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

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

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

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 .0300 m is attached to a 0.500 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 +0.015m and an initial velocity of +.400 m/s, determine the amplitude and phase angle of the motion. Write equations of position, velocity and acceleration as a function of time.

Energy in a SHO K U E x

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

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

Damped Harmonic Oscillator: (linear) friction Ff = -bv x vs t

Damped Harmonic Oscillator (cont’d) x vs t

Driven Harmonic Oscillator: F = Fmax cos(wdt) Fmax/k vs wd/ w0