Simple Harmonic Motion Calculator
Simple harmonic motion describes an oscillation where the restoring force is proportional to displacement, as in an ideal mass on a spring. The period, the time for one complete cycle, is T = 2 pi times the square root of the mass divided by the spring stiffness, and it does not depend on the amplitude. This calculator takes the mass in kilograms, the spring stiffness in newtons per metre, and the amplitude in metres, and returns the period, the frequency in hertz, the angular frequency in radians per second, the peak speed, and the peak acceleration. It assumes an ideal linear spring obeying Hooke's law with no damping.
Simple harmonic motion formula
Angular frequency omega = square root of (k / m)
Period T = 2 * pi / omega
Frequency f = 1 / T
Peak speed = A * omega
Peak acceleration = A * omega^2
The model assumes an ideal linear spring obeying Hooke's law with no damping or driving force. The period is independent of amplitude.
Oscillation context
- The restoring force in simple harmonic motion is proportional to displacement.
- Period depends only on mass and stiffness, not amplitude.
- Peak speed occurs at equilibrium; peak acceleration at the extremes.
- Angular frequency omega equals the square root of stiffness over mass.
- Real oscillators lose energy to damping, slowly reducing amplitude.
Simple harmonic motion: frequently asked questions
What is the period of a mass-spring oscillator?
For a mass on an ideal spring the period is T = 2 pi times the square root of mass divided by stiffness, T = 2 pi sqrt(m/k). The period is the time for one complete cycle and does not depend on the amplitude, a defining feature of simple harmonic motion.
How are frequency and angular frequency related?
Frequency f is the number of cycles per second, the reciprocal of the period, f = 1/T, measured in hertz. Angular frequency omega is the rate of change of phase in radians per second, omega = 2 pi f = square root of k over m. Both describe how fast the oscillation repeats.
What is the maximum speed in simple harmonic motion?
The peak speed occurs at the equilibrium point and equals the amplitude times the angular frequency, v_max = A times omega. The peak acceleration occurs at the extremes and equals the amplitude times omega squared, a_max = A times omega squared.
Does amplitude affect the period?
No. For an ideal linear spring the period depends only on mass and stiffness, not on how far the mass is displaced. This independence of amplitude is what makes the motion simple harmonic and is why pendulum clocks keep time over a range of swing sizes at small angles.
What assumptions does this calculator make?
It assumes an ideal, massless, linear spring obeying Hooke's law with no damping or driving force. Real systems lose energy to friction and air resistance, which slowly reduces amplitude without changing the natural period much for light damping.
Official sources
- NIST: Fundamental Physical Constants and units.
- NIST Digital Library of Mathematical Functions: trigonometric and harmonic functions.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.