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-by-Step Solution:

1. **Identify the Amplitudes and Phase Difference**:
- The amplitude of the first wave, \( a_1 = a \).
- The amplitude of the second wave, \( a_2 = a \).
- The phase difference \( \phi \) between the two waves is \( \frac{\pi}{3} \).

2. **Use the Formula for Resultant Amplitude**:
The resultant amplitude \( A \) when two waves of the same amplitude interfere can be calculated using the formula:
\[ A = \sqrt{a_1^2 + a_2^2 + 2 a_1 a_2 \cos(\phi)} \]

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

4. **Calculate \( \cos\left(\frac{\pi}{3}\right) \)**:
We know that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

5. **Simplify the Expression**:
Substitute \( \cos\left(\frac{\pi}{3}\right) \) into the equation:
\[ A = \sqrt{a^2 + a^2 + 2 \cdot a^2 \cdot \frac{1}{2}} \]
\[ A = \sqrt{a^2 + a^2 + a^2} \]
\[ A = \sqrt{3a^2} \]

6. **Final Result**:
Taking the square root gives:
\[ A = \sqrt{3} a \]

Thus, the resultant amplitude of the vibrating particle is \( \sqrt{3} a \).