The fundamental frequency of sonometer wire increases by 9 Hz, if its tension is increased by 69 % , keeping the length constant. The frequency of the wire is

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TestHub · Accepted Answer

We know that, frequency of vibration of a stretched string is given by
$v = \frac{1}{2 l} \sqrt{\frac{T}{m}}$
where, $T$ = tension in string
$m$ = mass /length
and $l$ = length of string.
New frequency,
$v^{'} = v + 9 = \frac{1}{2 l} \sqrt{\frac{T + (69 / 100) T}{m}}$
$\Rightarrow \frac{v}{v^{'}} = \sqrt{\frac{\frac{T}{169}}{100} T} \Rightarrow \frac{v}{v + 9} = \sqrt{\frac{100}{169}} = \frac{10}{13}$
$\Rightarrow 13 v = 10 v + 90$
$v = 30 H z$

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