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Damped Oscillations
(Serway 15.6-15.7)
Physics 1D03 - Lecture 35
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Simple Pendulum
Recall, for a simple pendulum we have the
following equation of motion:
L
d 2
g
 
2
dt
L
Which give us:
θ
T

g
L
------------------------------------------------------------------------Hence:
or:
gT 2
L 2 

4 2
g
mg
2
4

L
2
g  L 
T2
Application - measuring height
- finding variations in g → underground resources
Physics 1D03 - Lecture 35
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SHM and Damping
SHM: x(t) = A cos ωt
Motion continues indefinitely.
Only conservative forces act,
so the mechanical energy is
constant.
Damped oscillator: dissipative
forces (friction, air resistance, etc.)
remove energy from the oscillator,
and the amplitude decreases with
time.
x
t
x
t
Physics 1D03 - Lecture 35
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A damped oscillator has external nonconservative
force(s) acting on the system. A common example
is a force that is proportional to the velocity.
f = bv where b is a constant damping coefficient
F=ma give:
dx
d 2x
 kx  b  m 2
dt
dt
For weak damping (small b), the solution is:
x
x(t )  Ae

b
t
2m
eg: green water
(weak damping)
cos(t   )
A e-(b/2m)t
t
Physics 1D03 - Lecture 35
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k
Without damping: the angular frequency is  0 
m
2
k  b 
2
 b 

With damping:  
  0  

m  2m 
 2m 
2
Effectively, the frequency ω is slower with damping, and
the amplitude gets smaller (decays exponentially) as
time goes on:
A(t )  Ao e

b
t
2m
Physics 1D03 - Lecture 35
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Example:
A mass on a spring oscillates with initial
amplitude 10 cm. After 10 seconds, the
amplitude is 5 cm.
Question: What is the value of b/(2m)?
Question: What is the amplitude after 30 seconds?
Physics 1D03 - Lecture 35
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Example: A pendulum of length 1.0 m has an initial
amplitude 10°, but after a time of1000 s it is reduced to
5°. What is b/(2m)?
Physics 1D03 - Lecture 35
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Types of Damping
Since:
 b 
2
  0   
 2m 
2
water
We can have different cases for the value under
the root: >0, 0 or <0. This leads to three types of
damping !
Eg: Strong damping (b large): there is no
oscillation when:
b
 0
2m
axle grease?
Physics 1D03 - Lecture 35
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b  2m 0 : “Underdamped”, oscillations with decreasing amplitude
b  2m 0 : “Critically damped”
b  2m 0 : “Overdamped”, no oscillation
x(t)
Critical damping
provides the fastest
dissipation of energy.
overdamped
critical damping
t
underdamped
Physics 1D03 - Lecture 35
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Quiz: If you were designing a automobile suspensions:
Should they be:
a) Underdamped
b) Critically damped
c) Overdamped
d) I don’t need suspension !
Physics 1D03 - Lecture 35
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Example:
Show that the time rate of change of mechanical energy for a
damped oscillator is given by:
dE/dt=-bv2
and hence is always negative.
Physics 1D03 - Lecture 35
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Damped Oscillations and Resonance
(Serway 15.6-15.7)
• Forced Oscillations
• Resonance
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Forced Oscillations
A periodic, external force pushes on the mass
(in addition to the spring and damping):
Fext (t )  Fmax cos t
This transfers energy into the system
The frequency  is set by the machine applying
the force. The system responds by oscillating at
the same frequency . The amplitude can be
very large if the external driving frequency is
close to the “natural” frequency of the oscillator.
Physics 1D03 - Lecture 35
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k
0 
m
is called the natural frequency or resonant
frequency of the oscillation.
Newton’s 2nd Law:
dx
d 2x
F  Fmax cos t  kx  b  m 2
dt
dt
Assume that : x = A cos (t + ) ; then the
amplitude of a drive oscillator is given by:
Fmax
A

2
 0

m
2 2
 b 
2


 m
Physics 1D03 - Lecture 35
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Fmax
A
So, as

2
 0

m
2 2
 b 


 m
2
   0 the amplitude, A, increases !
If the external “push” has the same frequency
as the resonant frequency ω0. The driving
force is said to be in resonance with the system.
A
0

Physics 1D03 - Lecture 35
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Resonance occurs because the driving force changes direction at just
the same rate as the “natural” oscillation would reverse direction, so
the driving force reinforces the natural oscillation on every cycle.
Quiz: Where in the cycle should the driving force be at
its maximum value for maximum average power?
A) When the mass is at maximum x (displacement)
B) When the mass is at the midpoint (x = 0)
C) It matters not
Physics 1D03 - Lecture 35
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Example
A 2.0 kg object attached to a spring moves without
friction and is driven by an external force given by:
F=(3.0N)sin(8πt)
If the force constant of the spring is 20.0N/m, determine
a) the period of the motion
b) the amplitude of the motion
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Summary
• An oscillator driven by an external periodic force will oscillate
with an amplitude that depends on the driving frequency. The
amplitude is large when the driving frequency is close to the
“natural” frequency of the oscillator.
•For weak damping, the system oscillates, and the amplitude
decreases exponentially with time.
• With sufficiently strong damping, the system returns smoothly
to equilibrium without oscillation.
Problems, Chapter 15
42, 43, 70
(6th ed – Chapter 13)
42, 70
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