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, $\cos 2 \omega \tau=2 \cos ^{2} \omega \tau-1$
$\Rightarrow \quad \frac{A-3 a}{A}=2\left(\frac{A-a}{A}\right)^{2}-1$
$\therefore \quad \frac{A-3 a}{A}=\frac{2 A^{2}+2 a^{2}-4 A a-A^{2}}{A^{2}}$
$\therefore \quad A^{2}-3 a A=A^{2}+2 a^{2}-4 A a$
$\therefore \quad a^{2}=2 a A \Rightarrow A=2 a$
$\text { Now, } A-a=A \times \cos \times \omega \tau \ldots \ldots[\text { From }(1)]$
$\Rightarrow \quad \cos x \omega \tau=\frac{1}{2}$
$\therefore \quad \frac{2 \pi}{T} \tau=\frac{\pi}{3} \Rightarrow T=6 \tau$
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