After reading the chapter or superposition and standing waves, you decide to design a special wind chime. It consists of three light wires, of linear mass density $,$ and three heavy objects, each of mass $M=2\mathrm{.}05kg,$ as shown in the figure.

The entire system is suspended from the ceiling. When the wind blows, it sets all three wires into vibration. You want the system to play a C major chord in the just scale, with the upper wire of length $L_1=26\mathrm{.}0cm$ having a frequency of middle C in the equal$-$tempered scale, $f_1=261\mathrm{.}6Hz.$ In the just scale, musical intervals between two notes are defined by frequencies of the notes whose ratios are those of small whole numbers. In a C major chord, the first note in the chord is C and the second note is E $.$ In the just scale, the ratio of the frequencies in the C$-$E interval is $\frac{f_E}{f_C}=\frac{5}{4}.$ The third note in the C major chord is G $,$ so the chord also contains a C$-$G interval. In the just scale, the ratio of the frequencies in the $C-G$ interval is $\frac{f_G}{f_C}=\frac{3}{2}.$ Before embarking on building the system, you need to determine what the lengths of the other two wires should be $($in cm $).$

$L_2$

Relate the frequencies of the strings to their lengths and the weights supported by the strings. cm

$L_3$

Relate the frequencies of the strings to their lengths and the weights supported by the strings. cm

$[0\mathrm{.}22/0\mathrm{.}66$ Points$]$

SERPSE$1017.$A$.$OP$.035.$