For a single degree of freedom spring-mass system with \(m=1 \mathrm{~kg}\) and \(k=\) \(4 \times 10^{4} \mathrm{~N} / \mathrm{m}\), complete the following for the case of free vibration.

(a) Determine the natural frequency in \(\mathrm{Hz}\) and the corresponding period of vibration.

(b) Given an initial displacement of \(5 \mathrm{~mm}\) and zero initial velocity, write an expression for the time response of free vibration using the following form:

\[ x(t)=X_{1} e^{i \omega_{n} t}+X_{2} e^{-i \omega_{n} t} \mathrm{~mm} \]

(c) Plot the first ten cycles of motion for the result from part (b).