Question

A transverse wave of amplitude 0⋅50 mm and frequency 100 Hz is produced on a wire stretched to a tension of 100 N. If the wave speed is 100 m s⁻¹, what average power is the source transmitting to the wire?

Answer 1

Given,
Amplitude of the transverse wave, r = 0.5 mm $=0.5\times {10}^{-3}\mathrm{m}$
Frequency, f = 100 Hz
Tension, T = 100 N
Wave speed, v = 100 m/s
Thus, we have:
$$\begin{gathered} \nu =\sqrt{\left(\frac{T}{m}\right)} \\ \Rightarrow {\nu }^2=\left(\frac{T}{m}\right) \\ \Rightarrow m=\frac{T}{{\nu }^2}=\frac{100}{{\left(100\right)}^2} \\ =0.01\mathrm{kg}/\mathrm{m} \\ \mathrm{Average}\mathrm{power}\mathrm{of}\mathrm{the}\mathrm{source}: \\ {P}_{\mathrm{avg}}=2{\pi }^{2}m\nu r^2{f}^2 \\ =2{\left(3.14\right)}^2\left(0.01\right)\times 100\times {\left(0.5\times {10}^{-3}\right)}^2\times \left(100\right) \\ =2\times 9.86\times 0.25\times {10}^{-6}\times {10}^4 \\ =19.7\times 0.0025=0.049\mathrm{W} \\ =49\times {10}^{-3}\mathrm{W}=49\mathrm{mW} \end{gathered}$$