a
(a) \(U = k|x{|^3} \)

\(\Rightarrow F= - \frac{{dU}}{{dx}} = - 3k|x{|^2}\) ...(i)

Also, for \(SHM \) \(x = a\sin \omega \,t\) and \(\frac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = 0\)
\( \Rightarrow \) acceleration \( = \frac{{{d^2}x}}{{d{t^2}}} = - {\omega ^2}x\)

\(\Rightarrow F = ma\)
\( = m\frac{{{d^2}x}}{{d{t^2}}} = - m{\omega ^2}x\) ...(ii)
From equation (i) & (ii) we get \(\omega = \sqrt {\frac{{3kx}}{m}} \)
\( \Rightarrow T = \frac{{2\pi }}{\omega } = 2\pi \sqrt {\frac{m}{{3kx}}} = 2\pi \sqrt {\frac{m}{{3k(a\sin \omega \,t)}}} \)

\( \Rightarrow T \propto \frac{1}{{\sqrt a }}\).