A particle begins its motion at t = 0 according to the equation x =...

The equation of motion of the particle is given by:

$x = 5 \sin(20 t + \dfrac{\pi}{3})$

The velocity $v$ is the first derivative of displacement with respect to time:

$v = \dfrac{dx}{dt} = 100 \cos(20 t + \dfrac{\pi}{3})$

The acceleration $a$ is the first derivative of velocity with respect to time:

$a = \dfrac{dv}{dt} = -2000 \sin(20 t + \dfrac{\pi}{3})$

For the acceleration to be zero, we must have:

$-2000 \sin(20 t + \dfrac{\pi}{3}) = 0$

$\sin(20 t + \dfrac{\pi}{3}) = 0$

This implies that the phase must be an integer multiple of $\pi$. Since we need the first time $t > 0$ when acceleration is zero, we equate the phase to the next multiple of $\pi$ after $\dfrac{\pi}{3}$, which is $\pi$:

$20 t + \dfrac{\pi}{3} = \pi$

$20 t = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$

$t = \dfrac{2\pi}{60} = \dfrac{\pi}{30} \text{ s}$