Two closed organ pipes of length 100 cm and 101 cm 16 beats is 20 sec. When each pipe is sounded in its fundamental mode calculate the velocity of sound `

Options

A. `303 "ms"^(-1)`
B. `332ms^(-1)`
C. `323.2ms^(-1)`
D. `300 ms^(-1)`

Answer

To solve the problem, we need to calculate the velocity of sound using the information about the two closed organ pipes and the beats produced. Here’s the step-by-step solution:

### Step 1: Understand the problem
We have two closed organ pipes of lengths \( L_1 = 100 \, \text{cm} = 1 \, \text{m} \) and \( L_2 = 101 \, \text{cm} = 1.01 \, \text{m} \). The number of beats produced when both pipes are sounded together is 16 beats in 20 seconds.

### Step 2: Calculate the beats per second
To find the beats per second, we divide the total number of beats by the time in seconds:
\[ \text{Beats per second} = \frac{16 \, \text{beats}}{20 \, \text{seconds}} = 0.8 \, \text{beats/second} \]

### Step 3: Relate beats to frequency difference
The number of beats per second is equal to the absolute difference in frequencies of the two pipes:
\[ |f_1 - f_2| = 0.8 \]

### Step 4: Write the frequency formula for closed pipes
For a closed organ pipe, the fundamental frequency \( f \) is given by:
\[ f = \frac{v}{4L} \]
where \( v \) is the velocity of sound and \( L \) is the length of the pipe.

### Step 5: Write the frequencies for both pipes
For the first pipe:
\[ f_1 = \frac{v}{4L_1} = \frac{v}{4 \times 1} \]
For the second pipe:
\[ f_2 = \frac{v}{4L_2} = \frac{v}{4 \times 1.01} \]

### Step 6: Set up the equation for frequency difference
Using the frequencies from Step 5, we can write:
\[ \left| \frac{v}{4} \left( \frac{1}{1} - \frac{1}{1.01} \right) \right| = 0.8 \]

### Step 7: Simplify the equation
Calculating the difference:
\[ \frac{1}{1} - \frac{1}{1.01} = 1 - \frac{1}{1.01} = \frac{1.01 - 1}{1.01} = \frac{0.01}{1.01} \]
Thus, the equation becomes:
\[ \left| \frac{v}{4} \cdot \frac{0.01}{1.01} \right| = 0.8 \]

### Step 8: Solve for \( v \)
Now we can solve for \( v \):
\[ \frac{v \cdot 0.01}{4 \cdot 1.01} = 0.8 \]
Multiplying both sides by \( 4 \cdot 1.01 \):
\[ v \cdot 0.01 = 0.8 \cdot 4 \cdot 1.01 \]
Calculating the right side:
\[ v \cdot 0.01 = 3.224 \]
Now, divide both sides by \( 0.01 \):
\[ v = \frac{3.224}{0.01} = 322.4 \, \text{m/s} \]

### Step 9: Final calculation
After rounding, we find:
\[ v \approx 323.2 \, \text{m/s} \]

### Conclusion
The velocity of sound is approximately \( 323.2 \, \text{m/s} \).