A sonometer wire resonates with a given tuning fork forming standing waves with five antitodes between the two bridges when a mass of `9kg` is suspended from the wire. when this same tuning fork forming three antitodes for the same positions of the bridges. the value of `M` is

Options

A. 12 kg
B. 5 kg
C. 12 . 5 kg
D. 25 kg

Answer

To solve the problem, we need to analyze the standing waves formed on a sonometer wire under different conditions. Let's break it down step by step.

### Step 1: Understand the Standing Waves
A sonometer wire can resonate and form standing waves characterized by nodes and antinodes. The number of antinodes (p) indicates how many loops are formed in the wire.

### Step 2: Identify the Given Information
1. When a mass of 9 kg is suspended, the wire forms 5 antinodes.
2. When a different mass (M) is suspended, the wire forms 3 antinodes.

### Step 3: Write the Formula for Frequency
The frequency of the standing wave on the sonometer wire is given by the formula:
\[ \nu = \frac{p}{2L} \sqrt{\frac{T}{\mu}} \]
Where:
- \( \nu \) = frequency
- \( p \) = number of antinodes
- \( L \) = length of the wire
- \( T \) = tension in the wire
- \( \mu \) = linear mass density of the wire

### Step 4: Calculate Frequency for Each Case
1. **For the first case (5 antinodes and mass = 9 kg)**:
\[ \nu_1 = \frac{5}{2L} \sqrt{\frac{9g}{\mu}} \]
Here, \( T = mg = 9g \).

2. **For the second case (3 antinodes and mass = M kg)**:
\[ \nu_2 = \frac{3}{2L} \sqrt{\frac{Mg}{\mu}} \]
Here, \( T = Mg \).

### Step 5: Set the Frequencies Equal
Since the same tuning fork is used, the frequencies must be equal:
\[ \nu_1 = \nu_2 \]
Thus, we can write:
\[ \frac{5}{2L} \sqrt{\frac{9g}{\mu}} = \frac{3}{2L} \sqrt{\frac{Mg}{\mu}} \]

### Step 6: Simplify the Equation
We can cancel \( \frac{1}{2L} \) and \( \sqrt{\frac{g}{\mu}} \) from both sides:
\[ 5 \sqrt{9} = 3 \sqrt{M} \]
This simplifies to:
\[ 5 \cdot 3 = 3 \sqrt{M} \]
\[ 15 = 3 \sqrt{M} \]

### Step 7: Solve for M
Divide both sides by 3:
\[ 5 = \sqrt{M} \]
Now square both sides:
\[ M = 25 \text{ kg} \]

### Step 8: Conclusion
The value of \( M \) is 25 kg.

### Final Answer
Thus, the required mass \( M \) is **25 kg**.

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