Simple Harmonic Motion Calculator

What is simple harmonic motion?

An object performs simple harmonic motion when it moves back and forth about an equilibrium position under a restoring force proportional to its displacement. A mass on a spring and a pendulum swinging through small angles are the best known examples. Position varies with time as a cosine (or sine).

Formulas

• Angular frequency: ω = 2 × π / T = 2 × π × f
• Position: x = A × cos(ω × t + φ)
• Velocity: v = −A × ω × sin(ω × t + φ)
• Acceleration: a = −ω² × x
• Maximum speed: A × ω, maximum acceleration: A × ω²

A is the amplitude, T the period, f the frequency and φ the initial phase.

Where are speed and acceleration greatest?

As the object passes through equilibrium its speed is greatest and its acceleration is zero. At the extremes it is the other way round: speed is zero and acceleration is greatest, directed toward equilibrium. Acceleration always has the opposite sign to position.

Example

For an oscillation with amplitude 10 cm and period 2 s, ω = π rad/s. One third of a second after release from the extreme the phase is 60 degrees: position is 10 × cos 60° = 5 cm, velocity −0.1 × π × sin 60° = −0.272 m/s and acceleration −π² × 0.05 = −0.493 m/s². The maximum speed is 0.314 m/s.

Initial phase

The phase fixes where the object is at t = 0. For an object released from the extreme φ = 0. For one that starts at equilibrium moving in the positive direction φ = −90°. If your textbook writes x = A sin(ωt), that is the same as entering φ = −90° here.

Limits

The calculation is for undamped, linear oscillation. With friction the amplitude decays over time. For a pendulum swinging through large angles the period depends on amplitude and the motion is not exactly harmonic.