A stretched rope having linear mass density $5 \times 10^{-2} \text{ kgm}^{-1}$ is under a tension of $80…

A stretched rope having linear mass density $5 \times 10^{-2} \text{ kgm}^{-1}$ is under a tension of $80 \text{ N}$. The power that has to be supplied to the rope to generate harmonic waves at a frequency of $60 \text{ Hz}$ and an amplitude of $6 \text{ cm}$ is
  1. 362 W
  2. 251 W
  3. 511 W
  4. 416 W

Solution

Speed of the wave, $v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{80}{5 \times 10^{-2}}} = 40\text{ ms}^{-1}$ $\therefore \text{Power, } P = \frac{1}{2}\rho\omega^2 A^2 sv = \frac{1}{2}\mu(2\pi f)^2 vA^2 \quad (\text{As, } \rho s = \mu)$ $= \frac{1}{2} \times 5 \times 10^{-2}(2\pi \times 60)^2 \times 40 \times (0.06)^2$ $= 511\text{W}$

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