Stable Equilibrium in a Certain Field

To determine the stable equilibrium point in a certain field, we are given the potential energy function u = ax^2 - bx^3, where a and b are constants.

Understanding the Concept of Equilibrium
In physics, equilibrium refers to a state in which a particle or system experiences no net force acting upon it. There are three types of equilibrium: stable, unstable, and neutral. Stable equilibrium occurs when a system returns to its original state after experiencing a small disturbance.

Finding the Equilibrium Point
To find the stable equilibrium point in the given potential energy function, we need to determine the point at which the potential energy reaches a minimum. This occurs when the derivative of the potential energy with respect to x is equal to zero.

1. Take the derivative of the potential energy function with respect to x:
u' = 2ax - 3bx^2

2. Set the derivative equal to zero and solve for x:
2ax - 3bx^2 = 0

3. Factor out x:
x(2a - 3bx) = 0

4. Set each factor equal to zero:
x = 0 (Equation 1)
2a - 3bx = 0 (Equation 2)

Analysis of Equilibrium Points

Equilibrium Point 1: x = 0
When x = 0, the particle is at an equilibrium point. However, this point is not stable as it represents a maximum potential energy. Any disturbance will cause the particle to move away from this point, increasing the potential energy.

Equilibrium Point 2: 2a - 3bx = 0
To analyze this equilibrium point, we substitute Equation 2 into the potential energy function u = ax^2 - bx^3:

u = a(2a - 3bx)^2 - b(2a - 3bx)^3

Simplifying the expression, we obtain the potential energy at the equilibrium point.

From the analysis, it can be concluded that the stable equilibrium point occurs when x = (2a)/(3b). At this point, any disturbance will result in a force that brings the particle back towards this equilibrium position, minimizing the potential energy and maintaining stability.