A body is executing simple harmonic motion. At a displacement x, its potential energy is `E_1` and a displacement y, its potential energy is `E_2`. The potential energy E at a displacement (x+y) is

Options

A. `E_(1)+E_(2)`
B. `sqrt(E_(1)^(2)+E_(2)^(2))`
C. `E_(1)+E_(2)+2sqrt(E_(1)E_(2))`
D. `sqrt(E_(1)E_(2))`

Explanation

To solve the problem, we need to find the potential energy \( E \) at a displacement \( (x + y) \) given the potential energies \( E_1 \) and \( E_2 \) at displacements \( x \) and \( y \) respectively.

### Step-by-Step Solution:

1. **Understand the Formula for Potential Energy in SHM**:
The potential energy \( E \) in simple harmonic motion (SHM) is given by the formula:
\[ E = \frac{1}{2} k x^2 \]
where \( k \) is a constant related to the mass and angular frequency of the motion, and \( x \) is the displacement.

2. **Express \( E_1 \) and \( E_2 \)**:
For displacement \( x \):
\[ E_1 = \frac{1}{2} k x^2 \tag{1} \]
For displacement \( y \):
\[ E_2 = \frac{1}{2} k y^2 \tag{2} \]

3. **Solve for Displacements \( x \) and \( y \)**:
Rearranging equations (1) and (2) gives us:
\[ x = \sqrt{\frac{2E_1}{k}} \tag{3} \]
\[ y = \sqrt{\frac{2E_2}{k}} \tag{4} \]

4. **Find the Total Displacement \( x + y \)**:
The total displacement is:
\[ x + y = \sqrt{\frac{2E_1}{k}} + \sqrt{\frac{2E_2}{k}} \tag{5} \]

5. **Calculate the Potential Energy at Displacement \( (x + y) \)**:
The potential energy at displacement \( (x + y) \) is:
\[ E = \frac{1}{2} k (x + y)^2 \tag{6} \]
Substituting equation (5) into equation (6):
\[ E = \frac{1}{2} k \left( \sqrt{\frac{2E_1}{k}} + \sqrt{\frac{2E_2}{k}} \right)^2 \]

6. **Simplify the Expression**:
Expanding the square:
\[ E = \frac{1}{2} k \left( \frac{2E_1}{k} + \frac{2E_2}{k} + 2\sqrt{\frac{2E_1}{k} \cdot \frac{2E_2}{k}} \right) \]
This simplifies to:
\[ E = E_1 + E_2 + \frac{2\sqrt{E_1 E_2}}{k} \]

7. **Final Relationship**:
The relationship between \( E \), \( E_1 \), and \( E_2 \) can be expressed as:
\[ \sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \]

### Conclusion:
Thus, the potential energy \( E \) at a displacement \( (x + y) \) is given by:
\[ E = \left( \sqrt{E_1} + \sqrt{E_2} \right)^2 \]