A transverse sinusoidal wave moves along a string in the positive $x$-direction at a speed of $10…

A transverse sinusoidal wave moves along a string in the positive $x$-direction at a speed of $10 \mathrm{~cm} / \mathrm{s}$. The wavelength of the wave is $0.5 \mathrm{~m}$ and its amplitude is $10 \mathrm{~cm}$. At a particular time $t$, the snap-shot of the wave is shown in figure. The velocity of point $P$ when its displacement is $5 \mathrm{~cm}$ is
  1. $\frac{\sqrt{3} \pi}{50} \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}$
  2. $-\frac{\sqrt{3} \pi}{50} \hat{j} \mathrm{~m} / \mathrm{s}$
  3. $\frac{\sqrt{3} \pi}{50} \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}$
  4. $-\frac{\sqrt{3} \pi}{50} \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}$

Solution

Particle velocity $v_p=-v$ (slope of $y-x$ graph) Here,$v=+v e$, as the wave is travelling in positive $x$-direction. Slope at $P$ is negative. $\therefore$ Velocity of particle is in positive $y(+\overrightarrow{\mathbf{j}})$ direction. $\therefore$ correct option is (a).

Asked in: JEE Advanced 2008 (Paper 2)

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