A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $$9\,kg$$  is suspended from the wire. When this mass is replaced by a mass $$M,$$ the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $$M$$ is

Solution :
$$\eqalign{ & {f_0} = \frac{5}{{2\,\ell }}\sqrt {\frac{{9\,g}}{\mu }} = \frac{3}{{2\,\ell }}\sqrt {\frac{{Mg}}{\mu }} \cr & \Rightarrow \,\,\,M = 25\,kg \cr} $$
NOTE : Using the formula of a vibrating string,
$$f = \frac{p}{{2\,\ell }}\sqrt {\frac{T}{\mu }} $$    where $$p$$ = number of loops.
In each case, the wire vibrates, in resonance with the same tuning fork. Frequency of wire remains same while $$p$$ and $$T$$ change.
$$\eqalign{ & \therefore \,\,\frac{{{p_1}}}{{2\,\ell }}\sqrt {\frac{{{T_1}}}{\mu }} = \frac{{{p_2}}}{{2\,\ell }}\sqrt {\frac{{{T_2}}}{\mu }} \,\,{\text{or, }}{p_1}\sqrt {{T_1}} = {p_2}\sqrt {{T_2}} \cr & {\text{or, }}\sqrt {\frac{{{T_2}}}{{{T_1}}}} = \frac{{{p_1}}}{{{p_2}}} \cr & \sqrt {\frac{{M \times g}}{{9 \times g}}} = \frac{5}{3}\,{\text{or, }}M = \frac{{5 \times 5 \times 9}}{{3 \times 3}} \cr & {\text{or, }}M = 25\,kg. \cr} $$