The resultant amplitude of a vibrating particle by the superposition of the two waves
`y_(1)=asin[omegat+(pi)/(3)]` and `y_(2)=asinomegat` is

Options

A. `a`
B. `sqrt(2)a`
C. `2a`
D. `sqrt(3)a`

Explanation

To find the resultant amplitude of the two waves given by \( y_1 = a \sin\left(\omega t + \frac{\pi}{3}\right) \) and \( y_2 = a \sin(\omega t) \), we can use the principle of superposition. Here are the steps to solve the problem:

### Step 1: Identify the Amplitudes and Phase Difference
The amplitudes of both waves are the same:
- Amplitude of \( y_1 \) (denoted as \( a_1 \)) = \( a \)
- Amplitude of \( y_2 \) (denoted as \( a_2 \)) = \( a \)

The phase difference \( \phi \) between the two waves can be determined from their equations:
\[ \phi = \left(\omega t + \frac{\pi}{3}\right) - \omega t = \frac{\pi}{3} \]

### Step 2: Use the Formula for Resultant Amplitude
The formula for the resultant amplitude \( A \) when two waves of the same amplitude interfere is given by:
\[ A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2 \cos(\phi)} \]

### Step 3: Substitute the Values
Substituting \( a_1 = a \), \( a_2 = a \), and \( \phi = \frac{\pi}{3} \) into the formula:
\[ A = \sqrt{a^2 + a^2 + 2a \cdot a \cdot \cos\left(\frac{\pi}{3}\right)} \]

### Step 4: Calculate \( \cos\left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 5: Substitute \( \cos\left(\frac{\pi}{3}\right) \) into the Equation
Now substituting \( \cos\left(\frac{\pi}{3}\right) \) into the equation:
\[ A = \sqrt{a^2 + a^2 + 2a^2 \cdot \frac{1}{2}} \]
\[ A = \sqrt{a^2 + a^2 + a^2} \]
\[ A = \sqrt{3a^2} \]

### Step 6: Simplify the Result
Finally, simplifying gives:
\[ A = \sqrt{3}a \]

Thus, the resultant amplitude of the vibrating particle is:
\[ \boxed{\sqrt{3}a} \]