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Toolbox · SHM Period Calculator

SHM Period Calculator

Spring and pendulum periods with animation and an x-t graph

T=2\pi\sqrt{\frac{m}{k}}

The period depends only on mass m and stiffness k, not on amplitude.

Mass m

Stiffness k

Amplitude A (for animation and vmax)

Period T0.9935s

ω6.3246 rad/s

f1.0066 Hz

v_max0.6325 m/s

a_max4 m/s²

Speed x = 0.1 m · v = 0 m/s

Displacement–time graphx = A cos(ωt)

Steps

6 / 6

1Formula
$T=2\pi\sqrt{\frac{m}{k}}$
Period of a spring oscillator: T = 2π√(m/k).
2Units
$m=0.5\ \mathrm{kg},\quad k=20\ \mathrm{N/m}$
Convert to kg, N/m and m before substituting.
3Substitute
$T=2\pi\sqrt{\frac{0.5}{20}}$
Substitute the numbers into the period formula.
4Solve
$\omega=\sqrt{\frac{k}{m}}=6.3246\ \mathrm{rad/s}$
Angular frequency and period satisfy T = 2π/ω.
5Solve
$T=0.9935\ \mathrm{s},\quad f=\frac1T=1.0066\ \mathrm{Hz}$
Period and frequency are reciprocals: f = 1/T.
6Note
$v_{\max}=A\omega=0.6325\ \mathrm{m/s},\quad a_{\max}=A\omega^2=4\ \mathrm{m/s^2}$
Amplitude affects only the maximum speed and acceleration, not the period.

Concept

Simple harmonic motion is the basic vibration: the restoring force is proportional and opposite to displacement (F = −kx), so displacement follows a cosine.

A spring oscillator's period depends only on m and k; a small-angle pendulum's only on L and g — independent of bob mass and amplitude.

Going further

Isochronism: Galileo noticed that for small swings the period is independent of amplitude — the basis of early pendulum clocks. A 1 m pendulum takes about 2 s (a "seconds pendulum").
Large-angle correction: At larger amplitudes T = 2π√(L/g) underestimates the period; multiply by 1 + θ₀²/16 + … — about 1.7% at 30°.
Resonance and energy: When the driving frequency approaches the natural frequency the amplitude grows sharply (resonance). Kinetic and potential energy trade back and forth, conserving total mechanical energy.

Common mistakes

A spring oscillator's period is independent of amplitude — only m and k matter.
The pendulum formula holds only for small angles; do not use it for large swings.
k is in N/m; if given in N/cm multiply by 100 first.

Page keywordssimple harmonic motionpendulum periodspring periodoscillation periodpendulum period formula T = 2π√(L/g)spring mass period formulahow to calculate SHM periodperiod of a 1 m pendulumangular frequency and period

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