$\frac{d x}{d t}=a \omega \cos (\omega t+\pi / 6)$

Max. velocity $=a \omega$

$\therefore \quad \frac{a \omega}{2}=a \omega \cos (\omega t+\pi / 6)$

$\therefore \quad \cos (\omega t+\pi / 6)=\frac{1}{2}$

$\Rightarrow 60^{\circ}$ or $\frac{2 \pi}{6}$ radian $=\frac{2 \pi}{T} \cdot t+\pi / 6$

$\Rightarrow \frac{2 \pi}{T} \cdot t=\frac{2 \pi}{6}-\frac{\pi}{6}=+\frac{\pi}{6}$

$\therefore \quad t=+\frac{\pi}{6} \times \frac{T}{2 \pi}=\left|+\frac{T}{12}\right|$