What is simple harmonic motion?

SHM is periodic motion where the restoring force is directly proportional to and opposite the displacement from equilibrium. Examples include a mass on a spring (Hooke's law) and a simple pendulum (small-angle approximation).

What is angular frequency (ω)?

Angular frequency ω = 2πf = 2π/T, where f is frequency in hertz and T is the period in seconds. For a spring-mass system, ω = √(k/m).

What is the phase angle (φ)?

The phase angle (initial phase) shifts the starting position of the oscillation. φ = 0° means the object starts at maximum positive displacement; φ = 90° means it starts at the equilibrium moving in the negative direction.

Why is acceleration 180° out of phase with displacement?

Because a(t) = −ω² x(t). When displacement is maximum positive, acceleration is maximum negative — the restoring force always points back toward equilibrium.

How do I find the period or frequency from ω?

Period T = 2π / ω seconds. Frequency f = ω / (2π) hertz. Both are shown in the results panel once you enter ω.

What units does this calculator use?

Displacement and amplitude are in meters (or any consistent length unit you choose). Angular frequency is in rad/s. Time is in seconds. Velocity is in m/s and acceleration in m/s².

Simple Harmonic Motion Calculator

Enter amplitude, angular frequency, and time to instantly compute displacement, velocity, and acceleration for a simple harmonic oscillator.

Oscillator Parameters

Amplitude (A, meters)

Angular Frequency (ω, rad/s)

Initial Phase Angle (φ, degrees)

Time (t, seconds)

Derived Properties

Period (T)

—

seconds

Frequency (f)

—

Max Velocity

—

m/s

Max Acceleration

—

m/s²

Instantaneous Values at t

Displacement x(t)

A cos(ωt + φ)

—

Velocity v(t)

−Aω sin(ωt + φ)

—

m/s

Acceleration a(t)

−Aω² cos(ωt + φ)

—

m/s²

Waveform — One Period

Summary

Simple harmonic motion (SHM) describes oscillating systems — springs, pendulums, LC circuits — where the restoring force is proportional to displacement. This calculator uses the standard SHM equations x(t) = A cos(ωt + φ), v(t) = −Aω sin(ωt + φ), and a(t) = −Aω² cos(ωt + φ) to find instantaneous values at any point in the cycle. Results update in real time as you type, and a phase plot lets you visualize the full cycle at a glance.

How it works

Enter the amplitude (A) — the peak displacement from equilibrium.
Enter the angular frequency (ω) in radians per second.
Enter the initial phase angle (φ) in degrees (default 0).
Enter the time (t) at which to evaluate the motion.
Read the instantaneous displacement, velocity, and acceleration below.
Inspect the phase plot to visualize the complete oscillation cycle.

Use cases

Check textbook SHM problems for displacement, velocity, and acceleration.
Verify spring-mass system behavior at a specific instant.
Explore how phase angle shifts the entire oscillation waveform.
Understand the phase relationship between position, velocity, and acceleration.
Quickly convert between period, frequency, and angular frequency.
Visualize how amplitude and frequency affect peak velocity and acceleration.

Frequently Asked Questions

Last updated: 2026-09-30 · Reviewed by Nham Vu