A particle is placed at the origin and a force F=Kx is acting on it (where k is a positive constant). If `U_((0))=0`, the graph of `U (x)` verses x will be (where U is the potential energy function.)

Options

A.
B.
C.
D.

Explanation

To solve the problem of finding the potential energy function \( U(x) \) given the force \( F = kx \), we can follow these steps:

### Step 1: Understand the relationship between force and potential energy
The force acting on a particle is related to the potential energy by the equation:
\[ F = -\frac{dU}{dx} \]
This means that the force is equal to the negative gradient (or derivative) of the potential energy function.

### Step 2: Substitute the given force into the equation
We are given that:
\[ F = kx \]
Substituting this into the relationship gives:
\[ kx = -\frac{dU}{dx} \]

### Step 3: Rearrange the equation
Rearranging the equation to isolate \( dU \):
\[ dU = -kx \, dx \]

### Step 4: Integrate both sides
To find \( U(x) \), we need to integrate:
\[ U(x) = -\int kx \, dx \]
Calculating the integral:
\[ U(x) = -\left(\frac{k}{2} x^2\right) + C \]
where \( C \) is the constant of integration.

### Step 5: Determine the constant of integration
We are given that the potential energy at \( x = 0 \) is \( U(0) = 0 \). Substituting \( x = 0 \) into the equation:
\[ U(0) = -\frac{k}{2}(0)^2 + C = 0 \]
This implies that \( C = 0 \).

### Step 6: Write the final expression for potential energy
Thus, the potential energy function is:
\[ U(x) = -\frac{k}{2} x^2 \]

### Step 7: Analyze the graph of \( U(x) \)
The function \( U(x) = -\frac{k}{2} x^2 \) represents a downward-opening parabola. The vertex of this parabola is at the origin (0,0), and it decreases as \( |x| \) increases.

### Conclusion
The graph of \( U(x) \) versus \( x \) will be a downward-opening parabola.

---