This simple harmonic motion calculator solves the spring-mass oscillator: angular frequency, period, frequency and peak speed/amplitude relations for SHM.
The oscillator’s memoir: x(t) = A·cos(ωt), fastest at equilibrium (Aω), still at the extremes where acceleration peaks (Aω²). Amplitude never enters the period — the isochronism Galileo supposedly timed by cathedral lamps is why oscillators became clocks.
ω = √(k/m), T = 2π√(m/k) — Stiffness Speeds, Mass Slows
The gravitational cousin is our Pendulum Calculator (T = 2π√(L/g)) — the same equation wearing a different restoring force. Part of our Physics Calculators collection.
Worked Example (Verified)
- m = 0.5 kg, k = 200 N/m, A = 0.10 m (defaults): ω = 20 rad/s, T = 0.314 s, f = 3.18 Hz, vmax = 2.0 m/s, amax = 40 m/s²
- Double the mass → period × √2 (0.444 s): mass resists in square roots, which is why suspension tuning is a game of modest changes
- Amplitude independence, demonstrated: A = 0.05 or 0.20 m changes vmax (1.0 vs 4.0 m/s) but T never moves — for ideal springs; real springs add progressive stiffness at extremes
- Energy view: E = ½kA² = 1.0 J here, sloshing between spring PE (extremes) and KE (equilibrium) twice per cycle
SHM Is Everywhere
- Molecules vibrate (IR spectroscopy reads their ω), quartz crystals clock your watch at 32,768 Hz by design.
- Buildings and bridges have natural frequencies — resonance disasters (and tuned mass dampers) are SHM with stakes.
- LC circuits oscillate identically (ω = 1/√(LC) — see the LC Resonance Calculator): charge plays position’s role.
- Any stable equilibrium wiggles harmonically for small displacements — SHM is nature’s default small-print.
Related Tools
- Pendulum Calculator — period from length
- LC Resonance Calculator — the electrical oscillator
- Physics Calculators — the full collection
Frequently Asked Questions
How do you find the period of a spring-mass system?
T = 2π√(m/k). For m = 0.5 kg and k = 200 N/m: T = 2π√(0.0025) = 0.314 s (f = 3.18 Hz). Heavier mass or softer spring lengthens the period; amplitude has no effect in ideal SHM.
What is angular frequency in SHM?
ω = √(k/m), in rad/s — how fast the phase cycles, linked by ω = 2πf = 2π/T. The defaults give ω = 20 rad/s. It also scales the extremes: vmax = Aω and amax = Aω².
Does amplitude affect the period?
Not for an ideal spring obeying Hooke’s law — the restoring force grows exactly in proportion to displacement, so bigger swings take proportionally faster trips. Real springs deviate at large stretch (and pendulums do beyond small angles), which instrument-makers call circular error.
Where is the oscillator fastest and slowest?
Fastest crossing equilibrium (v = Aω), motionless at the turning points where acceleration and restoring force peak (a = Aω²). At the defaults: 2.0 m/s at center, 40 m/s² of acceleration at the extremes — extremes are where the violence is, at zero speed.
How much energy does the oscillator hold?
E = ½kA², constant in the ideal case — 1.0 J for the defaults — trading between elastic potential (at extremes) and kinetic (at equilibrium). Damping skims the total each cycle in real systems, which is why clocks need escapements and cars need shock absorbers replaced.
Is this SHM calculator free?
Yes — free, instant, no sign-up.