A particle executes S.H.M. of amplitude A along x -axis. At t = 0 ,...

Given, the instantaneous equation of motion of the particle is

$x = Asin\left(\omegat + \delta\right) . . . \left(1\right)$

As the particle is moving along positive $x$-axis, $v = \dfrac{dx}{dt} > 0$.

Substitute $0$ for $t$ and $\dfrac{A}{2}$ for $x$ into equation (1) and solve to calculate the required phase difference.

$\begin{cases}\dfrac{A}{2} & = & Asin\left(\omega \times 0 + \delta\right) \\ & ⇒ & sin\delta = \dfrac{1}{2} \\ & = & sin\dfrac{\pi}{6} \\ & ⇒ & \delta = \dfrac{\pi}{6}\end{cases}$