The minimum intensity of audibility of sound is `10^(-12) W//m^(2) s` and density of air ` = 1.293 kg//m^(3)`. If the frequency of sound in `1000 Hz` , then the corresponding amplitude the vibration of the air particles is
[ Take velocity of sound `= 332 m//s`]

Options

A. `1.1 xx10^(-7)`m
B. `1.1 xx10^(-9)`m
C. `1.1 xx10^(-11)` m
D. `1.1 xx10^(-8)`m

Answer

To find the amplitude of the vibration of air particles given the intensity of sound, density of air, frequency of sound, and velocity of sound, we can use the formula for intensity:

\[ I = 2 \pi^2 A^2 f^2 \rho V \]

Where:
- \( I \) = intensity of sound
- \( A \) = amplitude of vibration
- \( f \) = frequency of sound
- \( \rho \) = density of air
- \( V \) = velocity of sound

### Step 1: Rearranging the Formula
We need to rearrange the formula to solve for amplitude \( A \):

\[ A = \sqrt{\frac{I}{2 \pi^2 f^2 \rho V}} \]

### Step 2: Substituting the Given Values
Now, we can substitute the given values into the formula:

- \( I = 10^{-12} \, \text{W/m}^2 \)
- \( \rho = 1.293 \, \text{kg/m}^3 \)
- \( f = 1000 \, \text{Hz} \)
- \( V = 332 \, \text{m/s} \)

Substituting these values into the equation for \( A \):

\[ A = \sqrt{\frac{10^{-12}}{2 \pi^2 (1000)^2 (1.293)(332)}} \]

### Step 3: Calculating the Denominator
Now we will calculate the denominator step-by-step:

1. Calculate \( 2 \pi^2 \):
\[ 2 \pi^2 \approx 19.7392 \]

2. Calculate \( (1000)^2 \):
\[ (1000)^2 = 1000000 \]

3. Calculate \( 1.293 \times 332 \):
\[ 1.293 \times 332 \approx 429.756 \]

4. Now, multiply all the parts together:
\[ 2 \pi^2 (1000)^2 (1.293)(332) \approx 19.7392 \times 1000000 \times 429.756 \approx 8.487 \times 10^9 \]

### Step 4: Final Calculation of Amplitude
Now substitute this back into the equation for \( A \):

\[ A = \sqrt{\frac{10^{-12}}{8.487 \times 10^9}} = \sqrt{1.177 \times 10^{-21}} \approx 1.086 \times 10^{-11} \, \text{m} \]

### Step 5: Conclusion
Thus, the amplitude of the vibration of the air particles is approximately:

\[ A \approx 1.1 \times 10^{-11} \, \text{m} \]