we have, \(x=A \cos \omega t .\) At \(t=0, x=A\)

When \(t=\tau, x=A-a\) and

\(\text { when } t=2 \tau, x=A-3 a\)

\(\Rightarrow A-a=A \times \cos \times \omega \tau\) and              \(...(i)\)

\(A-3 a=A \times \cos \times 2 \omega \tau\)                    \(...(ii)\)

As, \(\cos 2 \omega \tau=2 \cos ^{2} \omega \tau-1\)

\(\Rightarrow \quad \frac{A-3 a}{A}=2\left(\frac{A-a}{A}\right)^{2}-1\)

\(\therefore \quad \frac{A-3 a}{A}=\frac{2 A^{2}+2 a^{2}-4 A a-A^{2}}{A^{2}}\)

\(\therefore \quad A^{2}-3 a A=A^{2}+2 a^{2}-4 A a\)

\(\therefore \quad a^{2}=2 a A \Rightarrow A=2 a\)

\(\text { Now, } A-a=A \times \cos \times \omega \tau \ldots \ldots[\text { From }(1)]\)

\(\Rightarrow \quad \cos x \omega \tau=\frac{1}{2}\)

\(\therefore \quad \frac{2 \pi}{T} \tau=\frac{\pi}{3} \Rightarrow T=6 \tau\)