Question

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.

Moderate

Solution

Instantaneous power is given by
$P = \frac{{FA}^{2} ω^{2}}{v} \cos^{2} (ωt - kx)$
We know that $v_{p} = Aωcos (ωt - kx)$
$\Rightarrow P = \frac{F}{v} {v_{p}}^{2}$
$\Rightarrow 40 \times 10^{- 3} = \frac{20}{20} {v_{p}}^{2} \Rightarrow v_{p} = 0.2 m / s = 20 cm / s$

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