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11. ⇒ (MHT CET 2021 20th September Morning Shift )

A sonometer wire resonates with 4 antinodes between 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 two bridges is as before). The value of M is

A. 2.5 kg

B. 3.5 kg

C. 4 kg

D. 1 kg

Correct Option is (C)

If $p$ is the number of loops (or antinodes) then we have,

${Tp}^{2} =$ constant where $T$ is the tension

$\begin{aligned} ∴ T_{1} P_{1}^{2} = T_{2}^{2} P_{2}^{2} \\ ∴ \frac{T_{2}}{T_{1}} = \frac{P_{1}^{2}}{P_{2}^{2}} = {(\frac{4}{2})}^{2} = 4 ∴ T_{2} = 4 T_{1} = 4 \times 1 = 4 kg - wt \end{aligned}$

12. ⇒ (MHT CET 2021 20th September Morning Shift )

Two wires of same material of radius 'r' and '2r' respectively are welded together end to end. The combination is then used as a sonometer wire under tension 'T'. The joint is kept midway between the two bridges. The ratio of the number of loops formed in the wires such that the joint is a node is

A. 1 : 5

B. 1 : 2

C. 1 : 4

D. 1 : 3

Correct Option is (B)

Frequency of vibration n will be same for both the segments. If p $_{1}$ and p $_{2}$ are the number of loops for the wires of radius r and 2r then we have

$n = \frac{p_{1}}{2 l r} \sqrt{\frac{T}{π p}} = \frac{p_{2}}{4 l r} \sqrt{\frac{T}{π p}}$

$∴$ $\frac{p_{1}}{p_{2}} = \frac{1}{2}$

⇒Superposition of Waves