A tuning fork of frequency 200 Hz is in unison with a sonometer wire . Tension is the wire of sonometer is increased by 1% without any change in its length . Find the number of beats heard in 9 s.

Options

A. 9
B. 3
C. 6
D. 12

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
- The frequency of the tuning fork (f₀) is 200 Hz.
- The tension in the sonometer wire is increased by 1%.

### Step 2: Determine the relationship between tension and frequency
The frequency of a vibrating wire is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( f \) = frequency,
- \( L \) = length of the wire (constant),
- \( T \) = tension in the wire,
- \( \mu \) = mass per unit length of the wire (constant).

Since the length \( L \) and mass per unit length \( \mu \) remain constant, the frequency is directly proportional to the square root of the tension:
\[ f \propto \sqrt{T} \]

### Step 3: Calculate the new tension
If the tension is increased by 1%, the new tension \( T' \) can be expressed as:
\[ T' = T + 0.01T = 1.01T \]

### Step 4: Calculate the new frequency
Using the proportionality of frequency and tension, we can express the new frequency \( f' \) as:
\[ f' = k \sqrt{T'} = k \sqrt{1.01T} = \sqrt{1.01} \cdot k \sqrt{T} = \sqrt{1.01} \cdot f \]
Substituting \( f = 200 \, \text{Hz} \):
\[ f' = \sqrt{1.01} \cdot 200 \]

### Step 5: Calculate \( \sqrt{1.01} \)
Using a calculator or approximation:
\[ \sqrt{1.01} \approx 1.005 \]
Thus:
\[ f' \approx 1.005 \cdot 200 \approx 201 \, \text{Hz} \]

### Step 6: Calculate the beat frequency
The beat frequency \( f_b \) is the difference between the frequencies of the tuning fork and the sonometer wire:
\[ f_b = |f' - f| = |201 - 200| = 1 \, \text{Hz} \]

### Step 7: Calculate the number of beats in 9 seconds
The number of beats heard in 9 seconds is given by:
\[ \text{Number of beats} = f_b \times \text{time} = 1 \, \text{Hz} \times 9 \, \text{s} = 9 \, \text{beats} \]

### Final Answer
The number of beats heard in 9 seconds is **9 beats**.

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