1. Periodic Motion Oscillations: Stable Equilibrium: U  ½kx2 F  kx
(small) displacement from equilibrium generates restoring force
inertia
apply F = ma
Amplitude A = maximum displacement from equilibrium
Period T = time for one full cycle
Frequency f = number of cycles per unit time = 1/T
Angular Frequency  = 2f = “natural units” for frequency A A U

2. The Simple Harmonic Oscillator (SHO):
A = amplitude
= phase angle

3. SHO (continued):
get A and  from initial state of motion

4. Example: A horizontal spring which produces a force of 6
Example: A horizontal spring which produces a force of 6.00 N when stretched by m is attached to a kg body which slides on a frictionless surface.
What is the force constant of the spring?
What is the frequency and natural frequency of oscillations?
If the mass is given an initial displacement of m and an initial velocity of m/s, determine the amplitude and phase angle of the motion.
Write equations of position, velocity and acceleration as a function of time.

5. K U E x
Energy in a SHO

6. Vertical SHO (spring unstretched at y = 0) Equilibrium at y0: ky0 = mg
SHO variations
Vertical SHO (spring unstretched at y = 0)
Equilibrium at y0: ky0 = mg
Fy = ky – mg = k (y y0)
Torsion Pendulum
=   = I 
= ?
Molecules: Potential energy is approximately quadratic y0

7. Physical Pendulum, center of gravity mg d sin I sin 
More SHO variations
Simple Pendulum
FT = mg sin 
mg L sin I
sin I = mL2
mg L mL2 
Physical Pendulum, center of gravity
mg d sin I
sin 
mg d I  d

8. Damped Harmonic Oscillator: (linear) friction Ff = bv
x vs t

9. Damped Harmonic Oscillator (cont’d)
x vs t

10. Driven Harmonic Oscillator: F = Fmax cos(dt)
Fmax/k vs d/ 0