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 w = 2pf = “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. Energy in a SHO K U E x

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
= k q = I a
= ?
Molecules: Potential energy is approximately quadratic y0

7. Physical Pendulum, center of gravity = - mg d sin q = Ia sin q  q ,
More SHO variations
Simple Pendulum
FT = mg sin q
= - mg L sin q = Ia
sin q  q , I = mL2
mg L q = mL2 a w = ?
Physical Pendulum, center of gravity
= - mg d sin q = Ia
sin q  q ,
mg d q = I a w = ? 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(wdt)
Fmax/k vs wd/ w0