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 ... $kg$
IIT 2002, Diffcult
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$(1)$ When only $S_2$ is emitting sound and it is $Q$, the frequency of sound measured by the detector in $Hz$ is $. . . . .$
$(2)$ Consider both sources emitting sound. When $S_2$ is at $R$ and $S_1$ approaches the detector with a speed $4 \ ms ^{-1}$, the beat frequency measured by the detector is $Hz$.
