The amplitude of the vibrating particle due to superposition of two `SHMs`,
`y_(1)=sin (omega t+(pi)/(3)) and y_(2)=sin omega t` is :

Options

A. `1`
B. `sqrt2`
C. `sqrt3`
D. `2`

Answer

To find the amplitude of the vibrating particle due to the superposition of two simple harmonic motions (SHMs) given by the equations \( y_1 = \sin(\omega t + \frac{\pi}{3}) \) and \( y_2 = \sin(\omega t) \), we can follow these steps:

### Step-by-Step Solution:

1. **Write the Superposition of the Two SHMs:**
The total displacement \( y \) due to the superposition of the two SHMs is given by:
\[ y = y_1 + y_2 = \sin(\omega t + \frac{\pi}{3}) + \sin(\omega t) \]

2. **Use the Trigonometric Identity for Sine Addition:**
We can use the identity for the sum of two sine functions:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Here, let \( A = \omega t + \frac{\pi}{3} \) and \( B = \omega t \).

3. **Calculate \( A + B \) and \( A - B \):**
- \( A + B = (\omega t + \frac{\pi}{3}) + \omega t = 2\omega t + \frac{\pi}{3} \)
- \( A - B = (\omega t + \frac{\pi}{3}) - \omega t = \frac{\pi}{3} \)

4. **Substitute into the Identity:**
Now substituting \( A + B \) and \( A - B \) into the identity:
\[ y = 2 \sin\left(\frac{2\omega t + \frac{\pi}{3}}{2}\right) \cos\left(\frac{\frac{\pi}{3}}{2}\right) \]
This simplifies to:
\[ y = 2 \sin\left(\omega t + \frac{\pi}{6}\right) \cos\left(\frac{\pi}{6}\right) \]

5. **Calculate \( \cos\left(\frac{\pi}{6}\right) \):**
We know that:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

6. **Substitute Back to Find the Amplitude:**
Thus, we have:
\[ y = 2 \sin\left(\omega t + \frac{\pi}{6}\right) \cdot \frac{\sqrt{3}}{2} \]
This simplifies to:
\[ y = \sqrt{3} \sin\left(\omega t + \frac{\pi}{6}\right) \]

7. **Identify the Amplitude:**
The amplitude of the resultant SHM is the coefficient of the sine function, which is:
\[ \text{Amplitude} = \sqrt{3} \]

### Final Answer:
The amplitude of the vibrating particle due to the superposition of the two SHMs is \( \sqrt{3} \).