Question

A transverse sinusoidal wave moves along a string in the positive $x-$direction at a speed of $10\text{cm/s}$. The wavelength of the wave is $0.5\text{m}$ and its amplitude is $10\text{cm}$. At a particular time $t$, the snapshot of the wave is shown in the figure. The velocity of a particle at $P$ when its displacement is $5\text{cm}$ along $y-$axis is

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

Answer 1

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

Angular frequency of the wave,
$\omega =vk=\frac{2\pi v}{\lambda }=\frac{2\pi \times 0.1}{0.5}$
$\Rightarrow \omega =\frac{2\pi }{5}\text{rad/s}$
So, speed of particle at $P$ when its displacement is $5\text{cm}$ along $y-$ axis is
$v_p=\omega \sqrt{A^2-y^2}=\frac{2\pi }{5}\sqrt{\frac{1}{100}-\frac{1}{400}}$
$\Rightarrow v_p=\frac{\sqrt{3}\pi }{50}\text{m/s}$
It is shown in the figure that particle is going in $+y$ axis direction, so velocity is ${\overrightarrow{v}}_p=\frac{\sqrt{3}\pi }{50}\hat{j}\text{m/s}$