Oscillations
Introduction
Oscillations are a common phenomenon in the mechanics of classical objects, and the concepts involved have proven to be fundamental in our modern understanding of physics. Mathematically, the quadratic form is the lowest order approximation of the potential function at its local minimum point (where the first derivative vanishes).
In these notes, we will attempt to showcase some of the diversity and beauty of the concepts involved, and introduce some interesting problems and solutions. We first look at the simple harmonic oscillator, then introduce concepts of damping (energy dissipation), resonance (characteristic response to external forces), and normal modes (behaviour of coupled systems).
1. Derivation for an idealised spring-mass system
Consider a system composed of a horizontal spring attached to a block of mass \(\)\(m\) (supported on a frictionless flat surface). Let \(x\) be the extension of the spring, and let \(k\) be the spring constant.
1.1 Using Newton's second law of motion, write down the equation of motion for the mass in terms of \(x\) and its time derivatives. Solve the differential equation (there will be two arbitrary constants involved, since this is a second order differential equation). Plot \(x\left(t\right)\), and label the amplitude and period on the graph.