1. PHY238Y Lecture 3 Damped oscillations Forced oscillations. Resonance References: Halliday, Resnick, Walker: Fundamentals of Physics, 6 th edition, John Wiley 2003, Chapter 16 (16.8) Lecture notes

2. PHY238Y Lecture 3 What we did last time: Studied a simple (geometrical) pendulum Studied a heavy (physical) pendulum Used:  Restoring force F or torque τ;  Newton’s Law:  F = ma or: τ = I α Wrote the equation of motion: Solution:

3. PHY238Y Lecture 3 Damped SHM : - a damping force (usually friction) is exerted upon the oscillator: - friction force is proportional to the velocity: F = -bv - Newton’s Second Law: -bv – kx = ma

4. PHY238Y Lecture 3 The amplitude of the cosine function decreases with time due to the exponential factor

5. PHY238Y Lecture 3 Forced oscillations. Resonance The spring-mass system is driven by an external force F d ; Equation of motion:

6. PHY238Y Lecture 3 Forced oscillations. Resonance Two angular frequencies associated with a system undergoing forced oscillations: -The natural angular frequency ω 0 : -The angular frequency ω of the external driving force F d :

7. PHY238Y Lecture 3 Resonance – examples: The Tacoma Narrows Bridge collapse Java applet: http://www.walter-fendt.de/ph14e/resonance.htm