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Mass–spring and pendulum

Two common systems oscillate harmonically: a mass attached to a spring, and a pendulum swinging with small amplitudes. The period depends on the mass and spring stiffness in the first case, and only on the length and gravitational acceleration in the second – not on the mass.

\omega = \sqrt{k/m},\ \ T = 2\pi\sqrt{m/k}

mass–spring system

T = 2\pi\sqrt{L/g}

simple pendulum, small swings

Symbols

$k$	spring stiffness	N/m
$m$	mass	kg
$L$	pendulum length	m

Example

A 2.0 kg block hangs from a spring with $k = 800$ N/m. $\omega = \sqrt{800/2.0} = 20$ rad/s, so $T = 2\pi/20 \approx 0.314$ s.

The period of a pendulum is independent of the mass and (for small swings) of the amplitude.

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Part of Physics: Oscillations and waves.