Equation of motion in the same direction is given by `y_(1) =A sin (omega t - kx), y_(2) = A sin ( omega t - kx- theta )`. The amplitude of the medium particle will be

Options

A. `2A cos.(theta)/(2)`
B. `2A cos theta`
C. `sqrt2A cos.(theta)/(2)`
D. `1.2f,1.2lambda`

Answer

To find the amplitude of the medium particle given the equations of motion, we can follow these steps:

### Step 1: Identify the given equations
The equations of motion are:
1. \( y_1 = A \sin(\omega t - kx) \)
2. \( y_2 = A \sin(\omega t - kx - \theta) \)

### Step 2: Understand the parameters
Here, \( A \) is the amplitude of each wave, \( \omega \) is the angular frequency, \( k \) is the wave number, and \( \theta \) is the phase difference between the two waves.

### Step 3: Use the formula for the resultant amplitude
The resultant amplitude \( A_{\text{net}} \) when two waves interfere can be calculated using the formula:
\[ A_{\text{net}} = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]
where \( \phi \) is the phase difference between the two waves.

### Step 4: Substitute the values
In this case, both amplitudes \( A_1 \) and \( A_2 \) are equal to \( A \), and the phase difference \( \phi \) is equal to \( \theta \). Thus, we have:
\[ A_{\text{net}} = \sqrt{A^2 + A^2 + 2 A A \cos(\theta)} \]

### Step 5: Simplify the expression
This simplifies to:
\[ A_{\text{net}} = \sqrt{2A^2 + 2A^2 \cos(\theta)} = \sqrt{2A^2(1 + \cos(\theta))} \]

### Step 6: Factor out the common terms
We can factor out \( 2A^2 \):
\[ A_{\text{net}} = A \sqrt{2(1 + \cos(\theta))} \]

### Step 7: Use the trigonometric identity
Using the identity \( 1 + \cos(\theta) = 2 \cos^2(\theta/2) \), we can rewrite the expression:
\[ A_{\text{net}} = A \sqrt{2 \cdot 2 \cos^2(\theta/2)} = A \cdot 2^{1/2} \cdot 2^{1/2} \cos(\theta/2) = 2A \cos(\theta/2) \]

### Final Result
Thus, the amplitude of the medium particle is:
\[ A_{\text{net}} = 2A \cos(\theta/2) \]