Question

A sinusoidal progressive wave is generated in a string. It's equation is given by$y = 2\sin (2\pi x - 100\pi t + \dfrac{\pi }{3})$ . The time when particle at $x = 4$ first passes through mean position will be
A) $\dfrac{1}{{150}}\sec $
B) $\dfrac{1}{{12}}\sec $
C) $\dfrac{1}{{300}}\sec $
D) $\dfrac{1}{{100}}\sec $

Answer 1

Hint: A sinusoidal wave means a wave that resembles a sine graph. It is a mathematical curve that is named after the trigonometric function ‘sine. It is a continuous wave and describes a smooth periodic oscillation.

Complete step by step answer:
The distance travelled by a wave from its mean position is represented by its amplitude. The sine wave travels minimum distance when the function is equal to zero.
Therefore it can be written that
$y = 0$
$\Rightarrow 2\sin (2\pi x - 100\pi t + \dfrac{\pi }{3}) = 0$
But ‘$2$’ is constant in the above equation.
Therefore the equation becomes
$\Rightarrow \sin (2\pi x - 100\pi t + \dfrac{\pi }{3}) = 0$ ---(i)
The value of sin is zero if it travels with a difference of n pi. Here n is any integer and can have values $n = 0,1,2,...$
Hence equation (i) becomes
$\Rightarrow \sin (2\pi x - 100\pi t + \dfrac{\pi }{3}) = \sin n\pi $
$\Rightarrow 2\pi x - 100\pi t + \dfrac{\pi }{3} = n\pi $
$\Rightarrow 100\pi t = 2\pi x - n\pi + \dfrac{\pi }{3}$
$\Rightarrow t = \dfrac{{2\pi x - n\pi + \dfrac{\pi }{3}}}{{100\pi }}$
Given that distance at $4\sec $is to be calculated. Therefore substitute $x = 4$
$\Rightarrow t = \dfrac{{8\pi - n\pi + \dfrac{\pi }{3}}}{{100\pi }}$
For t to be minimum the value of n should be equal to $8$ .
$\Rightarrow t = \dfrac{{8\pi - 8\pi + \dfrac{\pi }{3}}}{{100\pi }}$
$\Rightarrow t = \dfrac{{\dfrac{\pi }{3}}}{{100\pi }}$
$\Rightarrow t = \dfrac{1}{{300}}\sec $

Therefore, the time at which the particle first passes through mean position is $\dfrac{1}{{300}}\sec $. Hence, Option C is the right answer.

Note:
A wave that always travels continuously in a medium is called a progressive wave. A progressive wave moves in one direction only with a constant amplitude. A progressive wave keeps on moving away from the mean position. It is to be noted that in a progressive wave the motion is easily transferred among the particles in a forward direction. In a progressive wave, the energy gets propagates into the medium. The particles of the medium vibrate in a to and fro motion and pass the disturbance.