A closed air column `32 cm` long is in resonance with a tuning fork . Another open air column of length `66 cm` is in resonance with another tuning fork . If the two forks produce `8 beats//s` when sounded together , find
the speed of sound in the air

Options

A. `33792 cm//s`
B. `35790 cm//s`
C. `31890 cm//s`
D. `40980 cm//s`

Explanation

To find the speed of sound in air based on the given problem, we can follow these steps:

### Step 1: Identify the frequencies of the tuning forks
For a closed air column, the frequency \( f_1 \) can be calculated using the formula:
\[ f_1 = \frac{c}{4L_1} \]
where \( L_1 = 32 \, \text{cm} = 0.32 \, \text{m} \).

For an open air column, the frequency \( f_2 \) is given by:
\[ f_2 = \frac{c}{2L_2} \]
where \( L_2 = 66 \, \text{cm} = 0.66 \, \text{m} \).

### Step 2: Write the expressions for frequencies
Substituting the lengths into the formulas, we have:
\[ f_1 = \frac{c}{4 \times 0.32} = \frac{c}{1.28} \]
\[ f_2 = \frac{c}{2 \times 0.66} = \frac{c}{1.32} \]

### Step 3: Set up the equation for beats
The number of beats produced when the two forks are sounded together is given as 8 beats per second. The difference in frequencies is:
\[ |f_1 - f_2| = 8 \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \left|\frac{c}{1.28} - \frac{c}{1.32}\right| = 8 \]

### Step 4: Simplify the equation
To simplify the left side:
\[ \frac{c}{1.28} - \frac{c}{1.32} = c \left(\frac{1}{1.28} - \frac{1}{1.32}\right) \]
Finding a common denominator (which is \( 1.28 \times 1.32 \)):
\[ \frac{1.32 - 1.28}{1.28 \times 1.32} = \frac{0.04}{1.28 \times 1.32} \]
Thus, we have:
\[ c \cdot \frac{0.04}{1.28 \times 1.32} = 8 \]

### Step 5: Solve for \( c \)
Rearranging gives:
\[ c = 8 \cdot \frac{1.28 \times 1.32}{0.04} \]
Calculating \( 1.28 \times 1.32 \):
\[ 1.28 \times 1.32 = 1.6896 \]
Now substituting back:
\[ c = 8 \cdot \frac{1.6896}{0.04} = 8 \cdot 42.24 = 337.92 \, \text{m/s} \]

### Step 6: Final result
Thus, the speed of sound in air is:
\[ c \approx 337.92 \, \text{m/s} \]