A mass suspended from a spring performs vertical oscillations and the displacement $x($cm$)$ of the mass at time $t($s$)$ is given by

$\frac{1}{2}\frac{d^{2}x}{d{t}^2}=-48x$

If $x=\frac{1}{6}$ and $\frac{dx}{dt}=0$ when $t=0,$ determine the period and amplitude of the oscillations.

$(10$ marks$)$