1 Honors Physics 1 Class 18 Fall 2013 Harmonic motion Unit circle and the phasor Solution to the spring differential equation Initial and boundary conditions Conservation of energy The damped harmonic oscillator

2 Mass on Spring where x is displacement from equilibrium.

3 Second-order linear differential equation

4 Physical problems of the same form –Mass on spring –Torsional oscilator –Simple pendulum –Physical pendulum –Atoms –Molecules –LC oscillator

5 Differential Equations General solutions and Sums of solutions –Sturm-Liouville problems Sturm-Liouville problems have a standard form and have the unique characteristic that all possible solutions over the defined range can be generated from a linear sum of the orthogonal functions that are the general solutions to the equation. Orthogonality of solutions and Completeness of the set of solutions allows us to express any possible solution in terms of sums of the orthogonal functions.

6 Simple harmonic motion basics x m Amplitude (meters)  t +  Phase ([radians])  Initial phase ([radians])  Angular Frequency ([rad]/s, s -1 ) TPeriod (s) fFrequency (Hz, [oscillations]/s)

7 Circular motion – Phasor representation

8 Circular motion

9 Initial, or Boundary Conditions

10 Class Activity Initial conditions in harmonic motion

11 Mass on a spring Note: x m,  and  are determined by details of the specific system.

12 Energy of mass+spring system

13 Energy oscillates between U and K x=Max, v=0, K=0, U=Max x=0, v=Max, K=Max, U=0 x=Max, v=0, K=0, U=Max x=0, v=Max, K=Max, U=0 Time

14 Damping – a correction to many physical models