If the tension and diameter of a sonometer wire of fundamental frequency $n$ are doubled and density is halved then its fundamental frequency will become
AIPMT 2001, Medium
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(c) $n = \frac{1}{{2l}}\sqrt {\frac{T}{{\pi {r^2}\rho }}} \propto \sqrt {\frac{T}{{{r^2}\rho }}} $
==>$\frac{{{n_1}}}{{{n_2}}} = \sqrt {\left( {\frac{{{T_1}}}{{{T_2}}}} \right)\,{{\left( {\frac{{{r_2}}}{{{r_1}}}} \right)}^2}\left( {\frac{{{\rho _2}}}{{{\rho _1}}}} \right)} = \sqrt {\left( {\frac{1}{2}} \right)\,{{\left( {\frac{2}{1}} \right)}^2}\left( {\frac{1}{2}} \right)} = 1$
$ \therefore$ ${n_1} = {n_2}$
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