A sonometer wire resonates with a given tuning fork forming standing wave with 5 antinodes between two…

A sonometer wire resonates with a given tuning fork forming standing wave with 5 antinodes between two bridges when mass of $9 \mathrm{~kg}$ is suspended from the wire. When mass 'm' is suspended from the wire, with same fork and same length between two bridges 3 antinodes are formed. Mass $\mathrm{M}$ is
  1. $25 \mathrm{Kg}$
  2. $20 \mathrm{Kg}$
  3. $15 \mathrm{Kg}$
  4. $10 \mathrm{Kg}$

Solution

$\mathrm{n}=\frac{\mathrm{P}_{1}}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mathrm{m}_{1}}}=\frac{\mathrm{P}_{2}}{2 \ell} \sqrt{\frac{\mathrm{I}}{\mathrm{m}_{2}}}$ $\therefore \frac{\mathrm{P}_{1}}{\mathrm{P}_{2}}=\sqrt{\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}}}=\sqrt{\frac{\mathrm{m}}{9}}=\frac{\sqrt{\mathrm{m}}}{3}$ $\therefore \frac{5}{3}=\frac{\sqrt{\mathrm{m}}}{3} \quad \therefore \mathrm{m}=25 \mathrm{~kg}$ $\therefore \sqrt{\mathrm{m}}=5 \quad$

Asked in: MHT CET 2020 (14 Oct Shift 1)

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