Transcript Rotating Earth

Rotating Earth
Spinning Sphere
0
  
w 0
w 
 

A point fixed to the Earth is a
non-inertial system.

The surface is nearly
spherical.
RE
• Radius RE = 6.37 x 106 m

w
2
2

s 1
T
24  3600
The axis is fixed except for
very long intervals.
• Angular velocity w = 7.27 x
10-5 rad/s
Surface Point

A point on the surface is in
the body frame of the Earth.
• Define by radius vector
r’
RE
r

A point rotating with the
Earth can be defined from
the center or surface.
• Primed from center
• Unprimed from surface
 

r   RE  r
 
r  r
Rotating Equation of Motion

The equation of motion in the rotating Earth’s frame
includes external forces.
• Gravity mg
• Other forces F

 
 
 





mr  F  mg  mw  w  r   2mw  r 

This is transformed to the surface frame.
• Earth is large compared to displacements

 


mr  F   mg  mw 

 


mr  F   mg  mw 
 




 

w  RE  r  2mw  r
 
 
w  RE  2mw  r

Plumb Bob


r  0
r  0

  

0  F   mg  mw  w  RE

   
Ftension  m g  w  w  RE
A weight hanging at rest is
only subject to gravity and
tension.



• Effective force due to
rotation correction





   
g eff  g  w  w  RE
  
w  w  RE  w 2 RE sin q

g
r
q
RE
geff


The angle is the colatitude.
• Magnitude: 0.03 sinq m/s2
• About 0.3% of g
Centripetal Correction

The spin of the earth creates
a centripetal acceleration
toward the axis of the earth.
a = w2r

The material of the Earth is
pulled less at the equator.

Items weigh less at the
equator.
Latitude Changes

As the Earth spins the
equator slightly bulges.

The radius is about 22 km
bigger compared to the pole.

Expect a few Gal difference
from equator to pole.
• Equator: 9.780 m/s2
• North Pole: 9.832 m/s2
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