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```					  Fundamentals of Physics
Chapter 13 Oscillations
1.   Oscillations
2.   Simple Harmonic Motion
Velocity of SHM
Acceleration of SHM
3.   The Force Law for SHM
4.   Energy in SHM
5.   An Angular Simple Harmonic Oscillator
6.   Pendulums
The Simple Pendulum
The Physical Pendulum
Measuring “g”
7.   SHM & Uniform Circular Motion
8.   Damped SHM
9.   Forced Oscillations & Resonance

Physics 2111, 2008                          Fundamentals of Physics   1
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.

Physics 2111, 2008                         Fundamentals of Physics                      2
Oscillations

Oscillations - motions that repeat themselves.

Physics 2111, 2008                      Fundamentals of Physics   3
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)

1
T
f

T

T

Physics 2111, 2008                             Fundamentals of Physics                 4
Simple Harmonic Motion

Position

Physics 2111, 2008               Fundamentals of Physics   5
Simple Harmonic Motion

  2 f
                 cycles 
s              cycle  
        s

1
T 
f
T  2

Physics 2111, 2008              Fundamentals of Physics                                                  6
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 

Physics 2111, 2008              Fundamentals of Physics                                    7
Velocity & Acceleration of SHM

xt   xm cos t   

vt  
dx
dt
vt     xm sin  t   

vmax   xm

The phase of v(t) is shifted ¼ period relative to x(t),

at  
dv

amax    2 xm                             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.

Physics 2111, 2008                         Fundamentals of Physics                                     8
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

Physics 2111, 2008                          Fundamentals of Physics                            9
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                                                k
 2 x  0                       where  2 
dt 2                                                m
The general solution of this differential equation is:
xt   xm cos t   

Physics 2111, 2008                          Fundamentals of Physics                              10
What is the frequency?

k = 7580 N/m
m = 0.245 kg
f=?

Physics 2111, 2008               Fundamentals of Physics   11
xm without m falling off?

m = 1.0 kg
M = 10 kg
k = 200 N/m
ms = 0.40
Maximum xm without slipping

Physics 2111, 2008                      Fundamentals of Physics   12
Simple Harmonic Motion

SimpleHarmonicMotion/HorizSpring.html

Physics 2111, 2008                     Fundamentals of Physics   13
Vertical Spring Oscillations

Physics 2111, 2008                   Fundamentals of Physics   14
Energy in Simple Harmonic Motion

K    1
2   m v2        U          1
2   k x2
E  K U

K t         2 m   xm sin t                    U t              2 k  xm cos t   
1                                       2                                                 2
1

k  m 2                                   U t          1       k xm cos2  t   
2
2
K t     1
2   k xm sin 2  t   
2

 E            1
2 k xm  constant
2

Physics 2111, 2008                                    Fundamentals of Physics                                        15
Energy in Simple Harmonic Motion

U    1
2 k x2
range
of
motion

E    1
2 k xm  constant
2

turning            turning
point              point

Physics 2111, 2008                       Fundamentals of Physics                            16
Gravitational Pendulum

Simple Pendulum: a bob of mass m hung on an unstretchable massless string
of length L.

SimpleHarmonicMotion/pendulum

Physics 2111, 2008                Fundamentals of Physics                        17
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  m L2
I

acceleration ~ - displacement
SHM
a  t     2 x t 
2
T


L
T  2
g
SHM for small q

Physics 2111, 2008                     Fundamentals of Physics                                       18
A pendulum leaving a trail of ink:

Physics 2111, 2008               Fundamentals of Physics   19
Physical Pendulum

A rigid body pivoted about a point other than its center of mass (com).                 SHM for
small q

   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

2
T

I
T  2
quick method to measure g                             mgh

Physics 2111, 2008                     Fundamentals of Physics                                        20
Angular Simple Harmonic Oscillator

Torsion Pendulum:    ~ q

 q                 Hooke’s Law

d 2q
  I  I 2  q
dt
I
T  2


Spring:            m  I
k 

Physics 2111, 2008                     Fundamentals of Physics                     21
Simple Harmonic Motion

m                    L
T  2                      T  2                  T  2
I
T  2
I
k                    g                            mgh            

Any Oscillating System:
“inertia” versus “springiness”

inertia
T  2
springiness

Physics 2111, 2008                           Fundamentals of Physics                      22
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

Physics 2111, 2008                         Fundamentals of Physics                                 23
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.

angle   t  

x(t)

xt   xm cos t   

UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations

Physics 2111, 2008                                 Fundamentals of Physics                           24
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)                              v(t)                           a(t)

xt   xm cos t            vt     xm sin  t      at     2 xm cos t   
                               
a    2 xm
radius = xm                        v   xm

Physics 2111, 2008                          Fundamentals of Physics                                     25
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

Physics 2111, 2008                     Fundamentals of Physics                                  26
Damped SHM

SHM in which each oscillation is reduced by an external force.

Restoring Force
F  k x
SHM

FD   b v        Damping Force
In opposite direction to velocity
Does negative work
Reduces the mechanical energy

Physics 2111, 2008                  Fundamentals of Physics                                27
Damped SHM

Fnet  m a

 k x bv  m a

dx    d 2x
 k x b     m 2
dt    dt

differential equation

Physics 2111, 2008        Fundamentals of Physics                                   28
Damped Oscillations

2nd Order Homogeneous Linear Differential Equation:
d 2x       dx
m 2  kx b     0                                               Eq. 15-41
dt         dt

b

cos  t   
t
Solution of Differential Equation:
x ( t )  xm e       2m

k   b2
where:                    
m 4 m2

b = 0  SHM

Physics 2111, 2008                         Fundamentals of Physics                                            29
Damped Oscillations

b

cos  t   
t                                                2
x ( t )  xm e       2m                                        b 
   1        
 2 m 

k
 
m

b
 1        small damping                 “the natural frequency”
2m

b
 1          0 " critically damped "
2m
b
1             2  0 "overdamped "
2m
Exponential solution to the DE

Physics 2111, 2008                           Fundamentals of Physics                                 30
Auto Shock Absorbers

Typical automobile shock absorbers are designed to produce
slightly under-damped motion

Physics 2111, 2008                         Fundamentals of Physics                31
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

Physics 2111, 2008                  Fundamentals of Physics                               32
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”

Physics 2111, 2008                      Fundamentals of Physics                           33
Forced Oscillations & Resonance

2nd Order Homogeneous Linear Differential Equation:

d 2x            dx
m 2  k x  m 2     F0 cos d t 
dt              dt

x(t )  xm cos t   
F0
xm 
               
2
m   d                         b2 d
2       2       2                     2
where:
bd
 
k                     tan  
m                                  
m  2  d
2

 = natural frequency
d = driving frequency

Physics 2111, 2008                       Fundamentals of Physics                                          34
Forced Oscillations & Resonance

The natural frequency, , is the frequency of oscillation when
there is no external driving force or damping.

F0
xm                                                                     less damping

m2  2   d    
2 2
 b2  d
2

k
 
m

 = natural frequency
more damping
d = driving frequency

When  = d resonance occurs!

Physics 2111, 2008                             Fundamentals of Physics                              35
Oscillations

Physics 2111, 2008   Fundamentals of Physics   36
Resonance

Physics 2111, 2008   Fundamentals of Physics   37
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.

Physics 2111, 2008                     Fundamentals of Physics                            38
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

Physics 2111, 2008                       Fundamentals of Physics                   39

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