Time Instances for Rest and Zero Acceleration in SHM

The equation of motion is given by: $x = a \sin\left(50t + \frac{\pi}{3}\right)$

To find the time $t_1$ when the particle comes to rest, we set the velocity $v = \frac{dx}{dt}$ to zero: $v = 50a \cos\left(50t + \frac{\pi}{3}\right) = 0$ Setting $50t_1 + \frac{\pi}{3} = \frac{\pi}{2}$ gives: $50t_1 = \frac{\pi}{6} \implies t_1 = \frac{\pi}{300}\text{ s}$

To find the time $t_2$ when the acceleration is zero, we set $a = \frac{dv}{dt}$ to zero: $a = -2500a \sin\left(50t + \frac{\pi}{3}\right) = 0$ Setting $50t_2 + \frac{\pi}{3} = \pi$ gives: $50t_2 = \frac{2\pi}{3} \implies t_2 = \frac{\pi}{75}\text{ s}$

Therefore, the required times are $t_1 = \frac{\pi}{300}\text{ s}$ and $t_2 = \frac{\pi}{75}\text{ s}$, which corresponds to Option A.