Simple Harmonic Motion Calculator
Enter mass m, spring constant k and amplitude A to get angular frequency ω=√(k/m), period T=2π√(m/k), frequency, max velocity Aω, max acceleration Aω² and total energy ½kA².
Input Data
Results
At a glance:Simple harmonic motion (SHM) is oscillation where the net force is a restoring force proportional to displacement and always directed toward equilibrium: F=−kx, with k the restoring constant (e.g. spring stiffness, N/m) and x the displacement. Solving the equation gives sinusoidal motion x(t)=A·cos(ωt+φ), where A is the amplitude and ω the angular frequency. For a mass m on a spring of constant k, ω=√(k/m), period T=2π/ω=2π√(m/k), frequency f=1/T. Key feature: the period is independent of amplitude (isochronism) and depends only on k and m — shared by pendulums, LC circuits and atomic vibrations. Velocity v(t)=−Aω·sin(ωt+φ), max v_max=Aω (at equilibrium); acceleration a(t)=−Aω²·cos(ωt+φ), max a_max=Aω² (at the ends). Energy: kinetic K=½mv², potential U=½kx², total E=K+U=½kA² is conserved — all kinetic at equilibrium (½mv_max²=½kA²), all potential at the ends (½kA²). Example (defaults): m=1 kg, k=16 N/m, A=0.5 m → ω=√(16/1)=4 rad/s, T=2π/4≈1.571 s, f≈0.637 Hz, v_max=0.5×4=2 m/s, a_max=0.5×16=8 m/s², E=½×16×0.25=2 J. SHM is the most fundamental oscillation in physics, appearing in mechanics (springs, small-angle pendulums), electricity (LC oscillation), optics (electromagnetic fields) and quantum mechanics (harmonic oscillator).
Formula
Angular frequency: ω = √(k/m)
Period: T = 2π·√(m/k)
Frequency: f = 1/T = (1/2π)·√(k/m)
Max velocity: v_max = A·ω
Max acceleration: a_max = A·ω²
Total energy: E = ½·k·A²
$$\omega = \sqrt{\frac{k}{m}}, \quad T = 2\pi\sqrt{\frac{m}{k}}$$$$v_{max} = A\omega, \quad a_{max} = A\omega^2, \quad E = \frac{1}{2}kA^2$$How to Use
- Enter mass m (kg), spring constant k (N/m) and amplitude A (m).
- The tool gives ω, T, f, v_max, a_max and E.
- k=0 is treated as a free particle (ω=0, no oscillation).
Case Studies
Spring-mass oscillator
m=1 kg, k=16 N/m, A=0.5 m.
ω=√(16/1)=4 rad/s; T=2π/4≈1.571 s; f≈0.637 Hz.
v_max=0.5×4=2 m/s; a_max=0.5×16=8 m/s²; E=½×16×0.25=2 J.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.