PHY 711 Classical Mechanics and
Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 5:
1. Chapter 2 – Physics described
in an accelerated coordinate
frame
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PHY 711 Fall 2012 -- Lecture 5
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#5
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PHY 711 Fall 2012 -- Lecture 5
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Special seminar today: 11-11:50 AM Olin 103
SILICENE
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PHY 711 Fall 2012 -- Lecture 5
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Application of Newton’s laws in a coordinate system which
has an angular velocity ω and linear acceleration  d 2a 
 2 
 dt inertial
Newton’s laws;
Let r denote the position of particle of mass m:
 d 2r 
m 2 
 Fext
 dt inertial
 d 2r 
 d 2a 
dω
 dr 
m 2 
 Fext  m 2 
 2mω     m
 r  mω  ω  r
dt
 dt body
 dt body
 dt inertial
Coriolis
force
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PHY 711 Fall 2012 -- Lecture 5
Centrifugal
force
4
Motion on the surface of the Earth:

2

Fext  
 7.3  10 5 rad/s
GM e m
rˆ  F '
2
r
Main contributions :
 d 2r 
GM e m
dω
 dr 
ˆ
m 2 

r

F
'

2
m
ω


m
 r  mω  ω  r
 
2
r
dt
 dt  earth
 dt  earth
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PHY 711 Fall 2012 -- Lecture 5
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Non-inertial effects on effective gravitational “constant”
 d 2r 
GM e m
dω
 dr 
ˆ
m 2 

r

F
'

2
m
ω


m
 r  mω  ω  r


2
r
dt
 dt  earth
 dt  earth
 d 2r 
 dr 
For  
 0 and  2 
 0,
 dt  earth
 dt  earth
GM e m
0
rˆ  F '  mω  ω  r
2
r
F '   mg
GM e
 g   2 rˆ  ω  ω  r
r
r  Re
 GM e

2
2
2
ˆ


ˆ
 


R
sin

r

sin

cos

R
θ
e
e
2

 Re

0.03 m/s2
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9.80 m/s2
PHY 711 Fall 2012 -- Lecture 5
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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 2012 -- Lecture 5
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Equation of motion on Earth’s surface
 d 2r 
GM e m
dω
 dr 
ˆ


m 2 

r  F '  2mω   
m
 r  mω  ω  r
2
r
dt
 dt  earth
 dt  earth

y (east)
z (up)

x (south)
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PHY 711 Fall 2012 -- Lecture 5
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Foucault pendulum continued – keeping leading terms:
 d 2r 
GM e m
 dr 
ˆ


m 2 

r  F '  2mω   
2
Re
 dt  earth
 dt  earth
GM e m
rˆ   mgzˆ
2
r
F '  T sin cos xˆ  T sin sin yˆ  T coszˆ
ω   sin xˆ   cos zˆ

 dr 
ω 
   y cos xˆ   x cos   z sin  yˆ  y cos zˆ 
 dt  earth
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PHY 711 Fall 2012 -- Lecture 5
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Foucault pendulum continued – keeping leading terms:
 d 2r 
GM e m
 dr 
ˆ


m 2 

r  F '  2mω   
2
Re
 dt  earth
 dt  earth
mx  T sin cos   2my cos 
my  T sin sin   2m  x cos   z sin  
mz  T cos  mg  2my cos 
Further approximation :
  1; z  0; T  mg
mx   mg sin cos   2my cos 
my   mg sin sin   2mx cos 
Also note that :
x   sin cos 
y   sin sin 
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PHY 711 Fall 2012 -- Lecture 5
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Foucault pendulum continued – coupled equations:
g
x  2 cos y

g
y   y  2 cos x

x  
Try to find a solution of the form :
x(t )  Xe iqt
y (t )  Yeiqt
Denote    cos 
  q 2  g i 2 q  X 


  i 2 q  q 2  g  Y   0


 

Non - trivial solutions :
q           
2
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g

PHY 711 Fall 2012 -- Lecture 5
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Foucault pendulum continued – coupled equations:
Solution continued :
x(t )  Xe iqt
y (t )  Yeiqt
  q 2  g i 2 q  X 


  i 2 q  q 2  g  Y   0


 

Non - trivial solutions :
q           
2
g

Amplitude relationship : X  iY
General solution with complex amplitudes C and D :

y (t )  ReCe



x(t )  Re iCe i    t  iDe i    t
9/7/2012
i    t
 De i   
t
PHY 711 Fall 2012 -- Lecture 5
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General solution w ith complex amplitudes C and D :

y (t )  ReCe



x(t )  Re iCe i    t  iDe i    t
i    t
 De i   
q           
2
g

  7 10 5 cos  rad / s 
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t
g

PHY 711 Fall 2012 -- Lecture 5
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