• Simple pendulum

• Physical pendulum

• Diatomic molecule

• Damped oscillations

• Driven oscillations

Lecture 24: General Oscillations

SHO

𝑑2𝑥𝑑𝑡2

=−ω2𝑥 Equation for SHO

General solution:

𝑇=2𝜋𝜔

Simple Pendulum

= displacement coordinate (with sign) from vertical equilibrium position

Point mass at the end of a massless string of length

Simple Pendulum

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

Very complicated differential equation!But for small oscillations:

And

Differential equation of SHO

Simple pendulum oscillations

−𝑔𝐿θ=𝑑2θ

𝑑𝑡2Differential equation of simple harmonic oscillator

θ (𝑡 )=θ𝑚𝑎𝑥cos (ω 𝑡+φ)

With and 𝑇=2 π √ 𝐿𝑔Demo: Simple pendulum with different masses, lengths and amplitudes

𝑇=2 π √ 𝐿𝑔Demo: Simple pendulum with different masses, lengths and amplitudes

• Period independent of mass• Period independent of amplitude

Physical Pendulum

Extended object of mass that swings back and forth about an axis P that does not go through its center of mass CM.

For small oscillations:

SHO

Motion of the Physical Pendulum

−𝑚𝑔𝐷𝐼

θ=𝑑2θ𝑑 𝑡2

SHO

𝑇=2 π √ 𝐼𝑚𝑔𝐷

Demo: Meter stick pivoted at different positions

I is moment of inertia about axis PD is distance between P and CMParallel axis theorem:

𝐼𝑃=𝐼𝐶𝑀+𝑚𝐷2

𝑇=2 π √ 𝐼𝑚𝑔𝐷

Example

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

𝑇=2 π √ 𝐼𝑚𝑔𝐷

𝐼𝑃=𝐼𝐶𝑀+𝑚𝐷2

Vibrations of Molecules

Diatomic molecule: two atoms separated a distance r

Diatomic molecule near equilibrium

Near equilibrium (potential minimum): approximately quadratic approximately linear

Expansion about :

for small displacement from equilibrium:

⟹ harmonic oscillation about equilibrium position

Damped Oscillations

Damping   force   proportional   to   velocity 𝐹𝐷𝑥=−𝑏𝑣 𝑥

:

−𝑘𝑥−𝑏𝑣𝑥=𝑚𝑑2 𝑥𝑑𝑡 2

𝑑2𝑥𝑑𝑡2

=− 𝑘𝑚𝑥− 𝑏

𝑚𝑑𝑥𝑑𝑡

Characteristics of damped oscillation

Driven oscillations

Oscillating system with natural frequency

Externally applied periodic driving force

http://www.walter-fendt.de/ph14e/resonance.htm

Resonance

http://www.walter-fendt.de/ph14e/resonance.htm

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