A transverse sinusoidal wave moves along a string in the positive $x -$ direction at a speed of $20 c m / s$ . The wavelength of the wave is 0.5 m and its amplitude is 20 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 10 cm is

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$\frac{\sqrt{3} π}{50} \hat{j} m / s$

$\frac{2 \sqrt{3} π}{25} \hat{j} m / s$

$\frac{π}{25} \hat{i} m / s$

$- \frac{π}{25} \hat{i} m / s$

answer is B.

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Detailed Solution

$v_{p} = ω \sqrt{A^{2} - y^{2}} = 2 π f \sqrt{A^{2} - y^{2}}; v_{p} = 2 π (\frac{v}{λ}) \sqrt{A^{2} - y^{2}} v_{p} = \frac{2 π (0.2)}{0.5} \sqrt{{(20 \times 10^{- 2})}^{2} - {(10 \times 10^{- 2})}^{2}} v_{p} = 2 π (\frac{2}{5}) \sqrt{4 \times 10^{- 2} - 1 \times 10^{- 2}} = \frac{4 π}{5} \sqrt{3 \times 10^{- 2}} = \frac{4 π}{50} \sqrt{3} v_{p} = \frac{2 \sqrt{3} π}{25} \hat{j} m / s$