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(B) The power $P$ transmitted by a transverse wave in a string is given by $P = F v_p$,where $F$ is the transverse component of the tension force and

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$20 \ cm/s$

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(B) The power $P$ transmitted by a transverse wave in a string is given by $P = F v_p$,where $F$ is the transverse component of the tension force and $v_p$ is the transverse velocity of the particle at point $P$.For a wave traveling in the $x$-direction,the transverse force is $F = -T \frac{\partial y}{\partial x}$.Since $\frac{\partial y}{\partial x} = -\frac{v_p}{v}$,where $v$ is the wave speed,we have $F = T \frac{v_p}{v}$.Thus,the power is $P = (T \frac{v_p}{v}) v_p = \frac{T}{v} v_p^2$.Given $P = 40 \ mW = 40 	imes 10^{-3} \ W$,$T = 20 \ N$,and $v = 20 \ m/s$.Substituting the values: $40 	imes 10^{-3} = \frac{20}{20} v_p^2$.$40 	imes 10^{-3} = v_p^2$.$v_p = \sqrt{0.04} \ m/s = 0.2 \ m/s$.Converting to $cm/s$: $v_p = 0.2 	imes 100 \ cm/s = 20 \ cm/s$.

$A$ transverse wave is passing through a stretched string with a speed of $20 \ m/s$. The tension in the string is $20 \ N$. At a certain point $P$ on the string,it is observed that energy is being transferred at a rate of $40 \ mW$ at a given instant. Find the speed of point $P$.

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