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 $360Hz$, find the number of beats produced per seconds.

Answer 1

Answer: 10.00

We know that,
$v=\frac{1}{LD}\sqrt{\frac{T}{\rho }}$
$\therefore \frac{v_1}{v_2}=\frac{L_2{D}_2}{L_1{D}_1}\sqrt{\frac{T_1}{{\rho }_1}\times \frac{{\rho }_2}{T_2}}$
$=\left(\frac{L_2}{L_1}\right)\left(\frac{D_2}{D_1}\right)\sqrt{\frac{T_1}{T_2}\times \frac{{\rho }_2}{{\rho }_1}}$
$\frac{v_1}{360}=\left(\frac{36}{35}\right)\left(\frac{1}{4}\right)\sqrt{\left(\frac{8}{1}\right)\times \left(\frac{2}{1}\right)}$
$v_1=350Hz$
So, number of beats produced per second
$=v_2-v_1=360-350=10Hz$