Physics Tools
What It Does
Works out how fast something oscillates — its period, frequency, and angular frequency — for a mass on a spring or a simple pendulum, and (when you add an amplitude) its energy, peak speed, and peak acceleration. Everything runs in your browser.
How to Use It
- Pick a mode — Spring or Pendulum.
- Spring: enter any two of mass, spring constant, and period/frequency (add an amplitude for energy and peak speed). Pendulum: enter a length or a period/frequency and pick a planet for gravity (add a bob mass and angular amplitude for energy and peak speed).
- Read the derived values and the equation for each, and see the motion in the displacement-vs-time graph (and the pendulum schematic in pendulum mode).
- Click Copy on any value, or Copy all.
About Simple Harmonic Motion
Simple harmonic motion is the back-and-forth of anything pulled toward a resting point by a force proportional to how far it has moved — a mass on a spring, or a swinging pendulum. The motion repeats with a fixed period T (seconds per cycle); its frequency f = 1/T counts cycles per second (hertz), and its angular frequency ω = 2πf measures the same thing in radians per second. For a spring, ω = √(k/m), so a stiffer spring or a lighter mass oscillates faster; for a pendulum, ω = √(g/L), so a shorter pendulum or stronger gravity ticks faster. Two famous, counter-intuitive facts fall out: a pendulum’s period doesn’t depend on the mass of the bob, and (for small swings) it barely depends on the amplitude — Galileo’s observation that makes pendulum clocks keep time. A spring is isochronous: its amplitude sets the stored energy E = ½kA² and the peak speed v_max = Aω, but not the period. A real pendulum is not perfectly isochronous, though: this tool computes the exact large-amplitude period using the elliptic integral T = 2π·√(L/g)·(2/π)·K(sin(θ₀/2)), so it stays accurate for wide swings (a 90° swing takes about 18% longer than the ideal small-amplitude period) — no small-angle (sinθ ≈ θ) approximation. Enter an angular amplitude to see that correction; leave it blank for the ideal small-amplitude period. Damping and driving forces are beyond this model. Everything here is computed on your device.
Example
A 1 m pendulum on Earth (small swing) → T ≈ 2.006 s; swung to 90° → T ≈ 2.368 s. A m = 0.5 kg, k = 200 N/m spring → T ≈ 0.3142 s, f ≈ 3.183 Hz.
Frequently Asked Questions
Why doesn’t the pendulum’s mass matter?
Gravity pulls harder on a heavier bob, but a heavier bob also resists acceleration by exactly the same factor, so the two cancel. The period depends only on the length L and the local gravity g — not on the mass.
Does this tool use the small-angle approximation?
No. Many calculators assume sinθ ≈ θ, which is only accurate for tiny swings (to ~1% up to about 15°). This tool uses the exact period from the complete elliptic integral, so the period you see is correct for any swing angle up to 90°.
Does amplitude change the period?
For an ideal spring, no — it is isochronous, so amplitude changes only the energy and peak speed. For a pendulum it does: larger swings take longer. This tool shows that exact dependence when you enter an angular amplitude.
How do I model a different planet?
In pendulum mode, pick a planet from the gravity presets or type a custom g. The Moon’s gravity is about 1.62 m/s², so the same pendulum swings far more slowly there than on Earth.
Is my data sent to a server?
No. Every calculation runs locally in your browser; nothing you enter is transmitted or stored.