Chapter 13 Oscillations

1
Fundamentals of Physics

• Chapter 13 Oscillations
• Oscillations
• Simple Harmonic Motion
• Velocity of SHM
• Acceleration of SHM
• The Force Law for SHM
• Energy in SHM
• An Angular Simple Harmonic Oscillator
• Pendulums
• The Simple Pendulum
• The Physical Pendulum
• Measuring g
• SHM Uniform Circular Motion
• Damped SHM
• Forced Oscillations Resonance

2
Oscillations

• Oscillations - motions that repeat themselves.
• Oscillation occurs when a system is disturbed
  from a position of stable equilibrium.
• Clock pendulums swing
• Boats bob up and down
• Guitar strings vibrate
• Diaphragms in speakers
• Quartz crystals in watches
• Air molecules
• Electrons
• Etc.

3
Oscillations

• Oscillations - motions that repeat themselves.

4
Simple Harmonic Motion

• Harmonic Motion - repeats itself at regular
  intervals (periodic).
• Frequency - of oscillations per second
• 1 oscillation / s 1 hertz (Hz)
• Period - time for one complete oscillation
  (one cycle)

T
T
5
Simple Harmonic Motion
Position
Time
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Simple Harmonic Motion
Angles are in radians.
7
Amplitude, Frequency Phase
The frequency of SHM is independent of the
amplitude.
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Velocity Acceleration of SHM
The phase of v(t) is shifted ¼ period relative
to x(t),
In SHM, a(t) is proportional to x(t) but opposite
in sign.
9
The Force Law for SHM

• Simple Harmonic Motion is the motion
  executed by a particle of mass m subject to a
  force proportional to the displacement of the
  particle but opposite in sign.

Hookes Law
Linear Oscillator F - x
SimpleHarmonicMotion/HorizSpring.html
10
The Differential Equation that Describes SHM

• Simple Harmonic Motion is the motion
  executed by a particle of mass m subject to a
  force proportional to the displacement of the
  particle but opposite in sign. Hookes
  Law!

Newtons 2nd Law
The general solution of this differential
equation is
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What is the frequency?
k 7580 N/m m 0.245 kg f ?
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xm without m falling off?
m 1.0 kg M 10 kg k 200 N/m ms
0.40 Maximum xm without slipping
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Simple Harmonic Motion
SimpleHarmonicMotion/HorizSpring.html
14
Vertical Spring Oscillations
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Energy in Simple Harmonic Motion
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Energy in Simple Harmonic Motion
turning point
turning point
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Gravitational Pendulum
Simple Pendulum a bob of mass m hung on an
unstretchable massless string of length L.
SimpleHarmonicMotion/pendulum
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Simple Pendulum
Simple Pendulum a bob of mass m hung on an
unstretchable massless string of length L.
acceleration - displacement SHM
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A pendulum leaving a trail of ink
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Physical Pendulum
A rigid body pivoted about a point other than its
center of mass (com). SHM for small q
Pivot Point
acceleration - displacement SHM
Center of Mass
quick method to measure g
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Angular Simple Harmonic Oscillator
Torsion Pendulum t q
Hookes Law
Spring
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Simple Harmonic Motion
Any Oscillating System inertia versus
springiness
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SHM Uniform Circular Motion
The projection of a point moving in uniform
circular motion on a diameter of the circle in
which the motion occurs executes SHM.
The execution of uniform circular motion
describes SHM.
http//positron.ps.uci.edu/dkirkby/music/html/dem
os/SimpleHarmonicMotion/Circular.html
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SHM Uniform Circular Motion
The reference point P moves on a circle of
radius xm. The projection of xm on a diameter of
the circle executes SHM.
radius xm
x(t)
UC Irvine Physics of Music Simple Harmonic Motion
Applet Demonstrations
25
SHM Uniform Circular Motion
The reference point P moves on a circle of
radius xm. The projection of xm on a diameter of
the circle executes SHM.
radius xm
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SHM Uniform Circular Motion
The projection of a point moving in uniform
circular motion on a diameter of the circle in
which the motion occurs executes SHM.
Measurements of the angle between Callisto and
Jupiter Galileo (1610)
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Damped SHM
SHM in which each oscillation is reduced by an
external force.
Restoring Force SHM
Damping Force In opposite direction to
velocity Does negative work Reduces the
mechanical energy
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Damped SHM
differential equation
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Damped Oscillations
2nd Order Homogeneous Linear Differential
Equation
Eq. 15-41
Solution of Differential Equation
where
b 0 ? SHM
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Damped Oscillations
the natural frequency
Exponential solution to the DE
31
Auto Shock Absorbers
Typical automobile shock absorbers are
designed to produce slightly under-damped motion
32
Forced Oscillations
Each oscillation is driven by an external force
to maintain motion in the presence of damping
wd driving frequency
33
Forced Oscillations
Each oscillation is driven by an external force
to maintain motion in the presence of damping.
2nd Order Inhomogeneous Linear Differential
Equation
the natural frequency
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Forced Oscillations Resonance
2nd Order Homogeneous Linear Differential
Equation
Steady-State Solution of Differential Equation


where
w natural frequency wd driving frequency
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Forced Oscillations Resonance
The natural frequency, w, is the frequency of
oscillation when there is no external driving
force or damping.
w natural frequency wd driving frequency
When w wd resonance occurs!
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Oscillations
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Resonance
38
Stop the SHM caused by winds on a high-rise
building
400 ton weight mounted on a spring on a high
floor of the Citicorp building in New York.
The weight is forced to oscillate at the same
frequency as the building but 180 degrees out of
phase.
39
Forced Oscillations Resonance
Mechanical Systems
e.g. the forced motion of a mass on a spring
Electrical Systems
e.g. the charge on a capacitor in an LRC circuit