Two open pipes of length 25 cm and 25.5 cm produced 0.1 beat/second. The velocity of sound will be :-

Options

A. 255 cm/s
B. 250 cm/s
C. 350 cm/s
D. None of these

Explanation

To solve the problem, we need to find the velocity of sound using the information given about two open pipes and the beats produced.

### Step-by-Step Solution:

1. **Identify the lengths of the pipes**:
- Let \( L_1 = 25 \, \text{cm} = 0.25 \, \text{m} \)
- Let \( L_2 = 25.5 \, \text{cm} = 0.255 \, \text{m} \)

2. **Determine the fundamental frequencies of the pipes**:
- The frequency of a pipe is given by the formula:
\[ f = \frac{v}{2L} \]
- For pipe 1:
\[ f_1 = \frac{v}{2L_1} = \frac{v}{2 \times 0.25} = \frac{v}{0.5} \]
- For pipe 2:
\[ f_2 = \frac{v}{2L_2} = \frac{v}{2 \times 0.255} = \frac{v}{0.51} \]

3. **Use the beat frequency formula**:
- The beat frequency is given as \( 0.1 \, \text{beats/second} \):
\[ |f_1 - f_2| = 0.1 \]
- Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \left| \frac{v}{0.5} - \frac{v}{0.51} \right| = 0.1 \]

4. **Simplify the equation**:
- Finding a common denominator:
\[ \left| \frac{v \cdot 0.51 - v \cdot 0.5}{0.5 \cdot 0.51} \right| = 0.1 \]
- This simplifies to:
\[ \left| \frac{v(0.51 - 0.5)}{0.5 \cdot 0.51} \right| = 0.1 \]
- Which further simplifies to:
\[ \left| \frac{v(0.01)}{0.255} \right| = 0.1 \]

5. **Solve for \( v \)**:
- Rearranging gives:
\[ v(0.01) = 0.1 \cdot 0.255 \]
- Therefore:
\[ v = \frac{0.1 \cdot 0.255}{0.01} = 2.55 \, \text{m/s} \]
- Converting to meters per second:
\[ v = 255 \, \text{m/s} \]

### Final Answer:
The velocity of sound is \( 255 \, \text{m/s} \).

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