Question

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9\ \text{kg}$ 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

• A. $5\ \text{kg}$
• B. $12.5\ \text{kg}$
• C. $1.25\ \text{kg}$
• D. $25\ \text{kg}$

Hint:

From the relationship $p \sqrt{m} = \text{constant}$, squaring both sides tells us that the suspended mass is inversely proportional to the square of the number of loops ($m \propto \frac{1}{p^2}$). If the number of loops decreases, the tension mass must increase significantly! This relationship allows you to quickly set up the ratio $\frac{M}{9} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \implies M = 25\ \text{kg}$ entirely in your head.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A sonometer wire is held between fixed bridges under tension supplied by a suspended mass. The wire forms standing wave patterns driven by a fixed-frequency tuning fork. We are given the number of antinodes formed under two different mass weights and need to solve for the unknown mass $M$.

Step 2: Key Formula or Approach:
1. The frequency ($f$) of a stretched string vibrating with $p$ loops (or $p$ antinodes) is given by: $$f = \frac{p}{2L} \sqrt{\frac{T}{\mu}}$$ Where $L$ is the bridge length, $\mu$ is the linear density, and tension $T = mg$. 2. Since the tuning fork frequency ($f$), bridge length ($L$), and wire density ($\mu$) are kept perfectly constant across both experiments, we establish the constant relationship: $$p \sqrt{T} = \text{constant} \implies p_1 \sqrt{m_1} = p_2 \sqrt{m_2}$$

Step 3: Detailed Explanation:
Let's list our variables from the problem data: Initial number of antinodes, $p_1 = 5$ Initial mass, $m_1 = 9\ \text{kg}$ Secondary number of antinodes, $p_2 = 3$ Secondary mass, $m_2 = M$ Substitute these values into our constant frequency relation: $$5 \times \sqrt{9} = 3 \times \sqrt{M}$$ Since $\sqrt{9} = 3$: $$5 \times 3 = 3 \times \sqrt{M}$$ Divide both sides of the equation by 3: $$5 = \sqrt{M}$$ To solve for $M$, square both sides of the expression: $$M = 5^2 = 25\ \text{kg}$$

Step 4: Final Answer:
The required value of the mass $M$ is $25\ \text{kg}$, which matches option (D).