Simple pendulum Physical pendulum Diatomic molecule Damped oscillations Driven oscillations Lecture 24: General Oscillations

SHO Equation for SHO General solution:

Simple Pendulum

T does not produce a torque, since the line of action goes through point P

Simple pendulum oscillations Differential equation of simple harmonic oscillator Demo: Simple pendulum with different masses, lengths and amplitudes

Period independent of mass Period independent of amplitude

Physical Pendulum SHO

Motion of the Physical Pendulum SHO Demo: Meter stick pivoted at different positions I is moment of inertia about axis P D is distance between P and CM Parallel axis theorem:

Example A vertical rod of mass M and length L can freely oscillate about a pivot P located at the top end. A small sphere of mass M is rigidly attached to the bottom end. The sphere can be considered a point mass. What is the period of small oscillations for this system?

Example A uniform disk of mass M and radius R is pivoted at a point at the rim. Find the period for small oscillations.

Vibrations of Molecules Diatomic molecule: two atoms separated a distance r

Diatomic molecule near equilibrium

Damped Oscillations

Characteristics of damped oscillation

Driven oscillations

Resonance If driving frequency close to natural frequency catastrophic growth of amplitude = Resonance

Millennium Bridge, London Video: Spontaneous resonance

Tacoma Narrows Bridge Collapse Cause still disputed, not technically resonance – but the video is worth watching and fits very well here