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

Max. velocity $=a \omega$

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

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

$60^{\circ}$ or $\frac{2 \pi}{6}$ radian, $=\frac{2 \pi}{\mathrm{T}} \cdot \mathrm{t}+\pi / 6=\frac{1}{2}=\pi$

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

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