The potential energy of a particle with displacement X is `U(X)`. The motion is simple harmonic, when (K is a positive constant)

Options

A. `U=-(KX^2)/(2)`
B. `U=KX^2`
C. `U=K`
D. `U=KX`

Explanation

To determine the conditions under which the motion of a particle is simple harmonic motion (SHM) based on its potential energy \( U(X) \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Potential Energy in SHM**:
In simple harmonic motion, the potential energy \( U \) of a particle is typically expressed in terms of its displacement \( x \) from the mean position. The general form for potential energy in SHM is given by:
\[ U(x) = \frac{1}{2} k x^2 \]
where \( k \) is a positive constant known as the force constant.

2. **Identifying the Relationship**:
For SHM, the potential energy must be proportional to the square of the displacement. This means that if we have a potential energy function of the form:
\[ U(x) = k x^2 \]
where \( k \) is a positive constant, then the motion is simple harmonic.

3. **Deriving the Force from Potential Energy**:
The force \( F \) acting on the particle can be derived from the potential energy using the relation:
\[ F = -\frac{dU}{dx} \]
If we differentiate \( U(x) = \frac{1}{2} k x^2 \):
\[ F = -\frac{d}{dx}\left(\frac{1}{2} k x^2\right) = -k x \]
This indicates that the force is directly proportional to the displacement \( x \) and acts in the opposite direction (restoring force).

4. **Conclusion**:
Therefore, if the potential energy of the particle is given by \( U(x) = k x^2 \) (where \( k \) is a positive constant), the motion of the particle is simple harmonic.

5. **Final Expression**:
The final expression for potential energy in SHM can be summarized as:
\[ U(x) = \frac{1}{2} k x^2 \]
This confirms that the motion is simple harmonic when the potential energy is proportional to the square of the displacement.