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Stable Equilibrium Point

Let's discuss a problem where we find a stable equilibrium point, useful for Physics Olympiad.

The Problem:

Find the stable equilibrium point corressponding to the potential (U(x)=k(2x^3-5x^2+4x)).

Disucssion:

$$U(x)=k(2x^3-5x^2+4x)$$
Differentiating with respect to x
$$\frac{dU}{dx}=k(6x^2-10x+4)$$
Differentiating again,
$$\frac{d^2U}{dx^2}=k(12x-10)$$
Putting (x=1) $$\frac{d^2U}{dx^2}=+2k$$
This is positive because (k) is positive and it is the minimum hence the (x=1)
corresponds to stable equilibrium.