U is the PE of an oscillating particle and F is the force. if acting on it at a given instant. Which of the following true?

Options

A. `(U)/(F)+x=0`
B. `(2U)/(F)+x=0`
C. `(F)/(U)+x=0`
D. `(F)/(2U)+x=0`

Explanation

To solve the problem, we need to analyze the relationship between the potential energy (U), the force (F), and the displacement (x) of an oscillating particle in simple harmonic motion (SHM).

### Step-by-Step Solution:

1. **Understanding Potential Energy (U)**:
The potential energy of an oscillating particle in simple harmonic motion is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

**Hint**: Remember that potential energy in SHM is related to the displacement squared and the spring constant.

2. **Understanding Force (F)**:
The force acting on the particle in SHM is given by Hooke's law:
\[ F = -k x \]
The negative sign indicates that the force acts in the opposite direction of the displacement.

**Hint**: The force in SHM is directly proportional to the displacement and acts in the opposite direction.

3. **Expressing k in terms of F and x**:
From the force equation, we can express \( k \) as:
\[ k = -\frac{F}{x} \]

**Hint**: Isolate \( k \) to relate it to force and displacement.

4. **Substituting k into the Potential Energy Equation**:
Substitute the expression for \( k \) back into the potential energy equation:
\[ U = \frac{1}{2} \left(-\frac{F}{x}\right) x^2 \]
Simplifying this gives:
\[ U = -\frac{F}{2} x \]

**Hint**: When substituting, ensure to simplify carefully to maintain the relationships.

5. **Rearranging the Equation**:
Rearranging the equation gives:
\[ U + \frac{F}{2} x = 0 \]
This can be further manipulated to express a relationship between U and F:
\[ 2U + F x = 0 \]

**Hint**: Look for ways to rearrange the equation to find relationships between the variables.

6. **Final Expression**:
Dividing the entire equation by \( F \) (assuming \( F \neq 0 \)) gives:
\[ \frac{2U}{F} + x = 0 \]
This indicates that the relationship between potential energy, force, and displacement is established.

**Hint**: Always check if the final form matches any provided options in the question.

### Conclusion:
From the above steps, we find that the correct relationship derived from the equations is:
\[ \frac{2U}{F} + x = 0 \]
Thus, the correct option is the one that represents this relationship.