Two open organ pipes given 4 beats/sec, when sounded together in their fundamental notes. IF the length of the pipes are 100 cm and 102.5 cm respectively then the velocity of sound is

Options

A. 160 m/s
B. 240 m/s
C. 328 m/s
D. 496 m/s

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the problem
We have two open organ pipes with lengths:
- Pipe 1 (L1) = 100 cm
- Pipe 2 (L2) = 102.5 cm

When these pipes are sounded together, they produce 4 beats per second. We need to find the velocity of sound (V).

### Step 2: Use the formula for the fundamental frequency of an open pipe
The fundamental frequency (n0) of an open pipe is given by the formula:
\[ n_0 = \frac{V}{2L} \]
where:
- \( n_0 \) is the fundamental frequency,
- \( V \) is the velocity of sound,
- \( L \) is the length of the pipe.

### Step 3: Calculate the frequencies of both pipes
For Pipe 1 (length L1 = 100 cm):
\[ n_1 = \frac{V}{2L_1} = \frac{V}{2 \times 100} = \frac{V}{200} \]

For Pipe 2 (length L2 = 102.5 cm):
\[ n_2 = \frac{V}{2L_2} = \frac{V}{2 \times 102.5} = \frac{V}{205} \]

### Step 4: Set up the equation for beats
The number of beats per second is given by the difference in frequencies:
\[ |n_2 - n_1| = 4 \]
Substituting the expressions for \( n_1 \) and \( n_2 \):
\[ \left| \frac{V}{205} - \frac{V}{200} \right| = 4 \]

### Step 5: Simplify the equation
To simplify, we can find a common denominator:
\[ \left| \frac{200V - 205V}{205 \times 200} \right| = 4 \]
This simplifies to:
\[ \left| \frac{-5V}{205 \times 200} \right| = 4 \]
Removing the absolute value (since we are interested in the magnitude):
\[ \frac{5V}{205 \times 200} = 4 \]

### Step 6: Solve for V
Now, we can solve for V:
\[ 5V = 4 \times 205 \times 200 \]
\[ V = \frac{4 \times 205 \times 200}{5} \]

Calculating the right side:
\[ V = \frac{1640000}{5} = 328000 \text{ cm/s} \]

### Step 7: Convert to meters per second
To convert from cm/s to m/s, divide by 100:
\[ V = \frac{328000}{100} = 3280 \text{ m/s} \]

### Final Answer
The velocity of sound is:
\[ V = 3280 \text{ m/s} \]

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