Simple harmonic motion: the back-and-forth motion of an oscillator whose restoring force is proportional to its displacement. From the amplitude A and frequency f, this free calculator finds the angular frequency ω = 2πf, the period T, and the peak speed and acceleration — with every step of the working shown.
Simple harmonic motion (SHM) is the smooth, repeating oscillation you see in a mass on a spring or a small-angle pendulum. It is defined by a restoring force that always points back toward equilibrium and grows in proportion to the displacement. The position over time follows x = A·cos(ωt), where A is the amplitude — the furthest the object moves from the centre — and ω is the angular frequency.
The starting point for the numbers is the frequency f, the number of full oscillations completed each second, measured in hertz. From it you get the angular frequency directly: ω = 2πf, in radians per second. The angular frequency is the version of frequency that fits inside the sine and cosine functions, because one complete cycle corresponds to 2π radians. The time for a single cycle is the period, T = 1/f.
Once you know ω and the amplitude, the two extreme quantities follow at once. The speed is largest as the object whips through the equilibrium point, where v_max = Aω. The acceleration is largest at the turning points, where the displacement — and therefore the restoring force — is greatest, giving a_max = Aω². Because ω = 2πf, both peak values climb steeply as the frequency rises: doubling the frequency doubles the maximum speed and quadruples the maximum acceleration.
To use the calculator, enter the amplitude and frequency, choosing convenient units; everything is converted to SI base units (metres, hertz) behind the scenes. For the closely related case of a swinging pendulum, where the period instead depends on length and gravity, see the pendulum period calculator. We cover the underlying theory in depth in our guide to simple harmonic motion.
A mass on a spring oscillates with amplitude A = 0.1 m at a frequency of f = 2 Hz. The angular frequency is ω = 2πf = 2π × 2 = 12.57 rad/s, and the period is T = 1/f = 1/2 = 0.5 s. The maximum speed is v_max = Aω = 0.1 × 12.57 = 1.257 m/s, reached as the mass passes through equilibrium, and the maximum acceleration is a_max = Aω² = 0.1 × 12.57² = 15.79 m/s², reached at the ends of the swing.
SHM is the foundation for understanding springs, pendulums and clocks, the vibration of molecules and crystals, alternating current and electrical resonance, and the waves that carry sound and light. Almost any system disturbed slightly from a stable equilibrium oscillates in approximately simple harmonic motion, which is why these few equations appear across all of physics and engineering.
Angular frequency is ω = 2πf, where f is the ordinary frequency in hertz. It measures how many radians of the oscillation cycle pass each second, so one full cycle (frequency f) corresponds to 2π radians. Angular frequency is also the constant that appears in the SHM equation x = A·cos(ωt).
The period T is the time for one complete oscillation and is simply the reciprocal of the frequency: T = 1/f. It also relates to angular frequency through T = 2π/ω. For example, an oscillation at 2 Hz has a period of 0.5 s.
In simple harmonic motion the speed is greatest at the equilibrium point and the acceleration is greatest at the extremes of the motion. Their peak values are v_max = Aω and a_max = Aω², where A is the amplitude and ω the angular frequency. Both grow rapidly with frequency because ω = 2πf.
Frequency f counts complete cycles per second and is measured in hertz (Hz). Angular frequency ω measures the same motion in radians per second (rad/s) and is larger by a factor of 2π: ω = 2πf. Use frequency for everyday counting and angular frequency inside the sine and cosine functions that describe the motion.
No. For ideal simple harmonic motion the period and frequency depend only on the system (for example the mass and spring constant, or pendulum length and gravity), not on the amplitude. Changing the amplitude does change the maximum speed and acceleration, since both are proportional to A, but the period stays the same.