Transcript PHY238Y Lecture 3 - University of Toronto

PHY238Y
Lecture 3


Damped oscillations
Forced oscillations. Resonance
References:
Halliday, Resnick, Walker: Fundamentals of Physics, 6th edition, John
Wiley 2003, Chapter 16 (16.8)
Lecture notes
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:
d 2
2




2
dt
   m cost where  
g
m gh
or :  
L
I
PHY238Y
Lecture 3

Damped SHM:
-
a damping force (usually friction) is exerted upon the oscillator:
friction force is proportional to the velocity: F = -bv
-

Friction
-
Newton’s Second Law:
-bv – kx = ma
PHY238Y
Lecture 3

The amplitude of the cosine function decreases with time due to the
exponential factor
PHY238Y
Lecture 3

Forced oscillations.
Resonance

The spring-mass system is
driven by an external force Fd;

Motor

Equation of motion:
ma  Fspring ,  Fdriving
elastic
PHY238Y
Lecture 3

Forced oscillations.
Resonance

Two angular frequencies
associated with a system
undergoing forced oscillations:
-
The natural angular frequency ω0:
-
The angular frequency ω of the
external driving force Fd:
k
0 
m
Fd  F0 cost
PHY238Y
Lecture 3

Resonance – examples:

The Tacoma Narrows Bridge collapse

Java applet: http://www.walter-fendt.de/ph14e/resonance.htm