PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 17:
PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
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Transcript of PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
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PHY 711 Fall 2013 -- Lecture 4 19/4/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 4:
Chapter 2 – Physics described in an accelerated coordinate frame
1. Linear acceleration
2. Angular acceleration
3. Foucault pendulum
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PHY 711 Fall 2013 -- Lecture 4 29/4/2013
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PHY 711 Fall 2013 -- Lecture 49/4/20133
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PHY 711 Fall 2013 -- Lecture 4 49/4/2013
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PHY 711 Fall 2013 -- Lecture 4 59/4/2013
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PHY 711 Fall 2013 -- Lecture 4 69/4/2013
Physical laws as described in non-inertial coordinate systems
Newton’s laws are formulated in an inertial frame of reference 0ˆ ie
ˆ tie
For some problems, it is convenient to transform the the equations into a non-inertial coordinate system
01e
02e
03e
1e
2e
3e
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PHY 711 Fall 2013 -- Lecture 4 79/4/2013
Comparison of analysis in “inertial frame” versus “non-inertial frame”
dt
edV
dt
d
dt
d
edt
dV
dt
d
dt
edVe
dt
dVe
dt
dV
dt
d
eVeV
e
e
i
ii
bodyinertial
ii
i
body
i
iii
i
ii
i
i
inertial
ii
iii
i
i
i
ˆ
ˆ :Define
ˆˆˆ
ˆˆ
system coordinate moving a ˆby Denote
system coordinate fixed a ˆby Denote
3
1
3
1
3
1
3
1
03
1
0
3
1
03
1
0
0
VV
V
V
V
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PHY 711 Fall 2013 -- Lecture 4 89/4/2013
Properties of the frame motion (rotation only):
dW
dWye
ze
yed ˆ
zed ˆ
eωe
ee
ee
ˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
dt
ddt
d
dt
d
dd
eded
eded
yz
zy
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PHY 711 Fall 2013 -- Lecture 4 99/4/2013
VωVV
VV
bodyinertial
i
ii
bodyinertial
dt
d
dt
d
dt
edV
dt
d
dt
d ˆ3
1
Effects on acceleration (rotation only):
VωωVωV
ωVV
VωV
ωV
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
d
bodybodyinertial
bodybodyinertial
22
2
2
2
Properties of the frame motion (rotation only) -- continued
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PHY 711 Fall 2013 -- Lecture 4 109/4/2013
2
2
inertialdt
d
aApplication of Newton’s laws in a coordinate system which has an angular velocity and linear acceleration
Newton’s laws; Let r denote the position of particle of mass m:
rωωrωr
ωa
Fr
Fr
mdt
dm
dt
dm
dt
dm
dt
dm
dt
dm
bodyinertial
ext
body
ext
inertial
2 2
2
2
2
2
2
Coriolis force
Centrifugal force
ω
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PHY 711 Fall 2013 -- Lecture 4 119/4/2013
Motion on the surface of the Earth:
'ˆ
rad/s103.72
2
5
FrF
r
mGM eext
rωωrωr
ωFrr
m
dt
dm
dt
dm
r
mGM
dt
dm
earth
e
earth
2 'ˆ
:onscontributiMain
22
2
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PHY 711 Fall 2013 -- Lecture 4 129/4/2013
Non-inertial effects on effective gravitational “constant”
θr
rωωrg
gF
rωωFr
rr
rωωrωr
ωFrr
ˆcossinˆsin
ˆ
'
'ˆ0
,0 and 0For
2 'ˆ
2222
2
2
2
2
22
2
ee
e
e
Rr
e
e
earthearth
earth
e
earth
RRR
GM
r
GM
m
mr
mGM
dt
d
dt
d
mdt
dm
dt
dm
r
mGM
dt
dm
e
9.80 m/s2
0.03 m/s2
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PHY 711 Fall 2013 -- Lecture 4 139/4/2013
Foucault pendulum http://www.si.edu/Encyclopedia_SI/nmah/pendulum.htm
The Foucault pendulum was displayed for many years in the Smithsonian's National Museum of American History. It is named for the French physicist Jean Foucault who first used it in 1851 to demonstrate the rotation of the earth.
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PHY 711 Fall 2013 -- Lecture 4 149/4/2013
rωωrωr
ωFrr
m
dt
dm
dt
dm
r
mGM
dt
dm
earth
e
earth
2 'ˆ22
2
Equation of motion on Earth’s surface
w
z (up)
x (south)
y (east)
q
zxω ˆcosˆsin
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PHY 711 Fall 2013 -- Lecture 4 159/4/2013
zyxr
ω
zxω
zyxF
zr
ˆsinˆsincosˆcos
ˆcosˆsin
ˆcosˆsinsinˆcossin'
ˆˆ2
yzxydt
d
TTT
mgr
mGM
earth
e
Foucault pendulum continued – keeping leading terms:
earthe
e
earthdt
dm
R
mGM
dt
dm
rωFr
r2 'ˆ
22
2
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PHY 711 Fall 2013 -- Lecture 4 169/4/2013
Foucault pendulum continued – keeping leading terms:
earthe
e
earthdt
dm
R
mGM
dt
dm
rωFr
r2 'ˆ
22
2
symmgTzm
zxmTym
ymTxm
sin2cos
sincos2sinsin
cos2cossin
sinsin
cossin
: thatnote Also
cos2sinsin
cos2cossin
;0 ;1
:ionapproximatFurther
y
x
xmmgym
ymmgxm
mgTz
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PHY 711 Fall 2013 -- Lecture 4 179/4/2013
Foucault pendulum continued – coupled equations:
xyg
y
yxg
x
cos2
cos2
g
g
g
iqtiqt
q
Y
X
qqi
qiq
YetyXetx
2
2
2
:solutions trivial-Non
02
2
cos Denote
)( )(
:form theofsolution a find Try to
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PHY 711 Fall 2013 -- Lecture 4 189/4/2013
Foucault pendulum continued – coupled equations:
iYX
q
Y
X
qqi
qiq
YetyXetx
g
g
g
iqtiqt
:iprelationsh Amplitude
:solutions trivial-Non
02
2
)( )(
:continuedSolution
2
2
2
Re)(
Re)(
: and amplitudescomplex ith solution w General
titi
titi
DeCety
iDeiCetx
DC
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PHY 711 Fall 2013 -- Lecture 4 199/4/2013
g
gg
srad
q
/ cos107 since 5
2
Re)(
Re)(
: and amplitudescomplex ith solution w General
titi
titi
DeCety
iDeiCetx
DC
sincos )(
coscos)(
0 )0( )0( :Suppose
0
0
0
ttXty
ttXtx
yXx
g
g