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**Hint:**Sonometer is basically used to study the relationship between frequency, tension, and linear mass density and length of a stretched string. A Sonometer is a device based on the principle of Resonance. It is used to verify the laws of vibration of stretched string and also to determine the frequency of a tuning fork.

**Complete answer:**

Given,

The length of sonometer wire, \[AB = 100cm\]

Ratio of segments \[ = 1:2:6\]

Let ${L_1},{L_2}\,and\,{L_3}$be the lengths of three segments.

Then

$ {L_1} + {L_2} + {L_3} = 100$ ……………...(1)

Also the laws of vibrations of stretched strings

\[{f_1}{L_1} = {f_2}{L_2} = {f_3}{L_3}\]

Given

$ \Rightarrow {f_1}:{f_2}:{f_3} = 1:2:6$

Therefore

$ = {L_1}:2{L_2}:6{L_3}$

So

$ {L_2} = \dfrac{{{L_1}}}{2}$

$ {L_3} = \dfrac{{{L_1}}}{6}$

Put the value in equation (1) and we get

$ \Rightarrow {L_1} + \dfrac{{{L_1}}}{2} + \dfrac{{{L_1}}}{6} = 100$

Simplify

$ \Rightarrow \dfrac{{6{L_1} + 3{L_1} + {L_1}}}{6} = 100$

\[ \Rightarrow 10{L_1} = 100 \times 6\]

\[ \Rightarrow {L_1} = \dfrac{{600}}{{10}}\]

\[ \Rightarrow {L_1} = 60cm\]

Now

$ \Rightarrow {L_2} = \dfrac{{{L_1}}}{2}$

$ \Rightarrow {L_2} = \dfrac{{60}}{2}$

$ \Rightarrow {L_2} = 30cm$

Again

\[ \Rightarrow {L_3} = \dfrac{{{L_1}}}{6}\]

\[ \Rightarrow {L_3} = \dfrac{{60}}{6}\]

\[\Rightarrow {L_3} = 10cm\]

Now

The first bridge, \[{L_1} = 60cm\]

The second bridge, \[{L_1} + {L_2}\]

\[ = 60 + 30\]

\[ = 90cm\]

The second bridge is, \[{L_1} + {L_2} = 90cm\]

So the answer is (2) $60cm,\,90cm$.

**Additional Information:**

The monochord was used as a musical teaching tool in the 11th century by Guido of Arezzo (c. 990-1050), the musician who invented the first useful form of musical notation.

**Note:**This is known as resonance - when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion. The word resonance comes from Latin and means to "respond" - to sound out together with a loud sound. Resonance describes the phenomenon of increased amplitude that occurs when the frequency of a periodically applied force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts.

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