A particle moves with simple harmonic motion in a straight line. In first $\tau \,s,$ after starting from rest it travels a distance $a,$ and in next $\tau \,s$ it travels $2a,$ in same direction, then
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$A(1-\cos \omega \tau)=a$
$A(1-\cos 2 \omega \tau)=3 a$
$\cos \omega \tau=\left(1-\frac{a}{A}\right)$
$\cos 2 \omega \tau=\left(1-\frac{3 a}{A}\right)$
$2\left(1-\frac{a}{A}\right)^{2}-1=1-\frac{3 a}{A} \quad \because \cos 2 \theta=2 \cos ^{2} \theta-1$
Solving the equation $\frac{a}{A}=\frac{1}{2}$
$A=2 a$
$\cos \omega \tau=\frac{1}{2}$
$T=\frac{2 \pi}{\omega}$
$T=6 \tau$
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