A body executes simple harmonic motion. At a displacement x, its potential energy is `U_1`. At a displacement y, its potential energy is `U_2`. What is the potential energy of the body at a displacement (x + y)?

Options

A. `U_1+U_2`
B. `(sqrtU_1 + sqrtU_2)^2`
C. `sqrt(U_1^2 +U_2^2)`
D. `sqrt(U_1U_2)`

Answer

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

### Step-by-Step Solution:

1. **Understand the formula for potential energy in SHM**:
The potential energy \( U \) of a body in simple harmonic motion at a displacement \( x \) is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant.

2. **Write the expressions for potential energy at displacements \( x \) and \( y \)**:
- At displacement \( x \):
\[ U_1 = \frac{1}{2} k x^2 \]
- At displacement \( y \):
\[ U_2 = \frac{1}{2} k y^2 \]

3. **Find the potential energy at displacement \( x + y \)**:
The potential energy at displacement \( x + y \) can be expressed as:
\[ U = \frac{1}{2} k (x + y)^2 \]

4. **Expand the expression**:
Expanding \( (x + y)^2 \):
\[ (x + y)^2 = x^2 + 2xy + y^2 \]
Therefore, the potential energy becomes:
\[ U = \frac{1}{2} k (x^2 + 2xy + y^2) \]

5. **Substitute \( x^2 \) and \( y^2 \) using \( U_1 \) and \( U_2 \)**:
From the expressions for \( U_1 \) and \( U_2 \):
\[ x^2 = \frac{2U_1}{k} \quad \text{and} \quad y^2 = \frac{2U_2}{k} \]
Substituting these into the potential energy expression:
\[ U = \frac{1}{2} k \left( \frac{2U_1}{k} + 2xy + \frac{2U_2}{k} \right) \]

6. **Simplify the expression**:
The terms involving \( k \) will cancel out:
\[ U = U_1 + U_2 + kxy \]

7. **Express \( xy \) in terms of \( U_1 \) and \( U_2 \)**:
To express \( xy \), we can use the square root of the potential energies:
\[ xy = \sqrt{\left(\frac{2U_1}{k}\right) \left(\frac{2U_2}{k}\right)} = \frac{2\sqrt{U_1 U_2}}{k} \]

8. **Final expression for potential energy**:
Substitute \( xy \) back into the potential energy equation:
\[ U = U_1 + U_2 + \frac{2\sqrt{U_1 U_2}}{k} \]

9. **Rearranging the expression**:
This can be rewritten as:
\[ U = U_1 + U_2 + 2\sqrt{U_1 U_2} \]

### Conclusion:
The potential energy of the body at a displacement \( x + y \) is given by:
\[ U = U_1 + U_2 + 2\sqrt{U_1 U_2} \]