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## Problem

The frequency of a mass on a spring is 0.5 Hz. Determine the displacement of the spring prior to oscillation.

## Solution

The period of a mass-spring system is

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

The diagram above is a free-body diagram at the moment of release. Treating upwards as the positive direction:

${F}_{net} = ma$
$kx - mg = m(0)$
$k = \frac{mg}{x}$.

Thus

$0.5 = 2 \pi \sqrt{\frac{m}{mg/x}}$
$0.5 = 2 \pi \sqrt{\frac{x}{g}}$.

Solving for $x$ yields

$x = g \left(\frac{T}{2\pi}\right)^2$.

Therefore,

$x = 9.81 \left(\frac{0.5}{2\pi}\right)^2$
$x = 0.062 \: m$.
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