Lecture 22

Harmonic Oscillations

Need:
- Stable (equilibrium) location/state where forces balance
- Forces that push back toward stable state
"Harmonic" , as in music
- Because many oscillating systems oscillate with a sin function
- And a sin function in sound is a pure tone
Mass on a spring:
- Classic oscillation example

$Force\ (F) = -k* \Delta x = Mass * Acceleration$
Solve for motion ( using calculus )
$x(t) = A * cos(\omega_0*t)$
- $\omega_0 = \sqrt{\left(\frac{k}{m}\right)}$ = angular frequency
- $\frac{radians}{1\ second}$ $\frac{1}{1\ Second}$

F r e q u e n c y (f) = \frac{1}{S} = H e r t z

x (t) = A * c o s (2 * π * f (t))

ω_{0} = 2 * π * f

x (t) = A * c o s (ω_{0} * (t - t_{0}))

x (t) = A * c o s ((ω_{0} * t) - ϕ)

ϕ = ω_{0} * t_{0} = p h a s e s h i f t