Question

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of $10cm/s$. The wavelength of the wave is $0.5m$ and its amplitude is $10cm$. At a particular time $t$, the snap-shot of the wave is shown in figure. The velocity of the point $P$ when its displacement is $5cm$ is Figure:

• A. $\frac{\sqrt{3\pi }}{50}\hat{j}m/s$
• B. $-\frac{\sqrt{3\pi }}{50}\hat{j}m/s$
• C. $\frac{\sqrt{3\pi }}{50}\hat{i}m/s$
• D. $-\frac{\sqrt{3\pi }}{50}\hat{i}m/s$

Answer 1

The correct option is A $\frac{\sqrt{3\pi }}{50}\hat{j}m/s$

Particle displacement, $y=a\sin \left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]$

Putting $y=5cm;$

$a=10cm$

$\sin \left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]=\frac{1}{2}$

$\Rightarrow \frac{2\pi }{\lambda }(vt-x)]=\frac{\pi }{6}$

Particle velocity, $v=\frac{dy}{dt}=a\left(\frac{2\pi }{\lambda }v\right)$ $\cos \left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]$

Given, $a=10cm;v=0.1m{s}^{-1};\lambda =0.5m$

$\frac{2\pi }{\lambda }(vt-x)]=\frac{\pi }{6}$

$\therefore v=\frac{\sqrt{3}\pi }{50}m{s}^{-1}$ along y-axis

or $\overrightarrow{v}=\frac{\sqrt{3}\pi }{50}\hat{j}m{s}^{-1}$