Question

What is the meaning of frequency of small amplitude oscillations ?

Answer 1

One of the most important situations in physics is motion near equilibrium. Motion
near stable equilibrium can always be decomposed into the motion of harmonic oscillators.
From a pendulum in a clock, to a bridge swaying in the breeze, to photons in quantum
electrodynamics, the harmonic approximation for motion near equilibrium comes into play. This is the case of small oscillation.
We will now touch on some of the key ideas.

Example
Consider a mass m which is constrained to move on a straight line. The mass is bound
to a fixed point by harmonic force with potential energy
V=1/2K(r-R)^2
where K is a
constant, r is the distance of the particle to the fixed point. The distance from the point
to the line is l > R. A mechanical model of this system is a mass sliding on a straight
track; the mass being connected to a fixed point by a spring. Our goal is to find the
stable equilibrium position(s) and compute the frequency of small oscillations about the
equilibrium.
solution attached