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Q.

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. At a particular time t, the snap-shot of the wave is shown in figure. The speed of point P is $x \sqrt{2} π m / s$ when its displacement is $\frac{10}{\sqrt{2}} cm,$ then the value of x is____

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answer is 2.

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Detailed Solution

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$y = 10 \cos 4 π x$

So for displacement of $\frac{10}{\sqrt{2}}$

$\Rightarrow \frac{10}{\sqrt{2}} = 10 \cos 4 π x$
$\frac{\partial y}{\partial x} = - 40 π \sin 4 π x$
$\frac{\partial y}{\partial x} = - \frac{40 π}{\sqrt{2}}$
$V_{p a r t i c l e} = - V_{w a v e} \frac{\partial y}{\partial x}$
$V_{p a r t i c l e} = 10 \times (\frac{40 π}{\sqrt{2}}) = \frac{400 π}{\sqrt{2}} c m / s = 2 \sqrt{2} π m / s$

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