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
\[= 0 . 5 \times  {10}^{- 3}   m\]
Frequency, f = 100 Hz
Tension, T = 100 N
Wave speed, v = 100 m/s
Thus, we have:

\[\nu = \sqrt{\left( \frac{T}{m} \right)}\] 

\[ \Rightarrow  \nu^2  = \left( \frac{T}{m} \right)\] 

\[ \Rightarrow m = \frac{T}{\nu^2} = \frac{100}{\left( 100 \r = 0 . 01  kg/m\] 

Average  power  of  the  source: 

\[ P_{avg}  = 2 \pi^2 m\nu r^2  f^2 \] 

\[  = 2   \left( 3 . 14 \right)^2   \left( 0 . 01 \right) \times 100 \times  \left( 0 . 5 \times {10}^{- 3} \right)^2  \times \left( 100 \right)\] 

\[  = 2 \times 9 . 86 \times 0 . 25 \times  {10}^{- 6}  \times  {10}^4 \] 

\[ = 19 . 7 \times 0 . 0025 = 0 . 049  W\] 

\[ = 49 \times  {10}^{- 3}   W = 49  mW\]