A sonometer wire resonates with 4 antinodes between the two bridges for a given tuning fork when 1 kg mass is suspended from the wire. Using same fork, when mass M is suspended, the wire resonates pro

Question

A sonometer wire resonates with 4 antinodes between the two bridges for a given tuning fork when 1 kg mass is suspended from the wire. Using same fork, when mass M is suspended, the wire resonates producing 2 antinodes between the two bridges. (Distance between the bridges as before) The value of M is

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The frequency of a sonometer wire is given by $f = \frac{p}{2 L} \frac{T}{μ}$ , where $p$ is the number of antinodes, $L$ is the length, $T$ is the tension, and $μ$ is the linear mass density.

Since the frequency $f$ , length $L$ , and linear mass density $μ$ are constant, we have $p T = constant$ .

Therefore, $p_{1} T_{1} = p_{2} T_{2}$ .

Given $p_{1} = 4$ , $T_{1} = 1 \times g$ , $p_{2} = 2$ , and $T_{2} = M \times g$ .

Substituting the values, we get $4 1 \times g = 2 M \times g$ .

Dividing by $2 g$ , we get $2 = M$ .

Squaring both sides, $M = 4$ kg.

Answer: $4$ kg

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