Deriving the Expression for Force and Potential Energy at Equilibrium

To determine the force and potential energy at equilibrium, we need to find the point where the net force acting on the system is zero. This is the point where the potential energy is at a minimum or maximum value, depending on the nature of the force.

Finding the Equilibrium Position

To find the equilibrium position, we need to set the net force to zero. The net force is the negative of the derivative of the potential energy with respect to the position.

F = -dU/dx

= -d(ax^2 - bx)/dx

= -2ax + b

Setting this equal to zero gives:

2ax = b

x = b/2a

This is the equilibrium position where the net force acting on the system is zero.

Finding the Potential Energy at Equilibrium

To find the potential energy at equilibrium, we need to substitute the equilibrium position into the expression for potential energy.

U = a(b/2a)^2 - b(b/2a)

= b^2/4a - b^2/2a

= -b^2/4a

This is the potential energy at equilibrium.

Finding the Force at Equilibrium

To find the force at equilibrium, we need to substitute the equilibrium position into the expression for force.

F = -2a(b/2a) + b

= -b/2

This is the force at equilibrium.

Conclusion

In summary, the equilibrium position, potential energy, and force for a conservative force system with potential energy given by U=ax^2-bx are:

Equilibrium position: x = b/2a

Potential energy at equilibrium: U = -b^2/4a

Force at equilibrium: F = -b/2

These expressions can be used to analyze the behavior of the system near its equilibrium point.