1, is the design target for instrument needles, door closers and vehicle shock absorbers, which are tuned somewhat below it for comfort.

This calculator works through simple harmonic motion four ways: the state at any moment, the time a given position is reached (with the next time as well), the amplitude and phase that follow from a starting position and push, and a damped oscillation released from rest, with its regime, Q, ringing frequency and settling time. One box takes the frequency in hertz or rpm, the angular frequency in rad/s, or the period, chosen by its unit. The mass is drawn on its spring at the moment you asked about, and displacement, velocity and acceleration are plotted against time so you can see each one lead the next by a quarter-cycle.

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Related tools: Pendulum Calculator, Spring (Hooke’s Law) Calculator, and Standing Waves & Harmonics Calculator.