Math 5900 – Summer 2011 Lecture 1: Simple Harmonic Oscillations Gernot Laicher University of Utah - Department of Physics & Astronomy

Newton's Second Law

Acceleration

Velocity

Hook’s Law

The system can be described with Newton’s second law as follows: Differential equation (“Second Order”: Contains second derivative; “Linear”: The function and its derivatives appear as powers of 1)

Solution to this differential equation: Note: Alternatively, we could also have written the general solution in a different but equivalent form:

Reinserting solution into DE:

The amplitude A is determined by the initial conditions of the system (at “t=0”) and the resonance frequency  : The phase angle  is similarly determined by these initial conditions and the resonance frequency  as follows: f: frequency of oscillation T: period of oscillation A: amplitude of oscillation

Net restoring force directly proportional to displacement DE of that same form Simple harmonic (sinusoidal) oscillations In our example:  : fixed by k and m A and  imposed by the initial conditions Note: Changing  is equivalent to shifting the time when t=0

Spring Constantk=0.6N/m Massm=2kg AmplitudeA=1.2m Phase  0 degree s  s Frequencyf= Hz Period  s Example:

Energy of the oscillating mass (assuming no losses due to friction) Elastic Potential Energy: Total Energy: Kinetic Energy: