The potential energy of a body is given by `U = A - Bx^(2)` (where x is the displacement). The magnitude of force acting on the partical is

Options

A. constant
B. Proportional to x
C. proportional to `x^2`
D. `inversely proportional to x

Explanation

To solve the problem, we need to find the magnitude of the force acting on a particle given the potential energy function \( U = A - Bx^2 \).

### Step-by-Step Solution:

1. **Understand the relationship between potential energy and force**:
The force acting on a particle can be derived from the potential energy function using the formula:
\[ F = -\frac{dU}{dx} \]
where \( F \) is the force, \( U \) is the potential energy, and \( x \) is the displacement.

2. **Differentiate the potential energy function**:
Given the potential energy function:
\[ U = A - Bx^2 \]
we need to differentiate \( U \) with respect to \( x \):
\[ \frac{dU}{dx} = \frac{d}{dx}(A - Bx^2) \]
Since \( A \) is a constant, its derivative is 0. The derivative of \( -Bx^2 \) is:
\[ -B \cdot 2x = -2Bx \]

3. **Calculate the force**:
Now, substituting the derivative back into the force equation:
\[ F = -\frac{dU}{dx} = -(-2Bx) = 2Bx \]

4. **Determine the magnitude of the force**:
The magnitude of the force is given by:
\[ |F| = 2Bx \]

5. **Analyze the dependence of force on displacement**:
From the expression \( |F| = 2Bx \), we can conclude that the force is directly proportional to the displacement \( x \).

### Final Answer:
The magnitude of the force acting on the particle is:
\[ |F| = 2Bx \]
Thus, the force is directly proportional to \( x \).