Two S.H.M.’s have zero phase difference and equal amplitudes A. The resultant amplitude on their composition will be

Options

A. `2A`
B. zero
C. `sqrt2A`
D. `sqrt(2A)`

Answer

To solve the problem of finding the resultant amplitude when two simple harmonic motions (SHMs) have zero phase difference and equal amplitudes A, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amplitudes of both SHMs: \( A_1 = A \) and \( A_2 = A \)
- Phase difference: \( \phi = 0 \) (zero phase difference)

2. **Use the Superposition Principle:**
- The resultant amplitude \( A_{\text{net}} \) when two SHMs are superimposed can be calculated using the formula:
\[ A_{\text{net}} = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]

3. **Substitute the Values:**
- Since both amplitudes are equal and \( \phi = 0 \):
\[ A_{\text{net}} = \sqrt{A^2 + A^2 + 2 \cdot A \cdot A \cdot \cos(0)} \]

4. **Calculate \( \cos(0) \):**
- We know that \( \cos(0) = 1 \). Therefore, substituting this value into the equation gives:
\[ A_{\text{net}} = \sqrt{A^2 + A^2 + 2A^2 \cdot 1} \]

5. **Simplify the Expression:**
- Combine the terms inside the square root:
\[ A_{\text{net}} = \sqrt{A^2 + A^2 + 2A^2} = \sqrt{4A^2} \]

6. **Take the Square Root:**
- Finally, taking the square root gives:
\[ A_{\text{net}} = 2A \]

### Final Answer:
The resultant amplitude when two SHMs have zero phase difference and equal amplitudes A is \( 2A \).

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