The equation of motion of a particle is given by x = a sin (50t + π 3 ) cm. The particle will come to rest at time t_1 and it will have zero…

The equation of motion is given by $x = a \sin\left(50t + \dfrac{\pi}{3}\right)$x=asin(50t+3π​).

Velocity is the rate of change of displacement:

$$v = \dfrac{dx}{dt} = 50a \cos\left(50t + \dfrac{\pi}{3}\right)$$v=dtdx​=50acos(50t+3π​)

The particle comes to rest when $v = 0$v=0. For the first time it comes to rest:

$$\cos\left(50t_1 + \dfrac{\pi}{3}\right) = 0$$cos(50t1​+3π​)=0

$$50t_1 + \dfrac{\pi}{3} = \dfrac{\pi}{2}$$50t1​+3π​=2π​

$$50t_1 = \dfrac{\pi}{2} - \dfrac{\pi}{3} = \dfrac{\pi}{6}$$50t1​=2π​−3π​=6π​

$$t_1 = \dfrac{\pi}{300} \text{ s}$$t1​=300π​ s

Acceleration is the rate of change of velocity:

$$A = \dfrac{dv}{dt} = -2500a \sin\left(50t + \dfrac{\pi}{3}\right)$$A=dtdv​=−2500asin(50t+3π​)

The particle has zero acceleration when $A = 0$A=0. For the first time it has zero acceleration:

$$\sin\left(50t_2 + \dfrac{\pi}{3}\right) = 0$$sin(50t2​+3π​)=0

$$50t_2 + \dfrac{\pi}{3} = \pi$$50t2​+3π​=π

$$50t_2 = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$$50t2​=π−3π​=32π​

$$t_2 = \dfrac{2\pi}{150} = \dfrac{\pi}{75} \text{ s}$$t2​=1502π​=75π​ s

Answer: $\dfrac{\pi}{300} \, s, \dfrac{\pi}{75} \, s$300π​s,75π​s