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

Let L be the length of the wire between the bridges and let m be the mass per unit length of the wire.Five antinodes on a length L implies that L=52λ1 or λ1=2L5. Thus in this case we havev1=V1λ1=52LT1m, where T1=M1gThree antinodes on a length L implies that L=32λ2 or λ2=2L3. In this case, we have v2=V2λ2=32LT2m, where T2=M2gGiven v1=v2,M1=9kg and M2=M. Therefore, we have 52L9gm=32LMgmwhich gives M = 25 kg.