Question

The equation of motion of a particle started at $t = 0$ is given by $x = 5\sin (20t + \dfrac{\pi }{3})$, where $x$ is in centimeter and $t$ in second. When does the particle
(a) first come to rest

Answer 1

The given particle is in motion. When the particle is in motion, it’s velocity at any instant is given by the derivative of its position at that instant with respect to time. And when the particle comes to rest, its velocity becomes zero.

Formula Used:
The velocity is given by:$v = \dfrac{{dx}}{{dt}}$
where, $x$ is the particle's mean position and $t$ is the time.

Complete step by step answer:
It is given in the problem that the motion of the particle starts at $t = 0$ and the equation of motion of that particle is $x = 5\sin \left( {20t + \dfrac{\pi }{3}} \right)$.

The velocity is the derivative of a particle's mean position with respect to time.
v = \dfrac{{dx}}{{dt}}$$$$ \to (1)
Substituting the value of position $x$ in equation (1) and differentiating it

\Rightarrow v = \dfrac{d}{{dt}}[5\sin (20t + \dfrac{\pi }{3})] \\\ \Rightarrow v = 5 \times 20\cos (20t + \dfrac{\pi }{3})$$$$ \to (2)$$ When the particle is in motion and its velocity becomes zero, then that particle comes at rest. Therefore substituting the velocity as zero in equation (2) $$0 = 5 \times 20\cos (20t + \dfrac{\pi }{3})$$ Also, this can be written as $$0 = \cos (20t + \dfrac{\pi }{3})$$$$ \to (3)$$ The right hand side of equation (3) is a function of some value; it is a cosine of some value. Therefore, the left hand side should also be a cosine of something whose value is zero. And, $$\cos {90^ \circ } = 0$$. So, equation (3) becomes $$\cos 90 = \cos (20t + \dfrac{\pi }{3})$$ Equating both sides, $${90^ \circ } = 20t + \dfrac{\pi }{3} \\\ \Rightarrow 20t = \dfrac{\pi }{2} - \dfrac{\pi }{3} \\\ \Rightarrow 20t = \dfrac{\pi }{6} \\\ \therefore t = \dfrac{\pi }{{120}}$$ **Therefore, the time when the particle should come at rest is $$\dfrac{\pi }{{120}}$$ seconds.** **Note:** Velocity can also be zero for a moving particle only when the particle comes to rest after some time of being in motion (as given in question). But, velocity is also considered to be zero when a particle stops at one point and reverses its position. For example in a pendulum, the bob stops at the extreme position and reverses its direction. At that time, velocity is zero. Here, the acceleration due to gravity and the mass of the body are also considered.