Question

Two uniform wires are vibrating simultaneously in their fundamental notes. The tensions, lengths, diameters, and the densities of the two wires are in the ratios $8: 1,36: 35,4: 1$ and $1: 2$ respectively. If the note of the higher pitch has a frequency $360\,Hz$, find the number of beats produced per seconds.

Answer 1

We know that,
$v =\frac{1}{L D} \sqrt{\frac{T}{\rho}} $
$\therefore \frac{v_{1}}{v_{2}} =\frac{L_{2} D_{2}}{L_{1} D_{1}} \sqrt{\frac{T_{1}}{\rho_{1}} \times \frac{\rho_{2}}{T_{2}}} $
$=\left(\frac{L_{2}}{L_{1}}\right)\left(\frac{D_{2}}{D_{1}}\right) \sqrt{\frac{T_{1}}{T_{2}} \times \frac{\rho_{2}}{\rho_{1}}} $
$\frac{v_{1}}{360} =\left(\frac{36}{35}\right)\left(\frac{1}{4}\right) \sqrt{\left(\frac{8}{1}\right) \times\left(\frac{2}{1}\right)}$
$v_{1} =350\, Hz$
So, number of beats produced per second
$=v_{2}-v_{1}=360-350=10\, Hz$