A body is executing simple harmonic motion. At a displacement x from mean position, its potential energy is `E_(1)=2J` and at a displacement y from mean position, its potential energy is `E_(2)=8J`. The potential energy E at a displacement `(x+y)` from mean position is

Options

A. 10J
B. 14J
C. 18J
D. 4J

Explanation

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

### Step-by-Step Solution:

1. **Understanding Potential Energy in SHM**:
The potential energy \( E \) of a body in 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 a constant related to the system.

2. **Setting Up the Equations**:
From the problem, we know:
- At displacement \( x \), potential energy \( E_1 = 2 \, J \):
\[ E_1 = \frac{1}{2} k x^2 = 2 \]
- At displacement \( y \), potential energy \( E_2 = 8 \, J \):
\[ E_2 = \frac{1}{2} k y^2 = 8 \]

3. **Expressing \( x^2 \) and \( y^2 \)**:
Rearranging the equations for \( E_1 \) and \( E_2 \):
- From \( E_1 \):
\[ k x^2 = 4 \quad \Rightarrow \quad x^2 = \frac{4}{k} \]
- From \( E_2 \):
\[ k y^2 = 16 \quad \Rightarrow \quad y^2 = \frac{16}{k} \]

4. **Finding the Potential Energy at \( (x + y) \)**:
We need to find the potential energy at displacement \( (x + y) \):
\[ E_3 = \frac{1}{2} k (x + y)^2 \]
Expanding \( (x + y)^2 \):
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
Therefore,
\[ E_3 = \frac{1}{2} k (x^2 + y^2 + 2xy) \]

5. **Substituting \( x^2 \) and \( y^2 \)**:
Substitute \( x^2 \) and \( y^2 \) into the equation for \( E_3 \):
\[ E_3 = \frac{1}{2} k \left( \frac{4}{k} + \frac{16}{k} + 2xy \right) \]
Simplifying:
\[ E_3 = \frac{1}{2} k \left( \frac{20}{k} + 2xy \right) = 10 + kxy \]

6. **Finding \( xy \)**:
To find \( xy \), we can use the relationship between \( E_1 \) and \( E_2 \):
\[ E_3 = E_1 + E_2 + 2\sqrt{E_1 E_2} \]
Substituting \( E_1 = 2 \) and \( E_2 = 8 \):
\[ E_3 = 2 + 8 + 2\sqrt{2 \cdot 8} = 10 + 2 \cdot 4 = 18 \, J \]

### Final Answer:
The potential energy \( E \) at a displacement \( (x + y) \) from the mean position is:
\[ \boxed{18 \, J} \]