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 find the potential energy \( E \) at a displacement \( (x + y) \) for a body executing simple harmonic motion, we can follow these steps:

### Step 1: Understand the Potential Energy in Simple Harmonic Motion
In simple harmonic motion, the potential energy \( E \) 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.

### Step 2: Write the Potential Energies at Displacements \( x \) and \( y \)
Given:
- At displacement \( x \), the potential energy is \( E_1 \):
\[ E_1 = \frac{1}{2} k x^2 \]
- At displacement \( y \), the potential energy is \( E_2 \):
\[ E_2 = \frac{1}{2} k y^2 \]

### Step 3: Write the Potential Energy at Displacement \( (x + y) \)
Now, we need to find the potential energy \( E \) at the displacement \( (x + y) \):
\[ E = \frac{1}{2} k (x + y)^2 \]

### Step 4: Expand the Expression
Expanding the expression for \( E \):
\[ E = \frac{1}{2} k (x^2 + 2xy + y^2) \]
This can be rewritten as:
\[ E = \frac{1}{2} k x^2 + \frac{1}{2} k y^2 + \frac{1}{2} k (2xy) \]

### Step 5: Substitute \( E_1 \) and \( E_2 \)
From our earlier definitions, we can substitute \( E_1 \) and \( E_2 \):
\[ E = E_1 + E_2 + kxy \]

### Step 6: Express \( kxy \) in Terms of \( E_1 \) and \( E_2 \)
To express \( kxy \) in terms of \( E_1 \) and \( E_2 \), we can find \( x \) and \( y \) in terms of \( E_1 \) and \( E_2 \):
- From \( E_1 = \frac{1}{2} k x^2 \), we have \( x = \sqrt{\frac{2E_1}{k}} \)
- From \( E_2 = \frac{1}{2} k y^2 \), we have \( y = \sqrt{\frac{2E_2}{k}} \)

Now substituting these back into the expression for \( kxy \):
\[ kxy = k \left(\sqrt{\frac{2E_1}{k}}\right) \left(\sqrt{\frac{2E_2}{k}}\right) = \frac{2\sqrt{E_1 E_2}}{k} \]

### Final Expression for \( E \)
Combining everything, we get:
\[ E = E_1 + E_2 + 2\sqrt{E_1 E_2} \]

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