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

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$\frac{\sqrt{3} π}{50} \hat{i} m / s$

$- \frac{\sqrt{3} π}{50} \hat{i} m / s$

$- \frac{\sqrt{3} π}{50} \hat{j} m / s$

$\frac{\sqrt{3} π}{50} \hat{j} m / s$

answer is A.

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Detailed Solution

For a travelling wave at any point, we use Particle velocity $v_{p} = - (s l o p e o f y - x g r a p h)$
Here velocity is positive as wave is travelling in positive x direction and at point P slope
is negative. Here velocity of particle P is in positive y direction, given as
$v_{p} = ω \sqrt{A^{2} - y^{2}} \Rightarrow v_{p} = 2 π f \sqrt{A^{2} - y^{2}} \Rightarrow V_{P} = 2 π \frac{C}{λ} \sqrt{A^{2} - y^{2}} \Rightarrow v_{p} = \frac{2 π \times 0.1}{0.5} \sqrt{{(0.1)}^{2} - {(0.05)}^{2}} \Rightarrow v_{p} = \frac{\sqrt{3} π}{50} m / s$