PHY238Y
Lecture 2
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Pendulums:
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- simple, geometrical
- heavy, physical
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SHM and uniform circular motion: phasors
Damped SHM
References: Halliday, Resnick , Walker: Fundamentals of Physics, 6 th edition,
John Wiley 2003, Chapter 16 (16.5, 16.6, 16.8); Chapter 17 (17.10)
PHY238Y
Lecture 2
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What did we do last time:
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Studied a simple harmonic oscillator: a
spring -mass system
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Used:
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Hooke’s Law force: F = Fe = -kx
Newton’s Law: F = ma
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Wrote the equation of motion:
Tried solution:
Defined: angular frequency, period,
frequency
m
d2 x
= − kx
dt 2
x = Acos ωt
where ω =
k
m
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PHY238Y
Lecture 2
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Simple (geometrical
pendulum)
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Two forces:
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gravitational mg
tension T
Limit the swing to small angles
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Java applet:
http://www.walter-fendt.de/ph14e/pendulum.htm
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PHY238Y
Lecture 2
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Heavy (physical) pendulum:
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Mass is distributed and not
point-like;
A restoring torque appears
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PHY238Y
Lecture 2
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-
-
SHM and uniform
circular motion:
a reference particle P’
moving in uniform
circular motion in a
reference circle
projection of particle
onto the axis will be P
which will move in SHM
PHY238Y
Lecture 2
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Damped SHM :
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a damping force (usually friction) is exerted upon the oscillator:
friction force is proportional to the velocity: F = -bv
-
à
Friction
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Newton’s Second Law:
-bv – k x = ma
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PHY238Y
Lecture 2
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The amplitude of the cosine function decreases with time due to the
exponential factor
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