The length of a sonometer wire ' AB ' is 110 cm . Where should the two bridges be placed from end ' A ' to divide the wire in three segments whose fundamental frequencies are in the ratio \(1: 2: 3\) ?

\(\begin{aligned}
& \mathrm{L}_1+\mathrm{L}_2+\mathrm{L}_3=110 \mathrm{~cm} \text { and } \\
& \mathrm{n}_1 \mathrm{~L}_1=\mathrm{n}_2 \mathrm{~L}_2=\mathrm{n}_3 \mathrm{~L}_3 \\
& \text { Required condition, } \mathrm{n}_1: \mathrm{n}_2: \mathrm{n}_3:: 1: 2: 3
\end{aligned}\)
\(\begin{array}{ll}
\therefore \quad & \frac{n_1}{n_2}=\frac{1}{2}=\frac{L_2}{L_1} \Rightarrow L_2=\frac{L_1}{2} \text { and } \\
& \frac{n_1}{n_3}=\frac{1}{3}=\frac{L_3}{L_1} \Rightarrow L_3=\frac{L_1}{3} \\
\therefore \quad & L_1+\frac{L_1}{2}+\frac{L_1}{3}=110 \\
\therefore \quad & L_1=60 \mathrm{~cm} ; L_2=30 \mathrm{~cm}, L_3=20 \mathrm{~cm}
\end{array}\)
\(\therefore \quad\) The two bridges should be kept at 60 cm and \(60+30=90 \mathrm{~cm}\).