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Potential Energy System

An object is said to be in static equilibrium if it is (and remains) at rest in an inertial frame. A necessary condition for equilibrium is that the net force on the object vanishes. For conservative forces, the equilibrium implies that

where U represents the potential energy of the object. Equilibrium is stable if potential energy U is minimum at the point of equilibrium.

The potential energy is a function of position co-ordinates of the object. When the object is displaced from its position of stable equilibrium, its position of stable equilibrium, its potential energy changes. We shall consider only one-dimensional displacement of the object; that is, suppose only one position co-ordinate of the object is changed. Let us call this co-ordinate as q; q may be x or (y, z, r, θ, Ø, etc.) The potential energy is a function of q, and for q = q₀, U (q₀) is given to be minimum. Hence,

When the object is displaced from position of stable equilibrium, the co-ordinate q changes from q₀ to (q₀ + δ q). The potential energy of the object at q = q₀ + δ q can be expanded in a Taylor’s series as following:

Since q₀ is a definite number, the values of the derivatives after putting q = q₀ also become numbers or constants; i.e.

Now, we know that C₁ = 0, and C₂ > 0. Hence, we get

If we fix the origin of co-ordinates at position of equilibrium, i.e. q₀ = 0 and also choose the zero of potential energy there, i.e. U ( q₀ ) = 0, then we can write