[NOTES/ME-06005] Bounded Motion --- Oscillations Around Minimum

1. Frequency of small oscillations:

In the example under consideration we have seen that the body executes oscillatory,periodic motion about the point of stable equilibrium. This is true in general for any potential having a minimum. The frequency of small oscillations about the equilibrium can be found as follows. Let \(x=a\) be location of a minimum of potential \(V(x)\). Then the equation of motion of the body is \begin{equation}\label{EQ01} m \DD[x]{t} = -\dd[V(x)]{x}. \end{equation} Let \(u=x-a\) denote the displacement from the equilibrium. We write \(x=u+a\) and expand the potential in a Tailor series expansion \begin{equation}\label{EQ02} V(x) = V(a) + u \dd[V]{x}\Big|_{x=a} +\frac{1}{2} u^2 + \ldots. \end{equation} Since \(x=a\) corresponds to the minimum of the potential \(\dd[V]{x}\Big|_{x=a}=0\), neglecting \(O(u^3)\) terms, from the equation of motion, \eqRef{EQ01} leads to \begin{equation}\label{EQ03} m \DD[u]{t} = -\frac{K}{2} u^2, \end{equation} where \(K \DD[V]{x}\Big|_{x=a}\). the above equation implies that the frequency of small oscillations about the equilibrium is given by \begin{equation} \boxed{\omega = \sqrt{\frac{K}{M}},\quad \text{and} \quad K=\DD[V]{x}\Big|_{x=a}}. \end{equation} Note that \(K\) is positive because \(x=a\) is a minimum of the potential.