A sonometer wire vibrating in first overtone has length of `31.25cm`. Tensio in the wire is `256N` and mass per unit length is `10gm//m`. When it is sounded together with a tuning fork `10` beats are heard. If tension in the wire is slightly increased no beats are heard. What is the possible frequency of the tuning fork ?

Options

A. `502 Hz`
B. `510 Hz`
C. `505 Hz`
D. None of these

Explanation

To solve the problem, we need to find the possible frequency of the tuning fork based on the information given about the sonometer wire. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Parameters
- Length of the wire (L) = 31.25 cm = 0.3125 m
- Tension in the wire (T) = 256 N
- Mass per unit length (μ) = 10 g/m = 0.01 kg/m (since 10 g = 0.01 kg)

### Step 2: Calculate the Frequency of the Wire
The frequency of the first overtone (n=2) of a vibrating wire is given by the formula:
\[ f = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \]
For the first overtone, \( n = 2 \).

Substituting the values:
\[ f = \frac{2}{2 \times 0.3125} \sqrt{\frac{256}{0.01}} \]

### Step 3: Simplify the Expression
1. Calculate \( \frac{256}{0.01} = 25600 \).
2. Now, take the square root:
\[ \sqrt{25600} = 160 \]
3. Substitute back into the frequency formula:
\[ f = \frac{2}{0.625} \times 160 \]
\[ f = 3.2 \times 160 = 512 \text{ Hz} \]

### Step 4: Determine the Frequency Range
Given that 10 beats are heard when the tuning fork is sounded with the wire, the frequency of the tuning fork (f_t) can be either higher or lower than the frequency of the wire. The beat frequency (number of beats heard) is the absolute difference between the two frequencies:
\[ |f_t - f| = 10 \]

So, we have two possible equations:
1. \( f_t = f + 10 \)
2. \( f_t = f - 10 \)

Substituting \( f = 512 \text{ Hz} \):
1. \( f_t = 512 + 10 = 522 \text{ Hz} \)
2. \( f_t = 512 - 10 = 502 \text{ Hz} \)

### Step 5: Conclusion
The possible frequencies of the tuning fork are:
- 522 Hz
- 502 Hz

Since the problem states that when the tension in the wire is slightly increased, no beats are heard, it suggests that the tuning fork's frequency matches the new frequency of the wire. Therefore, the possible frequency of the tuning fork is:
**502 Hz** (as it is one of the options provided).