A particle is acted by `x` force `F = Kx` where `K` is `a( + ve)` constant its potential energy at `x = 0` is zero . Which curve correctly represent the variation of potential energy of the block with respect to `x`

Options

A. (a)
B. (b)
C. (c)
D. (d)

Explanation

To solve the problem, we need to determine the potential energy (U) of a particle acted upon by a force given by \( F = Kx \), where \( K \) is a positive constant. The potential energy at \( x = 0 \) is specified to be zero.

### Step-by-Step Solution:

1. **Understand the relationship between force and potential energy**:
The force acting on the particle is related to the potential energy by the equation:
\[ F = -\frac{dU}{dx} \]
This means that the force can be expressed as the negative gradient of the potential energy.

2. **Substitute the given force into the equation**:
We know that \( F = Kx \). Therefore, we can write:
\[ Kx = -\frac{dU}{dx} \]

3. **Rearranging the equation**:
Rearranging gives us:
\[ \frac{dU}{dx} = -Kx \]

4. **Integrate to find potential energy**:
To find \( U \), we integrate with respect to \( x \):
\[ U = -\int Kx \, dx \]
Performing the integration:
\[ U = -\left(\frac{K}{2} x^2\right) + C \]
where \( C \) is the constant of integration.

5. **Determine the constant of integration**:
We know that the potential energy at \( x = 0 \) is zero:
\[ U(0) = 0 = -\left(\frac{K}{2} (0)^2\right) + C \]
This implies that \( C = 0 \).

6. **Final expression for potential energy**:
Thus, the potential energy as a function of \( x \) is:
\[ U(x) = -\frac{K}{2} x^2 \]

7. **Interpret the result**:
The expression \( U(x) = -\frac{K}{2} x^2 \) indicates that the potential energy is a downward-opening parabola (an inverted parabola) with respect to \( x \).

### Conclusion:
The variation of potential energy \( U \) with respect to \( x \) is represented by a curve that is an inverted parabola.