Mass-Spring Period Calculator
A mass on an ideal spring oscillates in simple harmonic motion, and its period depends only on the mass and the spring's stiffness. This calculator returns the period, frequency and angular frequency from the mass and spring constant using T = 2 * pi * sqrt(m / k). A heavier mass or a softer spring oscillates more slowly. Gravity does not enter the period: it only shifts where the mass rests. The result assumes a massless, linear spring with no damping.
Mass-spring period formula
Period T = 2 * pi * sqrt(m / k)
Frequency f = 1 / T
Angular frequency omega = sqrt(k / m)
Oscillations per minute = 60 * f
The period grows with the square root of the mass and shrinks with the square root of the spring constant. Use SI units: mass in kilograms and spring constant in newtons per metre give the period in seconds.
Reading the result
- Quadrupling the mass doubles the period; quadrupling the spring constant halves it.
- The period is independent of amplitude for an ideal linear spring.
- Gravity changes the rest position but not the period.
- Real springs with mass or damping shift the result slightly.
Mass-spring period: frequently asked questions
What is the period of a mass on a spring?
For an ideal mass-spring system in simple harmonic motion, the period is T = 2 * pi * sqrt(m / k), where m is the mass and k is the spring constant. The period is the time for one full oscillation. A heavier mass or a softer spring gives a longer period.
How is frequency related to the period?
Frequency is the number of oscillations per second and is the reciprocal of the period: f = 1 / T, measured in hertz. The angular frequency is omega = sqrt(k / m), in radians per second, equal to 2 * pi * f.
Does gravity affect the period?
No. For a mass on an ideal spring, the period depends only on the mass and the spring constant, not on gravity. Gravity shifts the equilibrium position but does not change the oscillation period, which is why this works the same horizontally or vertically.
What are the assumptions?
This uses the ideal simple harmonic oscillator: a massless spring obeying Hooke's law, no damping or friction, and small enough amplitude that the spring stays linear. Real springs with mass, damping or nonlinearity will differ slightly.
Official sources
- NASA Glenn Research Center: Oscillations and simple harmonic motion.
- U.S. National Institute of Standards and Technology: SI units reference.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.