n Damped oscillations n Forced oscillations. Resonancesandra/PHY238Y/Lectures/... · 2003. 9. 11. · 1 PHY238Y Lecture 3 n Damped oscillations n Forced oscillations. Resonance References:

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PHY238YLecture 3

n Damped oscillationsn Forced oscillations. Resonance

References: Halliday, Resnick, Walker: Fundamentals of Physics, 6th edition, John

Wiley 2003, Chapter 16 (16.8)Lecture notes

PHY238YLecture 3

n What we did last time:

n Studied a simple (geometrical) pendulumn Studied a heavy (physical) pendulum

n Used:§ Restoring force F or torque t ;§ Newton’s Law:

§ F = ma or: t = I aWrote the equation of motion:

Solution:

θωθ 22

2

−=dtd

Imgh

orLg

wheretm === ωωωθθ :cos

2

PHY238YLecture 3

n 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

à

Friction

PHY238YLecture 3

n The amplitude of the cosine function decreases with time due to the exponential factor

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PHY238YLecture 3

n Forced oscillations. Resonance

n The spring-mass system isdriven by an external force Fd;

n Equation of motion:

Motor à

drivingelasticspring FFma += ,

PHY238YLecture 3

n Forced oscillations. Resonance

n Two angular frequencies associated with a system undergoing forced oscillations:

- The natural angular frequency ? 0:

- The angular frequency ? of the external driving force Fd:

mk

=0ω

tFFd ωcos0=

4

PHY238YLecture 3

n Resonance – examples:

n The Tacoma Narrows Bridge collapse

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