Physics 121,Lecture 21,Pg 1
Physics 121,Lecture 21
Today’s Agenda
? Announcements
? Homework 8,due Friday Nov,11 @ 6:00 PM.
?Chap,8,# 7,22,28,33,35,44,45,50,54,61,and 65.
? Today?s topics
?Fluids in motion
?Bernouilli?s equation
?Viscous fluids
?Simple oscillations
?Pendulum
Physics 121,Lecture 21,Pg 2
Review,Fluids at Rest
? What parameters do we use to describe fluids?
?Density Bulk Modulus
?Pressure
? For incompressible fluids ( )
? Pascal?s Principle:
?Any change in the pressure applied to an enclosed fluid is
transmitted to every portion of the fluid and to the walls of
the containing vessel.
)/V( V
pB
??
??
pB??
.const?? ygp)y(p 0 ???? ?(?y is depth)
??
? ? mV
??
p ? FA nF ?pA? A
n
Physics 121,Lecture 21,Pg 3
? The buoyant force is equal to the difference in the
pressures times the area.
W2?W1
)Ay-g ( y)( 12????? AppF 12B
l i q u i dl i q u i dl i q u i dl i q u i dB WgMgVF ???? ?
Archimedes:
The buoyant force is equal to the
weight of the liquid displaced.
? The buoyant force determines
whether an object will sink or float.
How does this work?
y
1
y
2
A
p
1
p
2
F
1
F
2
Archimedes’ Principle
Physics 121,Lecture 21,Pg 4
Fluids in Motion
? Up to now we have described fluids in terms
of their static properties:
?density ?
?pressure p
? To describe fluid motion,we need something
that can describe flow:
?velocity v
? There are different kinds of fluid flow of varying complexity
? non-steady / steady
? compressible / incompressible
? rotational / irrotational
? viscous / ideal
Physics 121,Lecture 21,Pg 5
? Simplest situation,consider
ideal fluid moving with steady
flow - velocity at each point in
the flow is constant in time
? In this case,fluid moves on
streamlines
A
1
A
2
v
1
v
2
streamline
Ideal Fluids
? Fluid dynamics is very complicated in general (turbulence,
vortices,etc.)
? Consider the simplest case first,the Ideal Fluid
?no,viscosity” - no flow resistance (no internal friction)
?incompressible - density constant in space and time
Physics 121,Lecture 21,Pg 6
? Flow obeys continuity equation
? volume flow rate Q = A·v is constant along flow
tube.
? follows from mass conservation if flow is
incompressible.
A
1
A
2
v
1
v
2
streamline
A1v1 = A2v2
Ideal Fluids
? streamlines do not meet or cross
? velocity vector is tangent to
streamline
? volume of fluid follows a tube of
flow bounded by streamlines
Physics 121,Lecture 21,Pg 7
Steady Flow of Ideal Fluids
(actually laminar flow of real fluid)
Physics 121,Lecture 21,Pg 8
1) Assuming the water moving in the pipe is an ideal
fluid,relative to its speed in the 1” diameter pipe,
how fast is the water going in the 1/2” pipe?
Lecture 21 Act 1
Continuity
? A housing contractor saves
some money by reducing the
size of a pipe from 1”
diameter to 1/2” diameter at
some point in your house,
v1 v1/2
a) 2 v1 b) 4 v1 c) 1/2 v1 c) 1/4 v1
Physics 121,Lecture 21,Pg 9
? Recall the standard work-energy relation
?Apply the principle to a section of flowing fluid with
volume dV and mass dm = ??dV (here W is work done
on fluid)
V)pp( xApxApW 21 222111pr e s s ur e d dd?? ??
)yy(gV )yy(gmW 12 12gr a v i t y ??? ??? d?d
)vv(VvmvmKW 21222121212221 ?????? d?dd
222212121211 gyvpgyvp ???? ?????? Bernoulli Equation
y
1
y
2
v
1
v
2
p
1
p
2
dV
p r e ssu r eg r a vi t y WWW ??
KW ??
Conservation of Energy for
Ideal Fluid
Physics 121,Lecture 21,Pg 10
Lecture 21 Act 2
Bernoulli’s Principle
? A housing contractor saves
some money by reducing the
size of a pipe from 1”
diameter to 1/2” diameter at
some point in your house,
2) What is the pressure in the 1/2” pipe relative to the
1” pipe?
a) smaller b) same c) larger
v1 v1/2
Physics 121,Lecture 21,Pg 11
Some applications
? Lift for airplane wing
? Enhance sport performance
? More complex phenomena,ex,turbulence
Physics 121,Lecture 21,Pg 12
More applications
? Vortices,ex,Hurricanes
? And much more …
Physics 121,Lecture 21,Pg 13
? Bernoulli says,high velocities go with low pressure
? Airplane wing
?shape leads to lower pressure on top of wing
?faster flow ? lower pressure ? lift
? air moves downward at downstream edge ? wing
moves up
Ideal Fluid,Bernoulli Applications
Physics 121,Lecture 21,Pg 14
? Warning,the explanations in text books are generally over-
simplified!
? Curve ball (baseball),slice or topspin (golf)
?ball drags air around (viscosity)
?air speed near ball higher on bottom
?lower pressure? force ? sideways acceleration or lift
v
F
Ideal Fluid,Bernoulli Applications
Physics 121,Lecture 21,Pg 15
? Bernoulli says,high velocities go with low pressure
?,Atomizer”
?moving air ?sweeps? air away from top of tube
?pressure is lowered inside the tube
?air pressure inside the jar drives liquid up into tube
Ideal Fluid,Bernoulli Applications
Physics 121,Lecture 21,Pg 16
? In ideal fluids mechanical energy is conserved (Bernoulli)
? In real fluids,there is dissipation (or conversion to heat) of
mechanical energy due to viscosity (internal friction of fluid)
Real Fluids,Viscosity
Viscosity measures the force
required to shear the fluid:
vA
yF
?
???
where F is the force required to move a fluid lamina
(thin layer) of area A at the speed v when the fluid is in contact
with a stationary surface a perpendicular distance y away.
v
y
area A
Physics 121,Lecture 21,Pg 17
? Viscosity arises from particle
collisions in the fluid
?as particles in the top layer
diffuse downward they
transfer some of their
momentum to lower layers
Real Fluids,Viscosity
???? ????????
Viscosity (Pa-s)
oilair glycerinH2O
???? ?????
v
y
area A
?lower layers get pulled along (F = ?p/?t)
Physics 121,Lecture 21,Pg 18
L
RpvAQ
?
?
8
4
???
p+?p Q r
L
p
R
? Because friction is involved,we know that mechanical energy
is not being conserved - work is being done by the fluid.
? Power is dissipated when viscous fluid flows,P = v·F = Q ·?p
the velocity of the fluid remains constant power goes into
heating the fluid,increasing its entropy
Real Fluids,Viscous Flow
? How fast can viscous fluid
flow through a pipe?
?Poiseuille?s Law
Physics 121,Lecture 21,Pg 19
1) Given that water is viscous,what is the ratio of the flow
rates,Q1/Q1/2,in pipes of these sizes if the pressure drop per
meter of pipe is the same in the two cases?
? Consider again the 1 inch diameter pipe and the 1/2 inch
diameter pipe.
a) 3/2 b) 2 c) 4
L/2 L/2
Lecture 21 Act 3
Viscous flow
Physics 121,Lecture 21,Pg 20
New topic (Ch,10)
Simple Harmonic Motion (SHM)
? We know that if we stretch a spring with a mass on the end
and let it go the mass will oscillate back and forth (if there is
no friction).
? This oscillation is called
Simple Harmonic Motion,
and is actually very easy
to understand...
? To describe SHM
?Amplitude A
?Period T
?Frequency f
k m
k m
k m
Physics 121,Lecture 21,Pg 21
SHM Dynamics
k
x
m
F = -kx
a
??
a ? ? km x ? ? ? 2 x
simple links between a(t) and x(t) !
? SHM occurs when motion follows Hooke?s law type force
?F = -k x
? At any given instant we know
that F = ma must be true
? So -kx = ma
?a= -k/m x
? Define ?2 = k/m
Physics 121,Lecture 21,Pg 22
Spring...
? Energy is stored in a spring only
?If stretched or compressed
? The potential energy stored is
?U = 1/2 k x2
? Recall conservation of energy
?Ki+Ui = Kf+Uf
?where K=1/2 mv2
? Total energy found
?when fully stretched/compressed (v=0)
?E = 1/2 k A2
Physics 121,Lecture 21,Pg 23
Spring...
??
v ? ? k
m
( A 2 ? x 2 )
? Recall conservation of energy
?E = Ki+Ui = Kf+Uf
? So
? Solving for v
? It is a function of x as well
??
E ? 12 kA 2 ? 12 mv 2 ? 12 kx 2
Physics 121,Lecture 21,Pg 24
Comparison of SHM and UCM
? We have
? Recall Uniform circular motion (UCM)
?|v | = v= ? R is constant
?The angle,?=?t
? The x motion is given
?x(t)=A cos ? = A cos ?t
?
??
v ? C ( A 2 ? x 2 )
Physics 121,Lecture 21,Pg 25
Comparison of SHM and UCM
? We have
? But
? So,as before
?
?
?
v
v0
A
x
??
s in ? ? ( A
2 ? x 2 )
A
??
( A 2 ? x 2 )
??
s in ? ? v
v 0
??
v ? v 0A ( A 2 ? x 2 ) ? C ( A 2 ? x 2 )
Physics 121,Lecture 21,Pg 26
Comparison of SHM and UCM
? Time (period) to make one revolution
?v0= 2?A / T
?T= 2?A / v0
? But when fully stretched
?1/2 k A2 = 1/2 m v02
? So
? Frequency is defined as
?f = 1 / T
? So,
??
A
v 0 ?
m
k ?
1
?
??
T ? 2 ? mk ? 2 ??
??
f ? 12 ? km ? ?2 ?
??
? ? 2 ? f ? km
Physics 121,Lecture 21,Pg 27
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
? From previous equations for x,v,and a
Physics 121,Lecture 21,Pg 28
Most general solution
x = Acos(?t+?) is equivalent to x = Bsin(?t)+ Ccos(?t)
x = Acos(?t+?)
= Acos(?t) cos?- Asin(?t) sin?
where C = Acos(?) and B = ?Asin(?)
It works!
= Ccos(?t) + Bsin(?t)
We want to use the most general solution:
So we can use x = Acos(?t+?) as the most general solution!
Physics 121,Lecture 21,Pg 29
SHM Solution...
? Drawing of Acos(?t )
? A = amplitude of oscillation
?? ? ???
T = 2?/?
A
A
Physics 121,Lecture 21,Pg 30
SHM Solution...
? Drawing of Acos(?t + ?)
?
?? ? ???
Physics 121,Lecture 21,Pg 31
SHM Solution...
? Drawing of Acos(?t - ?/2)
A
??????
?? ? ???
= Asin(?t) !
Physics 121,Lecture 21,Pg 32
Lecture 21 Act 4
Simple Harmonic Motion
? A mass oscillates up & down on a spring,It?s position as a
function of time is shown below,At which of the points shown
does the mass have positive velocity and negative acceleration?
t
y(t)
(a)
(b)
(c)
Physics 121,Lecture 21,Pg 33
SHM So Far
? The most general solution is x = Acos(?t + ?)
where A = amplitude
? = frequency
? = phase
? For a mass on a spring
?The frequency does not depend on the amplitude !!!
?We will see that this is true of all simple harmonic
motion !
? The oscillation occurs around the equilibrium point where
the force is zero!
? ? km
Physics 121,Lecture 21,Pg 34
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 21,Pg 35
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 21,Pg 36
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 21,Pg 37
Lecture 21 Act 5
Simple Harmonic Motion
? You are sitting on a swing,A friend gives you a small
push and you start swinging back & forth with period
T1.
? Suppose you were standing on the swing rather than
sitting,When given a small push you start swinging
back & forth with period T2,
?Which of the following is true:
(a) T1 = T2
(b) T1 > T2
(c) T1 < T2
Physics 121,Lecture 21,Pg 38
Recap for today:
? Todays? topics
? Homework 8,due Friday Nov,11 @ 6:00 PM.
?Chap,8,# 7,22,28,33,35,44,45,50,54,61,and 65.
? Today?s topics
?Fluids in motion
?Bernouilli?s equation
?Viscous fluids
?Simple oscillations
?Pendulum
Physics 121,Lecture 21
Today’s Agenda
? Announcements
? Homework 8,due Friday Nov,11 @ 6:00 PM.
?Chap,8,# 7,22,28,33,35,44,45,50,54,61,and 65.
? Today?s topics
?Fluids in motion
?Bernouilli?s equation
?Viscous fluids
?Simple oscillations
?Pendulum
Physics 121,Lecture 21,Pg 2
Review,Fluids at Rest
? What parameters do we use to describe fluids?
?Density Bulk Modulus
?Pressure
? For incompressible fluids ( )
? Pascal?s Principle:
?Any change in the pressure applied to an enclosed fluid is
transmitted to every portion of the fluid and to the walls of
the containing vessel.
)/V( V
pB
??
??
pB??
.const?? ygp)y(p 0 ???? ?(?y is depth)
??
? ? mV
??
p ? FA nF ?pA? A
n
Physics 121,Lecture 21,Pg 3
? The buoyant force is equal to the difference in the
pressures times the area.
W2?W1
)Ay-g ( y)( 12????? AppF 12B
l i q u i dl i q u i dl i q u i dl i q u i dB WgMgVF ???? ?
Archimedes:
The buoyant force is equal to the
weight of the liquid displaced.
? The buoyant force determines
whether an object will sink or float.
How does this work?
y
1
y
2
A
p
1
p
2
F
1
F
2
Archimedes’ Principle
Physics 121,Lecture 21,Pg 4
Fluids in Motion
? Up to now we have described fluids in terms
of their static properties:
?density ?
?pressure p
? To describe fluid motion,we need something
that can describe flow:
?velocity v
? There are different kinds of fluid flow of varying complexity
? non-steady / steady
? compressible / incompressible
? rotational / irrotational
? viscous / ideal
Physics 121,Lecture 21,Pg 5
? Simplest situation,consider
ideal fluid moving with steady
flow - velocity at each point in
the flow is constant in time
? In this case,fluid moves on
streamlines
A
1
A
2
v
1
v
2
streamline
Ideal Fluids
? Fluid dynamics is very complicated in general (turbulence,
vortices,etc.)
? Consider the simplest case first,the Ideal Fluid
?no,viscosity” - no flow resistance (no internal friction)
?incompressible - density constant in space and time
Physics 121,Lecture 21,Pg 6
? Flow obeys continuity equation
? volume flow rate Q = A·v is constant along flow
tube.
? follows from mass conservation if flow is
incompressible.
A
1
A
2
v
1
v
2
streamline
A1v1 = A2v2
Ideal Fluids
? streamlines do not meet or cross
? velocity vector is tangent to
streamline
? volume of fluid follows a tube of
flow bounded by streamlines
Physics 121,Lecture 21,Pg 7
Steady Flow of Ideal Fluids
(actually laminar flow of real fluid)
Physics 121,Lecture 21,Pg 8
1) Assuming the water moving in the pipe is an ideal
fluid,relative to its speed in the 1” diameter pipe,
how fast is the water going in the 1/2” pipe?
Lecture 21 Act 1
Continuity
? A housing contractor saves
some money by reducing the
size of a pipe from 1”
diameter to 1/2” diameter at
some point in your house,
v1 v1/2
a) 2 v1 b) 4 v1 c) 1/2 v1 c) 1/4 v1
Physics 121,Lecture 21,Pg 9
? Recall the standard work-energy relation
?Apply the principle to a section of flowing fluid with
volume dV and mass dm = ??dV (here W is work done
on fluid)
V)pp( xApxApW 21 222111pr e s s ur e d dd?? ??
)yy(gV )yy(gmW 12 12gr a v i t y ??? ??? d?d
)vv(VvmvmKW 21222121212221 ?????? d?dd
222212121211 gyvpgyvp ???? ?????? Bernoulli Equation
y
1
y
2
v
1
v
2
p
1
p
2
dV
p r e ssu r eg r a vi t y WWW ??
KW ??
Conservation of Energy for
Ideal Fluid
Physics 121,Lecture 21,Pg 10
Lecture 21 Act 2
Bernoulli’s Principle
? A housing contractor saves
some money by reducing the
size of a pipe from 1”
diameter to 1/2” diameter at
some point in your house,
2) What is the pressure in the 1/2” pipe relative to the
1” pipe?
a) smaller b) same c) larger
v1 v1/2
Physics 121,Lecture 21,Pg 11
Some applications
? Lift for airplane wing
? Enhance sport performance
? More complex phenomena,ex,turbulence
Physics 121,Lecture 21,Pg 12
More applications
? Vortices,ex,Hurricanes
? And much more …
Physics 121,Lecture 21,Pg 13
? Bernoulli says,high velocities go with low pressure
? Airplane wing
?shape leads to lower pressure on top of wing
?faster flow ? lower pressure ? lift
? air moves downward at downstream edge ? wing
moves up
Ideal Fluid,Bernoulli Applications
Physics 121,Lecture 21,Pg 14
? Warning,the explanations in text books are generally over-
simplified!
? Curve ball (baseball),slice or topspin (golf)
?ball drags air around (viscosity)
?air speed near ball higher on bottom
?lower pressure? force ? sideways acceleration or lift
v
F
Ideal Fluid,Bernoulli Applications
Physics 121,Lecture 21,Pg 15
? Bernoulli says,high velocities go with low pressure
?,Atomizer”
?moving air ?sweeps? air away from top of tube
?pressure is lowered inside the tube
?air pressure inside the jar drives liquid up into tube
Ideal Fluid,Bernoulli Applications
Physics 121,Lecture 21,Pg 16
? In ideal fluids mechanical energy is conserved (Bernoulli)
? In real fluids,there is dissipation (or conversion to heat) of
mechanical energy due to viscosity (internal friction of fluid)
Real Fluids,Viscosity
Viscosity measures the force
required to shear the fluid:
vA
yF
?
???
where F is the force required to move a fluid lamina
(thin layer) of area A at the speed v when the fluid is in contact
with a stationary surface a perpendicular distance y away.
v
y
area A
Physics 121,Lecture 21,Pg 17
? Viscosity arises from particle
collisions in the fluid
?as particles in the top layer
diffuse downward they
transfer some of their
momentum to lower layers
Real Fluids,Viscosity
???? ????????
Viscosity (Pa-s)
oilair glycerinH2O
???? ?????
v
y
area A
?lower layers get pulled along (F = ?p/?t)
Physics 121,Lecture 21,Pg 18
L
RpvAQ
?
?
8
4
???
p+?p Q r
L
p
R
? Because friction is involved,we know that mechanical energy
is not being conserved - work is being done by the fluid.
? Power is dissipated when viscous fluid flows,P = v·F = Q ·?p
the velocity of the fluid remains constant power goes into
heating the fluid,increasing its entropy
Real Fluids,Viscous Flow
? How fast can viscous fluid
flow through a pipe?
?Poiseuille?s Law
Physics 121,Lecture 21,Pg 19
1) Given that water is viscous,what is the ratio of the flow
rates,Q1/Q1/2,in pipes of these sizes if the pressure drop per
meter of pipe is the same in the two cases?
? Consider again the 1 inch diameter pipe and the 1/2 inch
diameter pipe.
a) 3/2 b) 2 c) 4
L/2 L/2
Lecture 21 Act 3
Viscous flow
Physics 121,Lecture 21,Pg 20
New topic (Ch,10)
Simple Harmonic Motion (SHM)
? We know that if we stretch a spring with a mass on the end
and let it go the mass will oscillate back and forth (if there is
no friction).
? This oscillation is called
Simple Harmonic Motion,
and is actually very easy
to understand...
? To describe SHM
?Amplitude A
?Period T
?Frequency f
k m
k m
k m
Physics 121,Lecture 21,Pg 21
SHM Dynamics
k
x
m
F = -kx
a
??
a ? ? km x ? ? ? 2 x
simple links between a(t) and x(t) !
? SHM occurs when motion follows Hooke?s law type force
?F = -k x
? At any given instant we know
that F = ma must be true
? So -kx = ma
?a= -k/m x
? Define ?2 = k/m
Physics 121,Lecture 21,Pg 22
Spring...
? Energy is stored in a spring only
?If stretched or compressed
? The potential energy stored is
?U = 1/2 k x2
? Recall conservation of energy
?Ki+Ui = Kf+Uf
?where K=1/2 mv2
? Total energy found
?when fully stretched/compressed (v=0)
?E = 1/2 k A2
Physics 121,Lecture 21,Pg 23
Spring...
??
v ? ? k
m
( A 2 ? x 2 )
? Recall conservation of energy
?E = Ki+Ui = Kf+Uf
? So
? Solving for v
? It is a function of x as well
??
E ? 12 kA 2 ? 12 mv 2 ? 12 kx 2
Physics 121,Lecture 21,Pg 24
Comparison of SHM and UCM
? We have
? Recall Uniform circular motion (UCM)
?|v | = v= ? R is constant
?The angle,?=?t
? The x motion is given
?x(t)=A cos ? = A cos ?t
?
??
v ? C ( A 2 ? x 2 )
Physics 121,Lecture 21,Pg 25
Comparison of SHM and UCM
? We have
? But
? So,as before
?
?
?
v
v0
A
x
??
s in ? ? ( A
2 ? x 2 )
A
??
( A 2 ? x 2 )
??
s in ? ? v
v 0
??
v ? v 0A ( A 2 ? x 2 ) ? C ( A 2 ? x 2 )
Physics 121,Lecture 21,Pg 26
Comparison of SHM and UCM
? Time (period) to make one revolution
?v0= 2?A / T
?T= 2?A / v0
? But when fully stretched
?1/2 k A2 = 1/2 m v02
? So
? Frequency is defined as
?f = 1 / T
? So,
??
A
v 0 ?
m
k ?
1
?
??
T ? 2 ? mk ? 2 ??
??
f ? 12 ? km ? ?2 ?
??
? ? 2 ? f ? km
Physics 121,Lecture 21,Pg 27
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
? From previous equations for x,v,and a
Physics 121,Lecture 21,Pg 28
Most general solution
x = Acos(?t+?) is equivalent to x = Bsin(?t)+ Ccos(?t)
x = Acos(?t+?)
= Acos(?t) cos?- Asin(?t) sin?
where C = Acos(?) and B = ?Asin(?)
It works!
= Ccos(?t) + Bsin(?t)
We want to use the most general solution:
So we can use x = Acos(?t+?) as the most general solution!
Physics 121,Lecture 21,Pg 29
SHM Solution...
? Drawing of Acos(?t )
? A = amplitude of oscillation
?? ? ???
T = 2?/?
A
A
Physics 121,Lecture 21,Pg 30
SHM Solution...
? Drawing of Acos(?t + ?)
?
?? ? ???
Physics 121,Lecture 21,Pg 31
SHM Solution...
? Drawing of Acos(?t - ?/2)
A
??????
?? ? ???
= Asin(?t) !
Physics 121,Lecture 21,Pg 32
Lecture 21 Act 4
Simple Harmonic Motion
? A mass oscillates up & down on a spring,It?s position as a
function of time is shown below,At which of the points shown
does the mass have positive velocity and negative acceleration?
t
y(t)
(a)
(b)
(c)
Physics 121,Lecture 21,Pg 33
SHM So Far
? The most general solution is x = Acos(?t + ?)
where A = amplitude
? = frequency
? = phase
? For a mass on a spring
?The frequency does not depend on the amplitude !!!
?We will see that this is true of all simple harmonic
motion !
? The oscillation occurs around the equilibrium point where
the force is zero!
? ? km
Physics 121,Lecture 21,Pg 34
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 21,Pg 35
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 21,Pg 36
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 21,Pg 37
Lecture 21 Act 5
Simple Harmonic Motion
? You are sitting on a swing,A friend gives you a small
push and you start swinging back & forth with period
T1.
? Suppose you were standing on the swing rather than
sitting,When given a small push you start swinging
back & forth with period T2,
?Which of the following is true:
(a) T1 = T2
(b) T1 > T2
(c) T1 < T2
Physics 121,Lecture 21,Pg 38
Recap for today:
? Todays? topics
? Homework 8,due Friday Nov,11 @ 6:00 PM.
?Chap,8,# 7,22,28,33,35,44,45,50,54,61,and 65.
? Today?s topics
?Fluids in motion
?Bernouilli?s equation
?Viscous fluids
?Simple oscillations
?Pendulum
