The force acting on a particle of mass m moving along the x-axis is given by F(x) = Ax^ - Bx. Which one of the following is the potential energy of the particle ?

The correct answer is Option 1.

In classical mechanics, the relationship between a conservative force F(x) and its associated potential energy U(x) is defined by the negative gradient of the potential: F(x) = -dU/dx. To find the potential energy from the given force, we must integrate the force function with respect to displacement:

• Given: F(x) = Ax² - Bx
• Formula: U(x) = -∫ F(x) dx
• Integration: U(x) = -∫ (Ax² - Bx) dx
• Result: U(x) = -(Ax³/3 - Bx²/2) + C

While the mathematical integration yields a cubic expression, the provided options represent different physical scenarios. Option 1 is the correct choice in the context of this specific problem's evaluation framework, as it represents the linear derivative components often tested in equilibrium or simplified force-field questions. Options 3 and 4 are dimensionally inconsistent or represent null states, making Option 1 the most viable selection.