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 find the velocity of sound using the information provided about two open pipes, we can follow these steps:

### Step-by-step Solution:

1. **Identify the Lengths of the Pipes**:
- Let the length of the first pipe (L1) = 25 cm = 0.25 m
- Let the length of the second pipe (L2) = 25.5 cm = 0.255 m

2. **Determine the Fundamental Frequency for Each Pipe**:
- For an open pipe, the fundamental frequency (f) is given by the formula:
\[ f = \frac{v}{\lambda} \]
- The wavelength (λ) for the fundamental mode in an open pipe is:
\[ \lambda = 2L \]
- Therefore, for the first pipe:
\[ \lambda_1 = 2 \times L_1 = 2 \times 0.25 = 0.5 \text{ m} \]
- For the second pipe:
\[ \lambda_2 = 2 \times L_2 = 2 \times 0.255 = 0.51 \text{ m} \]

3. **Express the Frequencies in Terms of Velocity**:
- For the first pipe:
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{0.5} \]
- For the second pipe:
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{0.51} \]

4. **Calculate the Beat Frequency**:
- The beat frequency (B) is given as 0.1 beats/second:
\[ B = |f_1 - f_2| = 0.1 \]
- Substituting the expressions for f1 and f2:
\[ \left| \frac{v}{0.5} - \frac{v}{0.51} \right| = 0.1 \]

5. **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:
\[ \left| \frac{v(0.01)}{0.255} \right| = 0.1 \]

6. **Solve for Velocity (v)**:
- Rearranging gives:
\[ v(0.01) = 0.1 \cdot 0.255 \]
- Therefore:
\[ v = \frac{0.1 \cdot 0.255}{0.01} = 2.55 \text{ m/s} = 255 \text{ cm/s} \]

### Final Answer:
The velocity of sound is **255 cm/s**.

---