Fundamentals of Physics
Chapter 13 Oscillations
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2.
3.
4.
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6.
7.
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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
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Oscillations
Oscillations - motions that repeat themselves.
Oscillation occurs when a system is disturbed from a position of
stable equilibrium.
–
–
–
–
–
–
–
–
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Clock pendulums swing
Boats bob up and down
Guitar strings vibrate
Diaphragms in speakers
Quartz crystals in watches
Air molecules
Electrons
Etc.
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Oscillations
Oscillations - motions that repeat themselves.
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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
1
f
T
T
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Simple Harmonic Motion
Position
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Simple Harmonic Motion
  2 f
radians
s
  cycles 
  radians
cycle
s

 
T 
1
f
T  2
Angles are in radians.
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Amplitude, Frequency & Phase
xm  xm
xt   xm cos t 
xt   xm cos t 
xt   xm cos2  t 
The frequency of SHM is
independent of the amplitude.
x  t   xm cos  t   4 
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Velocity & Acceleration of SHM
xt   xm cos t   
vt  

vmax   xm
dx
dt
vt     xm sin  t   
The phase of v(t) is shifted ¼ period relative to x(t),

amax    2 xm
at  
dv
dt
at     2 xm cos t   
at     2 x t 
In SHM, a(t) is proportional to x(t) but opposite in sign.
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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.
Hooke’s Law:
F t    k x t 
“Linear Oscillator”:
F ~ -x
SimpleHarmonicMotion/HorizSpring.html
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The Differential Equation that Describes SHM
F t    k xt 
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.
Hooke’s Law!
Newton’s 2nd Law:
F  ma
d 2x
m 2  k x
dt
d 2x
2


x  0
2
dt
where  2 
k
m
The general solution of this differential equation is:
xt   xm cos t   
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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
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Vertical Spring Oscillations
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Energy in Simple Harmonic Motion
K 
1
2
U 
m v2
1
2
k x2
E  K U
K t  
1
2 m   xm sin  t   
2
k  m 2
K t  
1
2
k xm2 sin 2  t   
 E 
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1
U t  
1
U t  
1
2 k  xm cos t   
2
2
k xm2 cos 2  t   
2
k
x
2
m  constant
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Energy in Simple Harmonic Motion
U 
1
E 
1
2
k
x
2
range
of
motion
turning
point
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2
k
x
2
m  constant
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.
   L Fg sin q   L Fg q
  I

mg L
q
I
I  m L2
acceleration ~ - displacement
SHM
a  t     2 x t 
T
2

T  2
SHM for small q
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L
g
18
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).
small q
SHM for
   h Fg sin q   h Fg q
  I
Pivot Point
mg h
q
I
acceleration ~ - displacement
SHM

a  t     2 x t 
Center of Mass
T
quick method to measure g
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2

T  2
I
mg h
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Angular Simple Harmonic Oscillator
Torsion Pendulum:
 ~ q
 q
Hooke’s Law
d 2q
  I  I 2  q
dt
T  2
Spring:
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
m  I
k 
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Simple Harmonic Motion
T  2
m
k
T  2
L
g
T  2
I
mg h
T  2
I

Any Oscillating System:
“inertia” versus “springiness”
T  2
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inertia
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/demos/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
angle   t  
x(t)
xt   xm cos t   
UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations
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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.
x(t)
xt   xm cos t   
radius = xm
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v(t)
vt     xm sin  t   

v   xm
Fundamentals of Physics
a(t)
at     2 xm cos t   

a    2 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)
planet
earth
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Damped SHM
SHM in which each oscillation is reduced by an external force.
F  k x
FD   b v
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Restoring Force
SHM
Damping Force
In opposite direction to velocity
Does negative work
Reduces the mechanical energy
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Damped SHM
Fnet  m a
 k x bv  ma
dx
d 2x
 k x b
 m 2
dt
dt
differential equation
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Damped Oscillations
2nd Order Homogeneous Linear Differential Equation:
d 2x
dx
m 2  kx b
 0
dt
dt
Solution of Differential Equation:
Eq. 15-41
x ( t )  xm e
where:

b
t
2m
 
cos  t   
b2
k

m 4 m2
b = 0  SHM
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Damped Oscillations
x ( t )  xm e

b
t
2m
cos  t   
 b 
   1  

 2 m 
 
b
 1
2m
    small damping
2
k
m
“the natural frequency”
b
 1
2m
   0 " critically damped "
b
1
2m
 2  0 "overdamped "
Exponential solution to the DE
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Auto Shock Absorbers
Typical automobile shock absorbers are designed to produce
slightly under-damped motion
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Forced Oscillations
Each oscillation is driven by an external force to maintain motion in the presence of
damping:
F0 cos  d t 
d = driving frequency
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Forced Oscillations
Each oscillation is driven by an external force to maintain motion in the presence of
damping.
Fnet  m a
 k x  b v  F0 cos d t   m a
2nd Order Inhomogeneous Linear Differential Equation:
d 2x
2 dx
m 2  k x  m
 F0 cos d t 
dt
dt
k
m
“the natural frequency”
 
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Forced Oscillations & Resonance
2nd Order Homogeneous Linear Differential Equation:
d 2x
dx
m 2  k x  m 2
 F0 cos d t 
dt
dt
Steady-State Solution of Differential Equation:
x (t )  xm cos  t   
xm 
 
k
m
tan  

m   d
2
where:
F0
2
bd
2
m  2  d

2

2
 b2 d
2

 = natural frequency
d = driving frequency
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Forced Oscillations & Resonance
The natural frequency, , is the frequency of oscillation when
there is no external driving force or damping.
xm 

F0
m2  2   d
 
2

2
 b2  d
less damping
2
k
m
 = natural frequency
d = driving frequency
more damping
When  = d resonance occurs!
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Oscillations
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Resonance
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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.
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Forced Oscillations & Resonance
Mechanical Systems
d 2x
dx
m 2  k x  m 2
 F0 cos d t 
dt
dt
e.g. the forced motion of a mass on a spring
Electrical Systems
d 2q
dq 1
L 2 R
 q  Em sin d t
dt
dt C
e.g. the charge on a capacitor in an LRC circuit
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