Spring Period Calculator
Enter mass m and spring constant k to compute the period T=2π√(m/k), frequency f=ω/(2π) and angular frequency ω=√(k/m). m=1, k=1 → T≈6.283 s. Period is amplitude-independent.
Input Data
Results
At a glance:The mass–spring oscillator is the archetypal simple-harmonic-motion (SHM) system. A mass m on a spring of constant k experiences a restoring force F=−kx (Hooke's law), so m·ẍ=−kx; the solution is x(t)=A·cos(ωt+φ), with angular frequency ω=√(k/m), period T=2π/ω=2π√(m/k) and frequency f=1/T=ω/(2π). Physical meaning: (1) the period depends only on mass and spring constant, not amplitude (isochronism, found by Galileo); (2) a larger k (stiffer spring) shortens the period (faster); (3) a larger m lengthens it (more inertia, slower); (4) energy exchanges between kinetic and elastic potential, E=½kA²=½mv_max²; (5) amplitude A sets the energy but not the period. History: Hooke discovered the elasticity law in 1676; Galileo observed pendulum isochronism. Applications: (1) mechanical-clock balance wheels; (2) car suspension (spring + damper); (3) building seismic tuned-mass dampers; (4) spring scales; (5) tuning forks and piezoelectric oscillators.
Formula
Angular frequency: ω = √(k/m)
Period: T = 2π/ω = 2π·√(m/k)
Frequency: f = 1/T = ω/(2π)
Restoring force: F = −k·x
Total energy: E = ½·k·A²
$$\omega = \sqrt{\frac{k}{m}}, \quad T = 2\pi\sqrt{\frac{m}{k}}, \quad f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$How to Use
- Enter mass m (kg) and spring constant k (N/m).
- The tool computes period, frequency and angular frequency.
Case Studies
Car suspension and tuned damper
Car suspension: k≈20000 N/m, body m≈400 kg (per axle) → T=2π√(400/20000)=0.889 s.
Too soft → wallow, too stiff → harsh; a damper (γ≈0.3) settles in 2–3 cycles.
Taipei 101 TMD: m=660 t, k≈1700 N/m → T≈124 s, cancelling the building's sway.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.