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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.

Mass-spring-system

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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