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Velocity in 2D

A sonometer w...

Question

A sonometer wire has a length of $114$ cms between two fixed ends; Where should two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio $1 : 3 : 4$ ?

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Solution

$l_{1} + l_{2} + l_{3} = 114$

$n_{1} = \frac{1}{2 l_{1}} \sqrt{T / m}; n_{2} = \frac{1}{2 l_{2}} \sqrt{T / m}; n_{3} = \frac{1}{2 l_{3}} \sqrt{T / m}$

$n_{1} : n_{2} : n_{3} = 1 : 3 : 4$

$\frac{n_{2}}{n_{1}} = \frac{l_{1}}{l_{2}} = 3$ ; or $l_{2} = l_{1} / 3$

Similarly, $l_{2} = l_{1} / 4$

$\Rightarrow l_{1} + l_{1} / 3 + l_{1} / 4 = 114$

or $l_{1} = 72$ cm; $l_{2} = 24$ cm; $l_{3} = 18$ cm.

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