Transcript Sect. 4.10 & Marion

Sect. 4.10: Coriolis & Centrifugal Forces
(Motion Relative to Earth, mainly from Marion)
• Summary: For motion in an accelerating frame (r), both
translating & rotating with respect to a fixed (f, inertial) frame:
Velocities: vf = V + vr + ω  r
Accelerations:
ar = Af + ar + ω  r + ω  (ω  r) + 2(ω  vr)
 Newton’s 2nd Law (inertial frame):
F = maf = mAf + mar + m(ω  r)
+ m[ω  (ω  r)] + 2m(ω  vr)
 “2nd Law” equation in the moving frame:
mar  Feff  F - mAf - m(ω  r)
- m[ω  (ω  r)] - 2m(ω  vr)
Motion Relative to Earth
 “2nd Law” in accelerating frame:
Feff  mar  F - mAf - m(ω  r)
- m[ω  (ω  r)] - 2m(ω  vr)
 Transformation gave:
Feff  F - (non-inertial terms)
• Interpretations:
- mAf : From translational acceleration of moving frame.
 ≈ 0 for
- m(ω  r): From angular acceleration of moving frame. motion near
Earth
- m[ω  (ω  r)]:  “Centrifugal Force”. If ω  r: Has
magnitude mω2r. Outwardly directed from center of rotation.
- 2m(ω  vr):  “Coriolis Force”. From motion of particle in
moving system (= 0 if vr = 0)
More discussion of last two now!
• Motion of Earth relative to inertial frame:
Rotation on axis causes small effects! However, this
dominates over other (much smaller!) effects:
Also, ω = (dω/dt) ≈ 0
ω = 7.292  10-5 s-1 ;
ω2Re = 3.38 cm/s2 = Centripetal acceleration at equator
2ωvr  1.5  10-4 v = max Coriolis acceleration
( 15 cm/s2 = 0.015g for v = 105 cm/s)
Even Smaller effects!
– Revolution about Sun
– Motion of Solar System in Galaxy
– Motion of Galaxy in Universe
• Coordinate systems (figure): z direction = local vertical
– Fixed: (x,y,z) At Earth center
– Moving: (x,y,z) On Earth surface
• Mass m at r in moving system.
• Physical forces in inertial system: F  S + mg0
S  Sum of non-gravitational forces
mg0  Gravitational force on m
g0  Gravitational field vector, vertical
(towards Earth center; along R in fig).
• From Newton’s Gravitation Law:
g0 = -[(GME)eR]/(R2)
G  Gravitational constant, R  Earth radius
ME  Earth mass, eR  Unit vector in R direction
– Assumes isotropic, spherical Earth
– Neglects gravitational variations due to oblateness;
non-uniformity; ...
• Effective force on m, measured in moving system
is thus: Feff  S + mg0 - mAf - m(ω  r)
- m[ω  (ω  r)] - 2m(ω  vr)
• Earth’s angular velocity ω is in z direction in
inertial system (North): ω  ωez
ez  unit vector along z
Earth rotation period T = 1 day
ω = (2π)/T = 7.3  10-5 rad/s
(Note: ω  365 ωes)
• ω  constant  ω  0  Neglect m(ω  r)
• Consider mAf term in Feff & use again formalism of
last time (rotation instead of translation):
Af = (ω  Vf ) = [ω  (ω  R)]
• Effective force on m is:
 Feff  S+mg0 - (mω)  [ω  (r + R)] - 2m(ω  vr)
• Rewrite as: Feff  S + mg - 2m(ω  vr)
Where, mg  Effective Weight
g  Effective gravitational field (= measured
gravitational acceleration, g on Earth surface!)
g  g0 - ω  [ω  (r + R)]
• Considering motion of mass m, at point r near
Earth surface. R = |R| = Earth radius.  |r| << |R|

ω  [ω  (r + R)]  ω  (ω  R)

Effective g near Earth surface:
g  g0 - ω  (ω  R)
• If m is at point r far from Earth surface, must consider
both R & r terms. Effective g for any r:
g = g0 - ω  [ω  (r + R)]
• Second term = Centrifugal force per unit mass
(Centrifugal acceleration).
• Centrifugal force:
– Causes Earth oblateness (g0 neglects). Goldstein
discussion, p 176
– Earth  Solid sphere. Earth  Viscous fluid with solid
crust.
– Rotation  “fluid” deforms,
Deviation of g from
 Requator - Rpole  21.4 km
local vertical direction!
gpole - gequator  0.052 m/s2
– Surface of calm ocean water is  g instead of g0.
• Summary: Effective force:
Feff = S + mg - 2m(ω  vr)
(1)
Where, g = g0 - ω  [ω  (r + R)]
(2)
Often,
g  g0 - ω  (ω  R)
(3)
These are all we need for motion near the Earth!
Direction of g
• Consider: g = g0 - (ω)  [ω  (r + R)]
(2)
• Effective g = Eqtn (2). Consider experiments.
Magnitude of g: Determined by measuring the period of a
pendulum (small θ). DIRECTION of g: Determined by the
direction of a “plumb bob” in equilibrium.
• Magnitude of 2nd term in (2):
ω2R  0.034 m/s2  (ω2R)/(g0)  0.35%
• Direction of 2nd term in (2): Outward from the axis
of the rotating Earth. Direction of g = Direction of
plumb bob = Direction of the vector sum in (2).
Slightly different from the “true” vertical  line to the
Earth’s center. (Figure next page!)
• Direction of plumb bob = Direction of
g = g0 - (ω)  [ω  (r + R)]
(2)
• Figure: (r in figure = r in previous figures!)
Deviation of g from g0 direction is exaggerated!
r=R+z
where z = altitude
Coriolis Effects
• Effective force on m near Earth:
Feff = S + mg - 2m(ω  vr)
- 2m(ω  vr) = Coriolis force. Obviously, = 0
unless m moves in the rotating frame (moving
with respect to Earth’s surface) with velocity vr.
• Figure again:
- 2m(ω  vr) = Coriolis force.
• Northern Hemisphere: Earth’s
angular velocity ω is in z direction
in inertial system (North) ω  ωez
ez  unit vector along z (Figures):
 In general, ω has components
along x, y, z axes of the rotating
system. All can have effects,
depending on the direction of vr.
• Most dominant is ω component
which is locally vertical in rotating
system, that is ωz  Component along local vertical.
- 2m(ω  vr) = Coriolis force, Northern hemisphere.
– Consider ωz only for now.
• Particle moving in locally horizontal plane (at Earth
surface): vr has no vertical component.
 Coriolis force has horizontal component only, magnitude =
2mωzvr & direction to right of particle motion (figure).
 Particle is deflected to right of the original direction:
• Magnitude of (locally) horizontal component of
Coriolis force  ωz = (locally)
vertical component of ω  (Local)
vertical component of ω depends on
latitude! Easily shown:
ωz = ω sin(λ), λ = latitude angle
(figure).  ωz = 0, λ =0 (equator);
ωz = ω, λ = 90 (N. pole)
 Horizontal component of Coriolis force, magnitude = 2m
ωzvr depends on latitude! 2mωzvr = 2mωvrsin(λ)
• All of this the in N. hemisphere! S. Hemisphere: Vertical
component ωz is directed inward along the local vertical. 
Coriolis force & direction of deflections are opposite of N.
hemisphere (left of the direction of velocity vr )
• Coriolis Deflections: Noticeable effects on:
• Flowing water (whirlpools)
• Air masses  Weather.
Air flows from high pressure
(HP) to low pressure (LP)
regions. Coriolis force deflects it. Produces
cyclonic motion. N. Hemisphere: Right
deflection: Air rotates with HP on right, LP
on left. HP prevents (weak) Coriolis force
from deflecting air further to right.
 Counterclockwise air flow!
S. Hemisphere: Left deflection.
(Falkland Islands story)
Bathtub drains!
• More Coriolis Effects on the Weather:
• Temperate regions: Airflow is not along pressure isobars due to the
Coriolis force (+ the centrifugal force due to rotating air mass).
• Equatorial regions: Sun heating the Earth causes hot surface air to
rise (vr has a vertical component).
 In Coriolis force need to account ALSO for (local) horizontal
components of ω
Northern hemisphere: Results in cooler air moving
South towards equator, giving vr a horizontal
component . Then, horizontal component of Coriolis
force deflects South moving air to right (West) Trade
winds in N. hemisphere are Southwesterly.
Southern hemisphere: The opposite!
No trade winds at equator because Coriolis force = 0 there
All is idealization, of course, but qualitatively correct!