# Simple Harmonic Motion Calculator

URL: https://speedycalculator.com/calculators/physics/simple-harmonic-motion/
Last updated: 2026-10-30

> The simple harmonic motion calculator finds the displacement, velocity and acceleration of an oscillator at any time from its amplitude and frequency, plus the maximum speed and acceleration.

## Inputs

- Amplitude A (m, cm, mm, km, in, ft, yd, mi)
- Frequency f (Hz, kHz, MHz, GHz, THz, rpm)
- Time t (s, ms, min, h, days)

## Results

- Angular frequency ω
- Displacement y
- Velocity v
- Acceleration a
- Maximum speed
- Maximum acceleration
- Period

## Formula

```
y = A·sin(ωt),  v = A·ω·cos(ωt),  a = −A·ω²·sin(ωt),  ω = 2πf
```

## Explanation

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

FieldWhat to enterAmplitude AUnits: m, cm, mm, km, in, ft …Frequency fUnits: Hz, kHz, MHz, GHz, THz, rpmTime tUnits: 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)0.050.04760.0750.07130.10.09510.1250.11890.150.14270.20.1902

## 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.

## FAQ

**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.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)