Two sources of sound `A` and `B` produces the wave of `350 Hz`. They vibrate in the same phase. The particel `P` is vibrating under the influence of these two waves, if the amplitudes at the point P produced by the two waves is `0.3mm` and `0.4mm` then the resultant amplitude of the point `P` will be when `AP-BP=25cm` and the velocity of sound is `350m//sec`.

Options

A. `0.7` mm
B. `0.1` mm
C. `0.2` mm
D. `0.5` mm

Answer

To find the resultant amplitude at point P due to the two sound waves from sources A and B, we can follow these steps:

### Step 1: Calculate the wavelength (λ)
Given:
- Velocity of sound (v) = 350 m/s
- Frequency (f) = 350 Hz

Using the formula for wavelength:
\[ \lambda = \frac{v}{f} \]
Substituting the values:
\[ \lambda = \frac{350 \, \text{m/s}}{350 \, \text{Hz}} = 1 \, \text{m} \]

### Step 2: Calculate the phase difference (Δφ)
Given:
- Path difference (AP - BP) = 25 cm = 0.25 m

Using the formula for phase difference:
\[ \Delta \phi = \frac{2\pi}{\lambda} \times \text{path difference} \]
Substituting the values:
\[ \Delta \phi = \frac{2\pi}{1 \, \text{m}} \times 0.25 \, \text{m} = 0.5\pi \, \text{radians} = 90^\circ \]

### Step 3: Find the resultant amplitude (A)
Given amplitudes:
- Amplitude from source A (A1) = 0.3 mm = 0.0003 m
- Amplitude from source B (A2) = 0.4 mm = 0.0004 m

Since the phase difference is 90 degrees, we can use the Pythagorean theorem to find the resultant amplitude:
\[ A = \sqrt{A_1^2 + A_2^2} \]
Substituting the values:
\[ A = \sqrt{(0.0003)^2 + (0.0004)^2} \]
Calculating:
\[ A = \sqrt{(0.00000009) + (0.00000016)} = \sqrt{0.00000025} = 0.0005 \, \text{m} = 0.5 \, \text{mm} \]

### Final Result
The resultant amplitude at point P is **0.5 mm**.

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