Simple Harmonic Motion Calculator
Period, frequency, maximum speed and acceleration for a mass on a spring — the oscillation behind every vibration problem.
How fast something bounces on a spring.
How the simple harmonic motion calculator works
When the restoring force is proportional to displacement — F = −kx — the motion is sinusoidal, and its period depends only on the mass and the spring constant: T = 2π√(m/k).
The amplitude does not appear. A larger swing moves faster and covers more distance in exactly compensating measure, which is why a pendulum clock keeps time as its swing decays.
Formula: T = 2π√(m/k); vmax = Aω; amax = Aω²
Worked examples
| Inputs | Period | Note |
|---|---|---|
| 0.5 kg on a 20 N/m spring | 0.99346 s | period 0.993 s |
| Four times the mass | 1.98692 s | double the period |
| Twice the amplitude | 0.99346 s | same period, four times the energy |
FAQFrequently asked questions
Why does amplitude not affect the period?
Because the restoring force grows in proportion to displacement. A bigger swing means a stronger push, and the two effects cancel exactly.
What is angular frequency?
ω = 2πf, in radians per second. It is the natural variable for the sine describing the motion.
Is a pendulum simple harmonic?
Only approximately, for small swings. Past about 15° the period starts to lengthen with amplitude.
Where does the energy go during a cycle?
It shuttles between kinetic and elastic potential, staying constant in total — until damping removes it.
Why does this matter in engineering?
Because every structure has natural frequencies. Driving one at its own frequency is resonance, and it is how bridges and machines fail.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Measurement Institute — Australia's measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.