Question

The length of the sonometer wire between two fixed ends is $100\text{cm}$. Three bridges are placed so as to divide the wire into four segments whose fundamental frequencies are in the ratio of $1:2:3:4$ from the left end. Then which of the following is/are correct.

• A. The position of the second bridge from the right fixed end is $28\text{cm}$
• B. The position of the first bridge from the left fixed end is $48\text{cm}$
• C. The separation between the first bridge and the third bridge is $40\text{cm}$
• D. The position of the third bridge from the left fixed end is $88\text{cm}$

Answer 1

Answer: (A), (B), (C), (D)

The position of the third bridge from the left fixed end is $88\text{cm}$

Let $n_1,n_2,n_3\text{and}{n}_4$ are fundamental frequencies of these segments respectively.


The fundamental frequency of the sonometer is given by :

$n=\frac{1}{2l}\sqrt{\frac{T}{\mu }}$

Or,

$n\propto \frac{1}{l}$

Given that

$n_1:n_2:n_3:n_4=1:2:3:4$

$\therefore \frac{1}{{\ell }_1}:\frac{1}{{\ell }_2}:\frac{1}{{\ell }_3}:\frac{1}{{\ell }_4}=1:2:3:4$

$\text{Or,}{\ell }_1:{\ell }_2:{\ell }_3:{\ell }_4=12:6:4:3$

Therefore,
${\ell }_1=\frac{12}{12+6+4+3}\times 100=48\text{cm}{\ell }_2=\frac{6}{12+6+4+3}\times 100=24\text{cm}{\ell }_3=\frac{4}{12+6+4+3}\times 100=16\text{cm}{\ell }_4=\frac{3}{12+6+4+3}\times 100=12\text{cm}$