Physics 121,Lecture 22,Pg 1
Physics 121,Lecture 22
Today’s Agenda
? Announcements
?No class next week
? Homework 9,due Friday Nov,18 @ 6:00 PM.
?Chap,9,# 1,7,9,13,17,23,24,27,29,37,39,and 40.
? Today’s topics
?SHM
?Pendulum
?Damped oscillations
?Resonance
?Waves
Physics 121,Lecture 22,Pg 2
The Simple Pendulum
? A pendulum is made by suspending a mass m at
the end of a string of length L,
? Find the frequency of
oscillation for small displacements.
? L
m
mg
z
Physics 121,Lecture 22,Pg 3
Aside,sin ? and cos ? for small ?
? A Taylor expansion of sin ? and cos? about ? = 0 gives:
...!5!3s in 53 ????????,..!4!21c os 42 ???????and
So for ? <<1,???sin 1cos ??and
Physics 121,Lecture 22,Pg 4
The Simple Pendulum...
? Recall that the torque due to gravity about the rotation (z)
axis is ? = Ia = -mgd,
d = Lsin ?? L? for small ?
so ? = -mg L?
? But ? = Ia???I?=?mL2
? L
d
m
mg
z
a? 2mLm g L ??
??a 2?? ? ?
g
Lwhere
Same equation as for simple harmonic motion !
? = ?0 cos(?t + ?)
Physics 121,Lecture 22,Pg 5
The Rod Pendulum
? A pendulum is made by suspending a thin rod of
length L and mass M at one end,
? Find the frequency of
oscillation for small displacements.
?
L
mg
z
x CM
Physics 121,Lecture 22,Pg 6
The Rod Pendulum...
? The torque about the rotation (z) axis is
? = -mgd = -mg{L/2}sin??? -mg{L/2}??????for small ?
? In this case
? So ? = Ia??becomes
?
Ld
mg
z
L/2
xCM
I ? 13 2mL
a? 2312 mLLmg ??
??a 2?? ? ? 32gLwhere
d I
Physics 121,Lecture 22,Pg 7
Lecture 22,Act 1
Period
(a) (b) (c)
? What length do we make the simple pendulum so that
it has the same period as the rod pendulum?
LR
LS
RS L3
2L ?RS L
2
3L ?
RS LL ?
Physics 121,Lecture 22,Pg 8
General Physical Pendulum
? Suppose we have some arbitrarily shaped
solid of mass M hung on a fixed axis,that we
know where the CM is located and what the
moment of inertia I about the axis is.
? The torque about the rotation (z) axis for
small ? is (sin ? ? )
? = -Mgd -MgR?? ?
d
Mg
z-axis
R
xCM
d
dt
2
2
2? ? ?? ?
? ? MgRIwhere
? = ?0 cos(?t + ?)
2
2
dt
dIM g R ????
? a
?
?
Physics 121,Lecture 22,Pg 9
Lecture 22,Act 2
Physical Pendulum
? A pendulum is made by hanging a thin hoola-hoop of
diameter D on a small nail,
?What is the angular frequency of oscillation of the hoop
for small displacements? (ICM = mR2 for a hoop)
(a)
(b)
(c)
?? gD
? ? 2gD
? ? g2D
D
pivot (nail)
Physics 121,Lecture 22,Pg 10
Torsion Pendulum
? Consider an object suspended by a wire
attached at its CM,The wire defines the
rotation axis,and the moment of inertia I
about this axis is known,
? The wire acts like a,rotational spring”.
?When the object is rotated,the wire is
twisted,This produces a torque that
opposes the rotation.
?In analogy with a spring,the torque
produced is proportional to the
displacement,? = -k?
I
wire
?
?
Physics 121,Lecture 22,Pg 11
Torsion Pendulum...
? Since ? = -k???? = Ia??becomes
??
? k ? ? Ia ? I d
2?
dt 2
I
wire
?
?
Similar to,mass on spring”,except
I has taken the place of m (no surprise)
d
dt
2
2
2? ? ?? ?
? ? kIwhere
Physics 121,Lecture 22,Pg 12
Lecture 22,Act 3
Period
? All of the following pendulum bobs have the
same mass,Which pendulum rotates the fastest,
i.e,has the smallest period? (The wires are
identical)
RRRR
A) B) C) D)
Physics 121,Lecture 22,Pg 13
SHM So Far
? The most general solution is x = Acos(?t + ?)
where A = amplitude
? = frequency
? = phase
? For a mass on a spring
? For a general pendulum
? For a torsion pendulum
? ? km
? ? MgRI
? ? kI
Physics 121,Lecture 22,Pg 14
Velocity and Acceleration
k
x
m
0
Position,x(t) = Acos(?t + ?)
Velocity,v(t) = -?Asin(?t + ?)
Acceleration,a(t) = -?2Acos(?t + ?)
Starting with
angle ? ant
t=0
xMAX = A
vMAX = ?A
aMAX = ?2A
? Equations for x,v,and a
Physics 121,Lecture 22,Pg 15
Energy of the Spring-Mass System
Add to get E = K + U
1/2 m (?A)2sin2(?t + ?) + 1/2 k (Acos(?t + ?))2
Remember that
? ?? ?
m
k
m
k ?? ? 2??
U~cos2 K~sin2
E = 1/2 kA2
so,E = 1/2 kA2 sin2(?t + ?) + 1/2 kA2 cos2(?t + ?)
= 1/2 kA2 [ sin2(?t + ?) + cos2(?t + ?)]
= 1/2 kA2
Active
Figure
Physics 121,Lecture 22,Pg 16
Energy in SHM
? For both the spring and the pendulum,we
can derive the SHM solution using energy
conservation,
? The total energy (K + U) of a
system undergoing SMH will
always be constant!
? This is not surprising since
there are only conservative
forces present,hence energy is conserved.
-A A0 s
U
U
K
E
Physics 121,Lecture 22,Pg 17
What about Friction?
? Friction causes the oscillations to get
smaller over time
? This is known as DAMPING.
? As a model,we assume that the force due
to friction is proportional to the velocity.
? The solution is dt
dxbF
f ??
)c o s ()2e x p ( ?? ??? tmbtAx
with,
??
? ? km ? b2 m?????? ??????
2
? ? o2 ? ? 2
??
?0 ? km
??
? ? b2mwhere and
Physics 121,Lecture 22,Pg 18
What about Friction?
)c o s ()2e x p ( ?? ??? tmbtAx
What does this function look like?
(You saw it in lab,it really works)
?? ?o ?? ?o ?? ?o
underdamped critically damped overdamped
Active
Figure
Physics 121,Lecture 22,Pg 19
Forced vibrations and Resonance
? To replace the energy lost to friction,we can drive the motion
with a periodic force
? The solution has the form
? The amplitude depends on ?
? Note,that A gets bigger if Fo does,and gets smaller if b or m
gets bigger,No surprise there.
? Something more surprising happens if you drive the pendulum
at exactly the frequency it wants to go,???k/m
? Then at least one of the terms in the denominator vanishes
and the amplitude gets real big,This is known as resonance.
F = F0 cos(?t)
??
x ( t ) ? A c o s (? t? ? )
??
A ? F 0 / m
(? 2 ? ? 02 ) 2 ? b ?
m
??
??
??
??
??
??
2
Physics 121,Lecture 22,Pg 20
Resonance
? Now,consider what b does,
?????0
b small
b middling
b large
???
??
A ? F 0 / m
(? 2 ? ? 02 ) 2 ? b ?
m
??
??
??
??
??
??
2
Physics 121,Lecture 22,Pg 21
Dramatic example of resonance
? In 1940,turbulent winds set up a torsional vibration in the Tacoma Narrow
Bridge
?
Physics 121,Lecture 22,Pg 22
Dramatic example of resonance
?
? when it reached the natural frequency
Physics 121,Lecture 22,Pg 23
Dramatic example of resonance
?
? it collapsed !
Other example,London Millenium Bridge
Physics 121,Lecture 22,Pg 24
Lecture 22,Act 4
Resonant Motion
? Consider the following set of pendula all attached to the
same string
D
A
B
CIf I start bob D swinging which of the
others will have the largest swing amplitude?
(A) (B) (C)
Physics 121,Lecture 22,Pg 25
Chapter 11,Waves
? A definition of a wave:
?A wave is a traveling disturbance that
transports energy but not matter.
? Examples:
?Sound waves (air moves back & forth)
?Stadium waves (people move up & down)
?Water waves (water moves up & down)
?Light waves (what moves)
Physics 121,Lecture 22,Pg 26
Types of Waves
? Transverse,The medium oscillates perpendicular
to the direction the wave is moving.
?Water (more or less)
?String waves
? Longitudinal,The medium oscillates in the
same direction as the wave is moving
?Sound
?Slinky
Physics 121,Lecture 22,Pg 27
Wave Properties
?
Wavelength
? Wavelength,The distance ? between identical points on the wave.
Amplitude A
? Amplitude,The maximum displacement A of a point on the wave.
A
Physics 121,Lecture 22,Pg 28
Wave Properties...
? Period,The time T for a point on the wave to
undergo one complete oscillation.
? Speed,The wave moves one wavelength ? in one
period T so its speed is v = ??/ T.
Tv
??
Physics 121,Lecture 22,Pg 29
Wave Properties...
? The speed of a wave is a constant that depends only
on the medium,not on amplitude,wavelength or
period
? and T are related !
v = ? / T
? ? = v T or ? = 2? v / ??????????????(since?T = 2? / ???
or ???v / f (since T = 1/ f )
? Recall f = cycles/sec or revolutions/sec
? ???rad/sec = 2?f
Physics 121,Lecture 22,Pg 30
Lecture 22,Act 5
Wave Motion
? The speed of sound in air is a bit over 300 m/s,and
the speed of light in air is about 300,000,000 m/s,
? Suppose we make a sound wave and a light wave
that both have a wavelength of 3 meters,
?What is the ratio of the frequency of the light wave
to that of the sound wave?
(a) About 1,000,000
(b) About,000,001
(c) About 1000
Physics 121,Lecture 22,Pg 31
Wave Forms
? So far we have examined
“continuous waves” that go
on forever in each direction !
v
v ? We can also have,pulses”
caused by a brief disturbance
of the medium:
v
? And,pulse trains” which are
somewhere in between.
Physics 121,Lecture 22,Pg 32
Mathematical Description
? Suppose we have some function y = f(x):
x
y
? f(x-d ) is just the same shape moved
a distance d to the right:
x
y
x=d0
0
? Let d =vt Then
f(x-vt) will describe the same
shape moving to the right with
speed v,x
y
x=vt 0
v
Physics 121,Lecture 22,Pg 33
Math...
? Consider a wave that is harmonic
in x and has a wavelength of ?.
? ? ?????? ??? x2c o sAxyIf the amplitude is maximum atx=0 this has the functional form:
y
x
?
A
? Now,if this is moving to
the right with speed v it will be
described by:
y
x
v
? ? ? ??????? ???? vtx2c osAt,xy
Physics 121,Lecture 22,Pg 34
Math...
? ? ? ??????? ???? vtx2c osAt,xy
? By using v T? ?? ???2from before,and by defining k ?2??
? So we see that a simple harmonic
wave moving with speed v in the x
direction is described by the equation:
we can write this as,? ? ? ?tkxc o sAt,xy ???
(what about moving in the -x direction?)
Physics 121,Lecture 22,Pg 35
Math Summary
? The formula
describes a harmonic wave of
amplitude A moving in the
+x direction,
? ? ? ?tkxc o sAt,xy ??? y
x
?
A
? Each point on the wave oscillates in the y direction with
simple harmonic motion of angular frequency ?.
? ??2k? The wavelength of the wave is
v k??? The speed of the wave is
? The quantity k is often called,wave number”.
Physics 121,Lecture 22,Pg 36
Lecture 22,Act 6
Wave Motion
? A harmonic wave moving in the positive x direction
can be described by the equation
y(x,t) = A cos ( kx - ?t )
? Which of the following equation describes a harmonic
wave moving in the negative x direction?
(a) y(x,t) = A sin ( kx ??t )
(b) y(x,t) = A cos ( kx ??t )
(c) y(x,t) = A cos (?kx ??t )
Physics 121,Lecture 22,Pg 37
Waves on a string
? What determines the speed of a wave?
? Consider a pulse propagating along a string:
??
Fv
? Making the tension bigger increases the speed.
? Making the string heavier decreases the speed.
? As we asserted earlier,this depends only on the nature of
the medium,not on amplitude,frequency etc of the wave.
v
tension F
mass per unit length ?
Physics 121,Lecture 22,Pg 38
Lecture 22,Act 7
Wave Motion
? A heavy rope hangs from the ceiling,and a small
amplitude transverse wave is started by jiggling the
rope at the bottom,
? As the wave travels up the rope,its speed will:
(a) increase
(b) decrease
(c) stay the same
v
Physics 121,Lecture 22,Pg 39
Superposition
? Q,What happens when two waves,collide”?
? A,They ADD together!
?We say the waves are,superposed”.
Physics 121,Lecture 22,Pg 40
Superposition & Interference
? We have seen that when colliding waves combine (add) the
result can either be bigger or smaller than the original
waves.
? We say the waves add,constructively” or,destructively”
depending on the relative sign of each wave.
will add constructively
will add destructively
? In general,we will have both happening
Physics 121,Lecture 22,Pg 41
Reflection
? Type of boundary affects the reflected wave
? Fixed end
?Wave is inverted
?Due to Newton’s 3rd law
?wall pulls down on the rope
? Free end
?Wave is not inverted
?The ring is accelerated
upward by the rope
?The downward component of
the tension brings it down
Physics 121,Lecture 22,Pg 42
Recap for today:
? Todays’ topics
? Homework 9,due Friday Nov,18 @ 6:00 PM.
?Chap,9,# 1,7,9,13,17,23,24,27,29,37,39,and 40.
? Today’s topics
?SHM
?Pendulum
?Damped oscillations
?Resonance
?Waves
Physics 121,Lecture 22
Today’s Agenda
? Announcements
?No class next week
? Homework 9,due Friday Nov,18 @ 6:00 PM.
?Chap,9,# 1,7,9,13,17,23,24,27,29,37,39,and 40.
? Today’s topics
?SHM
?Pendulum
?Damped oscillations
?Resonance
?Waves
Physics 121,Lecture 22,Pg 2
The Simple Pendulum
? A pendulum is made by suspending a mass m at
the end of a string of length L,
? Find the frequency of
oscillation for small displacements.
? L
m
mg
z
Physics 121,Lecture 22,Pg 3
Aside,sin ? and cos ? for small ?
? A Taylor expansion of sin ? and cos? about ? = 0 gives:
...!5!3s in 53 ????????,..!4!21c os 42 ???????and
So for ? <<1,???sin 1cos ??and
Physics 121,Lecture 22,Pg 4
The Simple Pendulum...
? Recall that the torque due to gravity about the rotation (z)
axis is ? = Ia = -mgd,
d = Lsin ?? L? for small ?
so ? = -mg L?
? But ? = Ia???I?=?mL2
? L
d
m
mg
z
a? 2mLm g L ??
??a 2?? ? ?
g
Lwhere
Same equation as for simple harmonic motion !
? = ?0 cos(?t + ?)
Physics 121,Lecture 22,Pg 5
The Rod Pendulum
? A pendulum is made by suspending a thin rod of
length L and mass M at one end,
? Find the frequency of
oscillation for small displacements.
?
L
mg
z
x CM
Physics 121,Lecture 22,Pg 6
The Rod Pendulum...
? The torque about the rotation (z) axis is
? = -mgd = -mg{L/2}sin??? -mg{L/2}??????for small ?
? In this case
? So ? = Ia??becomes
?
Ld
mg
z
L/2
xCM
I ? 13 2mL
a? 2312 mLLmg ??
??a 2?? ? ? 32gLwhere
d I
Physics 121,Lecture 22,Pg 7
Lecture 22,Act 1
Period
(a) (b) (c)
? What length do we make the simple pendulum so that
it has the same period as the rod pendulum?
LR
LS
RS L3
2L ?RS L
2
3L ?
RS LL ?
Physics 121,Lecture 22,Pg 8
General Physical Pendulum
? Suppose we have some arbitrarily shaped
solid of mass M hung on a fixed axis,that we
know where the CM is located and what the
moment of inertia I about the axis is.
? The torque about the rotation (z) axis for
small ? is (sin ? ? )
? = -Mgd -MgR?? ?
d
Mg
z-axis
R
xCM
d
dt
2
2
2? ? ?? ?
? ? MgRIwhere
? = ?0 cos(?t + ?)
2
2
dt
dIM g R ????
? a
?
?
Physics 121,Lecture 22,Pg 9
Lecture 22,Act 2
Physical Pendulum
? A pendulum is made by hanging a thin hoola-hoop of
diameter D on a small nail,
?What is the angular frequency of oscillation of the hoop
for small displacements? (ICM = mR2 for a hoop)
(a)
(b)
(c)
?? gD
? ? 2gD
? ? g2D
D
pivot (nail)
Physics 121,Lecture 22,Pg 10
Torsion Pendulum
? Consider an object suspended by a wire
attached at its CM,The wire defines the
rotation axis,and the moment of inertia I
about this axis is known,
? The wire acts like a,rotational spring”.
?When the object is rotated,the wire is
twisted,This produces a torque that
opposes the rotation.
?In analogy with a spring,the torque
produced is proportional to the
displacement,? = -k?
I
wire
?
?
Physics 121,Lecture 22,Pg 11
Torsion Pendulum...
? Since ? = -k???? = Ia??becomes
??
? k ? ? Ia ? I d
2?
dt 2
I
wire
?
?
Similar to,mass on spring”,except
I has taken the place of m (no surprise)
d
dt
2
2
2? ? ?? ?
? ? kIwhere
Physics 121,Lecture 22,Pg 12
Lecture 22,Act 3
Period
? All of the following pendulum bobs have the
same mass,Which pendulum rotates the fastest,
i.e,has the smallest period? (The wires are
identical)
RRRR
A) B) C) D)
Physics 121,Lecture 22,Pg 13
SHM So Far
? The most general solution is x = Acos(?t + ?)
where A = amplitude
? = frequency
? = phase
? For a mass on a spring
? For a general pendulum
? For a torsion pendulum
? ? km
? ? MgRI
? ? kI
Physics 121,Lecture 22,Pg 14
Velocity and Acceleration
k
x
m
0
Position,x(t) = Acos(?t + ?)
Velocity,v(t) = -?Asin(?t + ?)
Acceleration,a(t) = -?2Acos(?t + ?)
Starting with
angle ? ant
t=0
xMAX = A
vMAX = ?A
aMAX = ?2A
? Equations for x,v,and a
Physics 121,Lecture 22,Pg 15
Energy of the Spring-Mass System
Add to get E = K + U
1/2 m (?A)2sin2(?t + ?) + 1/2 k (Acos(?t + ?))2
Remember that
? ?? ?
m
k
m
k ?? ? 2??
U~cos2 K~sin2
E = 1/2 kA2
so,E = 1/2 kA2 sin2(?t + ?) + 1/2 kA2 cos2(?t + ?)
= 1/2 kA2 [ sin2(?t + ?) + cos2(?t + ?)]
= 1/2 kA2
Active
Figure
Physics 121,Lecture 22,Pg 16
Energy in SHM
? For both the spring and the pendulum,we
can derive the SHM solution using energy
conservation,
? The total energy (K + U) of a
system undergoing SMH will
always be constant!
? This is not surprising since
there are only conservative
forces present,hence energy is conserved.
-A A0 s
U
U
K
E
Physics 121,Lecture 22,Pg 17
What about Friction?
? Friction causes the oscillations to get
smaller over time
? This is known as DAMPING.
? As a model,we assume that the force due
to friction is proportional to the velocity.
? The solution is dt
dxbF
f ??
)c o s ()2e x p ( ?? ??? tmbtAx
with,
??
? ? km ? b2 m?????? ??????
2
? ? o2 ? ? 2
??
?0 ? km
??
? ? b2mwhere and
Physics 121,Lecture 22,Pg 18
What about Friction?
)c o s ()2e x p ( ?? ??? tmbtAx
What does this function look like?
(You saw it in lab,it really works)
?? ?o ?? ?o ?? ?o
underdamped critically damped overdamped
Active
Figure
Physics 121,Lecture 22,Pg 19
Forced vibrations and Resonance
? To replace the energy lost to friction,we can drive the motion
with a periodic force
? The solution has the form
? The amplitude depends on ?
? Note,that A gets bigger if Fo does,and gets smaller if b or m
gets bigger,No surprise there.
? Something more surprising happens if you drive the pendulum
at exactly the frequency it wants to go,???k/m
? Then at least one of the terms in the denominator vanishes
and the amplitude gets real big,This is known as resonance.
F = F0 cos(?t)
??
x ( t ) ? A c o s (? t? ? )
??
A ? F 0 / m
(? 2 ? ? 02 ) 2 ? b ?
m
??
??
??
??
??
??
2
Physics 121,Lecture 22,Pg 20
Resonance
? Now,consider what b does,
?????0
b small
b middling
b large
???
??
A ? F 0 / m
(? 2 ? ? 02 ) 2 ? b ?
m
??
??
??
??
??
??
2
Physics 121,Lecture 22,Pg 21
Dramatic example of resonance
? In 1940,turbulent winds set up a torsional vibration in the Tacoma Narrow
Bridge
?
Physics 121,Lecture 22,Pg 22
Dramatic example of resonance
?
? when it reached the natural frequency
Physics 121,Lecture 22,Pg 23
Dramatic example of resonance
?
? it collapsed !
Other example,London Millenium Bridge
Physics 121,Lecture 22,Pg 24
Lecture 22,Act 4
Resonant Motion
? Consider the following set of pendula all attached to the
same string
D
A
B
CIf I start bob D swinging which of the
others will have the largest swing amplitude?
(A) (B) (C)
Physics 121,Lecture 22,Pg 25
Chapter 11,Waves
? A definition of a wave:
?A wave is a traveling disturbance that
transports energy but not matter.
? Examples:
?Sound waves (air moves back & forth)
?Stadium waves (people move up & down)
?Water waves (water moves up & down)
?Light waves (what moves)
Physics 121,Lecture 22,Pg 26
Types of Waves
? Transverse,The medium oscillates perpendicular
to the direction the wave is moving.
?Water (more or less)
?String waves
? Longitudinal,The medium oscillates in the
same direction as the wave is moving
?Sound
?Slinky
Physics 121,Lecture 22,Pg 27
Wave Properties
?
Wavelength
? Wavelength,The distance ? between identical points on the wave.
Amplitude A
? Amplitude,The maximum displacement A of a point on the wave.
A
Physics 121,Lecture 22,Pg 28
Wave Properties...
? Period,The time T for a point on the wave to
undergo one complete oscillation.
? Speed,The wave moves one wavelength ? in one
period T so its speed is v = ??/ T.
Tv
??
Physics 121,Lecture 22,Pg 29
Wave Properties...
? The speed of a wave is a constant that depends only
on the medium,not on amplitude,wavelength or
period
? and T are related !
v = ? / T
? ? = v T or ? = 2? v / ??????????????(since?T = 2? / ???
or ???v / f (since T = 1/ f )
? Recall f = cycles/sec or revolutions/sec
? ???rad/sec = 2?f
Physics 121,Lecture 22,Pg 30
Lecture 22,Act 5
Wave Motion
? The speed of sound in air is a bit over 300 m/s,and
the speed of light in air is about 300,000,000 m/s,
? Suppose we make a sound wave and a light wave
that both have a wavelength of 3 meters,
?What is the ratio of the frequency of the light wave
to that of the sound wave?
(a) About 1,000,000
(b) About,000,001
(c) About 1000
Physics 121,Lecture 22,Pg 31
Wave Forms
? So far we have examined
“continuous waves” that go
on forever in each direction !
v
v ? We can also have,pulses”
caused by a brief disturbance
of the medium:
v
? And,pulse trains” which are
somewhere in between.
Physics 121,Lecture 22,Pg 32
Mathematical Description
? Suppose we have some function y = f(x):
x
y
? f(x-d ) is just the same shape moved
a distance d to the right:
x
y
x=d0
0
? Let d =vt Then
f(x-vt) will describe the same
shape moving to the right with
speed v,x
y
x=vt 0
v
Physics 121,Lecture 22,Pg 33
Math...
? Consider a wave that is harmonic
in x and has a wavelength of ?.
? ? ?????? ??? x2c o sAxyIf the amplitude is maximum atx=0 this has the functional form:
y
x
?
A
? Now,if this is moving to
the right with speed v it will be
described by:
y
x
v
? ? ? ??????? ???? vtx2c osAt,xy
Physics 121,Lecture 22,Pg 34
Math...
? ? ? ??????? ???? vtx2c osAt,xy
? By using v T? ?? ???2from before,and by defining k ?2??
? So we see that a simple harmonic
wave moving with speed v in the x
direction is described by the equation:
we can write this as,? ? ? ?tkxc o sAt,xy ???
(what about moving in the -x direction?)
Physics 121,Lecture 22,Pg 35
Math Summary
? The formula
describes a harmonic wave of
amplitude A moving in the
+x direction,
? ? ? ?tkxc o sAt,xy ??? y
x
?
A
? Each point on the wave oscillates in the y direction with
simple harmonic motion of angular frequency ?.
? ??2k? The wavelength of the wave is
v k??? The speed of the wave is
? The quantity k is often called,wave number”.
Physics 121,Lecture 22,Pg 36
Lecture 22,Act 6
Wave Motion
? A harmonic wave moving in the positive x direction
can be described by the equation
y(x,t) = A cos ( kx - ?t )
? Which of the following equation describes a harmonic
wave moving in the negative x direction?
(a) y(x,t) = A sin ( kx ??t )
(b) y(x,t) = A cos ( kx ??t )
(c) y(x,t) = A cos (?kx ??t )
Physics 121,Lecture 22,Pg 37
Waves on a string
? What determines the speed of a wave?
? Consider a pulse propagating along a string:
??
Fv
? Making the tension bigger increases the speed.
? Making the string heavier decreases the speed.
? As we asserted earlier,this depends only on the nature of
the medium,not on amplitude,frequency etc of the wave.
v
tension F
mass per unit length ?
Physics 121,Lecture 22,Pg 38
Lecture 22,Act 7
Wave Motion
? A heavy rope hangs from the ceiling,and a small
amplitude transverse wave is started by jiggling the
rope at the bottom,
? As the wave travels up the rope,its speed will:
(a) increase
(b) decrease
(c) stay the same
v
Physics 121,Lecture 22,Pg 39
Superposition
? Q,What happens when two waves,collide”?
? A,They ADD together!
?We say the waves are,superposed”.
Physics 121,Lecture 22,Pg 40
Superposition & Interference
? We have seen that when colliding waves combine (add) the
result can either be bigger or smaller than the original
waves.
? We say the waves add,constructively” or,destructively”
depending on the relative sign of each wave.
will add constructively
will add destructively
? In general,we will have both happening
Physics 121,Lecture 22,Pg 41
Reflection
? Type of boundary affects the reflected wave
? Fixed end
?Wave is inverted
?Due to Newton’s 3rd law
?wall pulls down on the rope
? Free end
?Wave is not inverted
?The ring is accelerated
upward by the rope
?The downward component of
the tension brings it down
Physics 121,Lecture 22,Pg 42
Recap for today:
? Todays’ topics
? Homework 9,due Friday Nov,18 @ 6:00 PM.
?Chap,9,# 1,7,9,13,17,23,24,27,29,37,39,and 40.
? Today’s topics
?SHM
?Pendulum
?Damped oscillations
?Resonance
?Waves
