A body is executing simple harmonic motion. At a displacement x (from its mean position) its potential energy is `E_(1)` and at a displacement y its potential energy is `E_(2)`. The potential energy is E at displacement (x+y) . Then:

Options

A. `sqrt(E ) = sqrt(E_(1))-sqrt(E_(2))`
B. `sqrt(E ) = sqrt(E_(1))+sqrt(E_(2))`
C. `E = E_(1)-E_(2)`
D. `E=E_(1)+E_(2)`

Answer

To solve the problem, we need to establish the relationship between the potential energies \(E_1\), \(E_2\), and \(E\) at different displacements in simple harmonic motion.

### Step-by-Step Solution:

1. **Understanding Potential Energy in SHM**:
The potential energy \(E\) of a body executing simple harmonic motion (SHM) at a displacement \(x\) from the mean position is given by the formula:
\[ E = \frac{1}{2} k x^2 \]
where \(k\) is the spring constant.

2. **Expressing \(E_1\) and \(E_2\)**:
- At displacement \(x\), the potential energy \(E_1\) can be expressed as:
\[ E_1 = \frac{1}{2} k x^2 \]
- At displacement \(y\), the potential energy \(E_2\) can be expressed as:
\[ E_2 = \frac{1}{2} k y^2 \]

3. **Expressing Potential Energy at Displacement \(x + y\)**:
- At displacement \(x + y\), the potential energy \(E\) can be expressed as:
\[ E = \frac{1}{2} k (x + y)^2 \]

4. **Expanding the Expression for \(E\)**:
- Expanding the square in the expression for \(E\):
\[ E = \frac{1}{2} k (x^2 + 2xy + y^2) \]

5. **Relating \(E\), \(E_1\), and \(E_2\)**:
- Substitute the expressions for \(E_1\) and \(E_2\):
\[ E = E_1 + E_2 + kxy \]
- Rearranging gives:
\[ kxy = E - E_1 - E_2 \]

6. **Finding the Relationship**:
- We can express \(x\) and \(y\) in terms of \(E_1\) and \(E_2\):
\[ x = \sqrt{\frac{2E_1}{k}}, \quad y = \sqrt{\frac{2E_2}{k}} \]
- Thus, substituting these values into the equation for \(E\):
\[ E = \frac{1}{2} k \left(\sqrt{\frac{2E_1}{k}} + \sqrt{\frac{2E_2}{k}}\right)^2 \]

7. **Final Relationship**:
- After simplification, we find:
\[ \sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \]

### Conclusion:
The relationship between the potential energies is:
\[ \sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \]
Thus, the correct answer is option number 2.