Title: Simple Harmonic Motion
1Section 5.1
2SIMPLE HARMONIC MOTION
The second-order differential equation where ?2
k/m is the equation that describes simple
harmonic motion, or free undamped motion.
3INITIAL CONDITIONS
The initial conditions for simple harmonic motion
are x(0) a, x'(0) ß. NOTES 1. If a gt 0, ß
lt 0, the mass starts from a point below the
equilibrium position with an imparted upward
velocity. 2. If a lt 0, ß 0, the mass is
released from rest from a point a units above
the equilibrium position.
4SOLUTION
The general solution of the equation for simple
harmonic motion is x(t) c1 cos ?t c2 sin
?t. The period of the free vibrations is T
2p/?, and the frequency of the vibrations is f
1/T ?/2p.
5ALTERNATIVE FORM OF x(t)
When c1 ? 0 and c2 ? 0, the actual amplitude A of
the vibrations is not obvious from the equation
on the previous slide. We often convert the
equation to the simpler form x(t) A sin (?t
f), where and f is a phase
angle defined by
6HOMEWORK
125 odd