Two wires are fixed on a sonometer wire. Their tensions are in the ratio 8 : 1, the lengths are in the ratio 36 : 35, the diameters in the ratio 4 : 1 and the densities are in the ratio 1 : 2. If the note of the higher pitch has frequency of 360 Hz, then the frequency of the beats produced is 5 (x) Hz. Then x to the nearest integer?

For the two wires, we have

$n_1=\frac{1}{2l_1}\sqrt{\left(\frac{T_2}{\pi r_2^2{\rho }_2}\right)}\sqrt{\left(\frac{T_1}{\pi r_1^2{\rho }_1}\right)}$

         And        $n_2=\frac{1}{2l_1}\sqrt{\left(\frac{T_2}{\pi r_2^2{\rho }_2}\right)}$
    Or    $=\frac{l_2}{l_1}\sqrt{\left(\frac{T_1}{T_2}\left(\frac{r_2}{r_1}\right)\frac{{\rho }_2}{{\rho }_1}\right)}$                         …..(6.242)
    According to the given problem
                                                 $\frac{l_2}{l_1}=\frac{36}{35};\frac{T_1}{T_2}=\frac{8}{1};\frac{r_1}{r_2}=\frac{4}{1}and\frac{{\rho }_1}{{\rho }_2}=\frac{1}{2}$
   Substituting these values in equation-(6.242)
   Or          $\frac{n_1}{n_2}=\frac{35}{36}\sqrt{[\left(\frac{8}{1}\right)\left(\frac{1}{16}\right).\left(\frac{2}{1}\right)]}=\frac{35}{36}$                            …… (6.243)

It is c1ear from equation-(6.243) that $n_2$ has higher pitch so
                                       $n_2=360$
   Now                           $\frac{n_1}{360}=\frac{35}{36}$
   Or                                 $n_1=350Hz$
Thus beat frequency is given as
                                              $\Delta n=360-350=10Hz$