1. Pendulums and Resonance
Lecture 3
Pendulums and Resonance
Pre-reading: §14.5–14.8

2. SHM: Vertical Springs IMPORTANT: Set up coordinate system!!
Motion of vertical spring is described by simple harmonic motion with ω = √(k/m)
§14.4

3. SHM: Pendulums
Simple pendulum: point mass on a massless, unstretched string
Exhibits SHM with ω = √(g/L) provided the amplitude (angle) is small (θ≲15°)
For real pendulum, need to know mass (m), distance to center of mass d, and moment of inertia I about rotation axis ω = √(mgd/I) provided the amplitude (angle) is small (θ≲15°)
§14.5–14.6

4. with b describing the amount of damping
Damped Oscillations
In real world, friction causes oscillations to decrease in amplitude
with b describing the amount of damping
If friction force varies linearly with speed, we can solve for motion:
§14.7

5. Effects of damping Amplitude decays exponentially.
Energy decays exponentially.
Angular frequency decreases.
Critical damping if b=2√(km) ⇒ ωʹ=0 returns to equilibrium without oscillating!
Underdamped b < 2√(km)
Overdamped b > 2√(km)

6. From the account of Leonard Coatsworth, a Tacoma News Tribune editor who was the last person to drive on the bridge, he was driving with his dog, Tubby, over the bridge when the bridge started to vibrate violently. Coatsworth was forced to flee his car:
Just as I drove past the towers, the bridge began to sway violently from side to side. Before I realized it, the tilt became so violent that I lost control of the car...I jammed on the brakes and got out, only to be thrown onto my face against the curb...Around me I could hear concrete cracking...The car itself began to slide from side to side of the roadway.
On hands and knees most of the time, I crawled 500 yards (460 m) or more to the towers...My breath was coming in gasps; my knees were raw and bleeding, my hands bruised and swollen from gripping the concrete curb...Toward the last, I risked rising to my feet and running a few yards at a time...Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.[6]
No human life was lost in the collapse of the bridge. Tubby, a black male cocker spaniel, was the only fatality of the Tacoma Narrows Bridge disaster; he was lost along with Coatsworth's car. Professor Farquharson[7] and a news photographer[8] attempted to rescue Tubby during a lull, but the dog was too terrified to leave the car and bit one of the rescuers. Tubby died when the bridge fell, and neither his body nor the car were ever recovered

7. Forced Oscillations
Now we add a driving force to a damped harmonic oscillator
Suppose driving force is sinusoidal with driving frequency ωd
Compare driving frequency with ‘natural frequency’
If ωd ≃ ωʹ then system oscillates with resonant behaviour: amplitude gets very large
§14.8

9. Next lecture
Mechanical waves
Read §15.1–15.2