The fundamental frequency of a sonometre wire is $n.$ If its radius is doubled and its tension becomes half, the material of the wire remains same, the new fundamental frequency will be
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(d) $n = \frac{1}{{2l}}\sqrt {\frac{T}{{\pi {r^2}\rho }}} $
==> $n \propto \frac{{\sqrt T }}{r}$
==> $\frac{{{n_2}}}{{{n_1}}} = \frac{{{r_1}}}{{{r_2}}}\sqrt {\frac{{{T_2}}}{{{T_1}}}} $
$ = \frac{1}{2} \times \sqrt {\frac{1}{2}} $=$\frac{1}{{2\sqrt 2 }}$
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