A particle which is constant to move along the `x- axis` , is subjected to a force in the same direction which varies with the distance `x` of the particle from the origin as `F(x) = -Kx + ax^(3)` . Hero `K` and `a` are positive constant . For `x ge 0`, the fanctional from of the patential every `U(x) of the particle is

Options

A.
B.
C.
D.

Explanation

To find the functional form of the potential energy \( U(x) \) of the particle subjected to the force \( F(x) = -Kx + ax^3 \), we will follow these steps:

### Step 1: Understand the relationship between force and potential energy
The relationship between force and potential energy is given by:
\[ F(x) = -\frac{dU}{dx} \]
This means that the force acting on the particle is equal to the negative derivative of the potential energy with respect to position.

### Step 2: Write down the expression for force
Given the force:
\[ F(x) = -Kx + ax^3 \]

### Step 3: Set up the equation for potential energy
From the relationship between force and potential energy, we have:
\[ -\frac{dU}{dx} = -Kx + ax^3 \]
This can be rearranged to:
\[ \frac{dU}{dx} = Kx - ax^3 \]

### Step 4: Integrate to find potential energy
To find \( U(x) \), we need to integrate the expression for \( \frac{dU}{dx} \):
\[ U(x) = \int (Kx - ax^3) \, dx \]

### Step 5: Perform the integration
Calculating the integral:
\[ U(x) = \int Kx \, dx - \int ax^3 \, dx \]
\[ U(x) = \frac{K}{2} x^2 - \frac{a}{4} x^4 + C \]
where \( C \) is the constant of integration.

### Step 6: Determine the constant of integration
To find the constant \( C \), we can set the potential energy to zero at the origin (i.e., \( U(0) = 0 \)):
\[ U(0) = \frac{K}{2} (0)^2 - \frac{a}{4} (0)^4 + C = 0 \]
This implies \( C = 0 \).

### Step 7: Write the final expression for potential energy
Thus, the functional form of the potential energy \( U(x) \) is:
\[ U(x) = \frac{K}{2} x^2 - \frac{a}{4} x^4 \]