# How to find if equilibrium points of a force is un/stable?

U = Ax2 - Bx3

## Homework Equations

du/dx = 2Ax - 3Bx2

## The Attempt at a Solution

If I was given a potential energy function U = Ax2 - Bx3 and am asked to find:

1) The expression for the force as a function of x.

2) The equilibrium points and determine if are they stable or unstable?

So, for 1):
Would I differential the function giving like so?

U' = f'(x) = 2Ax - 3Bx2

Now for 2):
Would I set f'(x) = 0 to find the equilibrium points?

f'(x) = 2Ax - 3Bx2 = 0

In return I get the points of x through the quadratic equation:
x = 0 and X = A/B

If this is all correct how can I determine if a equilibrium point is stable or unstable?

U = Ax2 - Bx3

## Homework Equations

du/dx = 2Ax - 3Bx2

## The Attempt at a Solution

If I was given a potential energy function U = Ax2 - Bx3 and am asked to find:

1) The expression for the force as a function of x.

2) The equilibrium points and determine if are they stable or unstable?

So, for 1):
Would I differential the function giving like so?

U' = f'(x) = 2Ax - 3Bx2

Recall the force is negative derivative of the potential energy.
Now for 2):
Would I set f'(x) = 0 to find the equilibrium points?

f'(x) = 2Ax - 3Bx2 = 0

In return I get the points of x through the quadratic equation:
x = 0 and X = A/B

If this is all correct how can I determine if a equilibrium point is stable or unstable?
X=0 is correct, but you have a mistake in the other equilibrium point.

The equilibrium is stable if the potential energy is minimum in that point and unstable if it is maximum.

Recall the force is negative derivative of the potential energy.

X=0 is correct, but you have a mistake in the other equilibrium point.

The equilibrium is stable if the potential energy is minimum in that point and unstable if it is maximum.

Oh sorry, x = 2A/(3B).
How I find the max and min of potential energy?

Oh sorry, x = 2A/(3B).
How I find the max and min of potential energy?
You found the positions of the extremes, at x=0 and at x=2A/3B.
Have you learned what should be the second derivative at a maximum and at a minimum?