Simple Harmonic Motion Calculator — Period, Frequency & Energy
Solve a spring-mass simple harmonic oscillator. Enter the mass, spring constant and amplitude — the calculator returns the period, frequency, angular frequency, maximum velocity, maximum acceleration and total mechanical energy.
kg
N/m
m
Time for one complete oscillation: T = 2π√(m/k)
- 1
Angular frequency ω = √(k/m)
√(10 ÷ 1) = 3.1623 - 2
Period T = 2π ÷ ω
2π ÷ 3.1623 = 1.9869Period is independent of amplitude — it depends only on mass and spring constant.
How does this calculator work?
For a spring-mass system: T = 2π√(m/k), f = 1/T, ω = √(k/m). Maximum speed is ωA and maximum acceleration is ω²A, both at equilibrium. Total energy is ½kA², constant throughout the oscillation. The period is independent of amplitude — doubling A doubles the energy but leaves T unchanged.
Formula
How this is calculated
Simple Harmonic Motion (SHM) describes the idealised oscillation of any system where the restoring force is proportional to displacement — the most common example being a mass on a spring obeying Hooke's law: F = −kx. The motion repeats with a period T = 2π√(m/k) that depends only on the mass m and spring constant k, not on the amplitude. This is the isochronous property of SHM.
The angular frequency ω = √(k/m) (in rad/s) is the natural oscillation rate. Ordinary frequency f = ω/(2π) in hertz is how many complete cycles occur per second. At any displacement x, the instantaneous velocity is v = ω√(A² − x²) and the acceleration is a = −ω²x. The extremes occur at the endpoints (x = ±A): velocity is zero and acceleration is at its maximum magnitude a_max = ω²A. At the equilibrium (x = 0): velocity peaks at v_max = ωA and acceleration is zero.
Total mechanical energy E = ½kA² is constant throughout the motion (conservation of energy) — it transfers between kinetic (½mv²) and elastic potential (½kx²) as the mass oscillates. This calculator assumes ideal conditions: no damping (friction or air resistance), massless spring, and small-amplitude motion where Hooke's law holds. Real systems always have some damping, which reduces amplitude over time and slightly shifts the resonant frequency.
Frequently asked questions
In ideal SHM the restoring force is perfectly proportional to displacement (F = −kx). This linearity makes the period purely a function of m and k. Doubling the amplitude doubles both the distance to travel and the average restoring force, so the two effects cancel exactly. This breaks down for large amplitudes where springs stop obeying Hooke's law.
The spring constant k must be in N/m (newtons per metre). To find k experimentally, hang the spring vertically and measure the displacement x when a known weight mg is applied: k = mg/x. Standard lab springs range from about 1 N/m (very soft) to 1000 N/m (stiff). Automotive coil springs are often 10,000–50,000 N/m.
A simple pendulum undergoing small-angle oscillations also exhibits SHM, but with period T = 2π√(L/g) where L is the pendulum length and g is gravitational acceleration (≈ 9.81 m/s²). The pendulum's effective "spring constant" is mg/L and its "mass" is m, giving the same SHM framework — but the inputs are different from a spring-mass system.
Also known as
TG we-Calculate Editorial Team. (2026). Simple Harmonic Motion Calculator — Period, Frequency & Energy [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/simple-harmonic-motion-calculator
TG we-Calculate Editorial Team. "Simple Harmonic Motion Calculator — Period, Frequency & Energy." TG we-Calculate. 2026. https://we-calculate.com/calculator/simple-harmonic-motion-calculator.
TG we-Calculate Editorial Team, "Simple Harmonic Motion Calculator — Period, Frequency & Energy," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/simple-harmonic-motion-calculator
@misc{wecalculate_simple_harmonic_motion_calculator, title = {Simple Harmonic Motion Calculator — Period, Frequency & Energy}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/simple-harmonic-motion-calculator}}, year = {2026}, note = {TG we-Calculate} }
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