Simple harmonic motion is the smooth back-and-forth of a mass on a spring, a pendulum at small angles or a vibrating string. Its position follows a sine wave in time, and the restoring acceleration is proportional to the displacement.
How to use the Simple Harmonic Motion Calculator
- Amplitude A
- Frequency f
- Time t
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the amplitude, the frequency and the time since the object passed through the centre moving in the positive direction.
Inputs explained
| Field | What to enter |
|---|---|
| Amplitude A | Units: m, cm, mm, km, in, ft … |
| Frequency f | Units: Hz, kHz, MHz, GHz, THz, rpm |
| Time t | Units: s, ms, min, h, days |
What each result means
- Angular frequency ω = 2×pi×[Frequency f]
- Displacement y = [Amplitude A]×sin([Angular frequency ω]×[Time t])
- Velocity v = [Amplitude A]×[Angular frequency ω]×cos([Angular frequency ω]×[Time t])
- Acceleration a = -[Amplitude A]×[Angular frequency ω]^2×sin([Angular frequency ω]×[Time t])
- Maximum speed = [Amplitude A]×[Angular frequency ω]
- Maximum acceleration = [Amplitude A]×[Angular frequency ω]^2
- Period = 1÷[Frequency f]
Example values
How the main result changes when Amplitude A changes and everything else stays at the example values above:
| Amplitude A (m) | Displacement y (m) |
|---|---|
| 0.05 | 0.0476 |
| 0.075 | 0.0713 |
| 0.1 | 0.0951 |
| 0.125 | 0.1189 |
| 0.15 | 0.1427 |
| 0.2 | 0.1902 |
Formula
y = A·sin(ωt), v = A·ω·cos(ωt), a = −A·ω²·sin(ωt), ω = 2πf
Worked example
Inputs — Amplitude A: 0.1 m; Frequency f: 2 Hz; Time t: 0.1 s.
Results — Angular frequency ω = 12.56637 rad/s; Displacement y = 0.095106 m; Velocity v = 0.388322 m/s; Acceleration a = -15.018483 m/s²; Maximum speed = 1.25664 m/s; Maximum acceleration = 15.79137 m/s²; Period = 0.5 s.
Key terms
- Amplitude
- Maximum displacement from equilibrium.
Frequently asked questions
Where is the speed greatest?
Passing through the centre (equilibrium position).
Where is the acceleration greatest?
At the extremes, where the displacement is largest.
Does the period depend on amplitude?
Not for ideal SHM.
What makes motion simple harmonic?
A restoring force proportional to displacement, F = −kx.