Chapter 14
Oscillations
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14-7 Damped Harmonic Motion
Damped harmonic motion is harmonic
motion with a frictional or drag force. If the
damping is small, we can treat it as an
envelope that modifies the undamped
oscillation.
If
then
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14-7 Damped Harmonic Motion
This gives
If b is small, a solution of the form
will work, with
Try it; like we did in class yesterday.
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14-7 Damped Harmonic Motion
If b2 > 4mk, ω becomes imaginary, and the
system is overdamped (C).
For b2 = 4mk, the system is critically damped (B)
—this is the case in which the system reaches
equilibrium in the shortest time.
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14-7 Damped Harmonic Motion
There are systems in which
damping is unwanted, such as
clocks and watches.
Then there are systems in
which it is wanted, and often
needs to be as close to critical
damping as possible, such as
automobile shock absorbers
and earthquake protection for
buildings.
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14-7 Damped Harmonic Motion
A damped harmonic oscillator loses 6.0% of
its mechanical energy per cycle.
a)  By what percentage does its frequency
differ from the natural frequency
(
f0 = 1 2!
)
k m ? !f f0 = 0.0012%
b) After how many periods will the amplitude
have decreased to 1/e of its original value?
(
)
n = !1 ln 0.94 " 16
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14-8 Forced Oscillations; Resonance
Forced vibrations occur when there is a periodic
driving force. This force may or may not have the
same period as the natural frequency of the
system.
If the frequency is the same as the natural
frequency, the amplitude can become quite large.
This is called resonance.
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14-8 Forced Oscillations; Resonance
The sharpness of the
resonant peak depends
on the damping. If the
damping is small (A) it
can be quite sharp; if
the damping is larger
(B) it is less sharp.
Like damping, resonance can be wanted or
unwanted. Musical instruments and TV/radio
receivers depend on it.
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14-8 Forced Oscillations; Resonance
The equation of motion for a forced
oscillator is:
The solution after a a long time is:
where
and
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14-8 Forced Oscillations; Resonance
The width of the
resonant peak can be
characterized by the Q
factor:
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Summary of Chapter 14
•  For SHM, the restoring force is proportional to
the displacement.
•  The period is the time required for one cycle,
and the frequency is the number of cycles per
second.
•  Period for a mass on a spring:
•  SHM is sinusoidal.
•  During SHM, the total energy is continually
changing from kinetic to potential and back.
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Summary of Chapter 14
•  A simple pendulum approximates SHM if its
amplitude is not large. Its period in that case is:
•  When friction is present, the motion is
damped.
•  If an oscillating force is applied to a SHO, its
amplitude depends on how close to the natural
frequency the driving frequency is. If it is close,
the amplitude becomes quite large. This is
called resonance.
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Group question:
A 2.00-kg mass oscillates on the end of a
spring with spring constant 12.0 N/m. Its
amplitude of oscillation decreases from
10.0 cm to 1.0 cm in 4.00 minutes. What is
the linear damping coefficient of this
oscillator?
A) 134 N·s/m
B) 1.76 N·s/m
C) 0.311 N·s/m
D) 0.0384 N·s/m
E) 0.622 N·s/m
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Group question:
A mass of 2.0 kg hangs from a spring with
a force constant of 50 N/m. An oscillating
force F = (4.8 N) cos[(3.0 rad/s)t] is applied
to the mass. What is the amplitude of the
resulting oscillations? Neglect damping.
A) 0.15 m
B) 0.30 m
C) 1.6 m
D) 2.4 m
E) 0.80 m
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Chapter 15
Wave Motion
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Units of Chapter 15
•  Characteristics of Wave Motion
•  Types of Waves: Transverse and Longitudinal
•  Energy Transported by Waves
•  Mathematical Representation of a Traveling
Wave
• The Principle of Superposition
•  Reflection and Transmission
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Units of Chapter 15
•  Interference
•  Standing Waves; Resonance
•  Refraction
•  Diffraction
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15-1 Characteristics of Wave Motion
All types of traveling waves transport energy.
Study of a single wave
pulse shows that it is
begun with a vibration
and is transmitted
through internal forces in
the medium.
Continuous waves start
with vibrations, too. If the
vibration is SHM, then the
wave will be sinusoidal.
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15-1 Characteristics of Wave Motion
Wave characteristics:
•  Amplitude, A
•  Wavelength, λ
•  Frequency, f and period, T
•  Wave velocity,
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15-2 Types of Waves: Transverse and
Longitudinal
The motion of particles in a wave can be either
perpendicular to the wave direction (transverse)
or parallel to it (longitudinal).
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15-2 Types of Waves: Transverse and
Longitudinal
Sound waves are longitudinal waves:
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15-2 Types of Waves: Transverse and
Longitudinal
The velocity of a transverse wave on a
cord is given by:
As expected, the
velocity increases
when the tension
increases, and
decreases when
the mass
increases.
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Assignment
Chapter 14 – 60, 87
HAND IN FRIDAY
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