Download Coriolis Force

The Coriolis Force
QMUL Interview
7th July 2016
Introduction
You all know Newton’s 2nd law, F = ma, but this is only valid in this form in an
inertial frame of reference
Inertial frame = at rest or in uniform motion
In a non-inertial frame, such as a rotating body, it needs to be modified by
introducing so-called fictitious or apparent forces
Non-inertial = accelerating w.r.t inertial frame
^ | = ma
Feff = |F – mω2r ^ur – 2mωvsinθu
θ
Fcentrifugal
FCoriolis
You’re all familiar with the first term:
the centrifugal force
Objects in a rotating frame feel
an outwards push, perp. to axis
This push has no physical origin
but is a consequence of inertia
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Coriolis force
The Coriolis force is another apparent force that only
acts when objects are moving in a rotating frame
Appears to deviate objects to right (left)
for anti-clockwise (clockwise) rotation
E.g. a ball thrown on a roundabout
In inertial frame ball travels straight
In rotating frame introduce Coriolis
force to explain deviation
ω
Coriolis force = 2mωvsinθu^ θ
Only acts on objects moving in rotating frame
Points perpendicular to velocity, v, & ω vectors
Proportional to velocity perpendicular to axis
FCoriolis
v
ω
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Coriolis force
The Coriolis force is another apparent force that only
acts when objects are moving in a rotating frame
Appears to deviate objects to right (left)
for anti-clockwise (clockwise) rotation
Inertial frame
E.g. a ball thrown on a roundabout
In inertial frame ball travels straight
In rotating frame introduce Coriolis
force to explain deviation
Coriolis force = 2mωvsinθu^ θ
Rotating frame
Only acts on objects moving in rotating frame
Points perpendicular to velocity, v, & ω vectors
Proportional to velocity perpendicular to axis
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Coriolis effect on Earth
In N hemisphere, earth rotates
anti-clockwise
Object going E (in to paper)
has a larger rotational velocity
Horizontal component of the
centrifugal (CF) and gravity (g)
forces balance for object at rest
ω
gh
Increases horizontal component of
CF  deviated S (to right) due to
net Coriolis force
ω
CFv
gh
CFv
CF
CF
g
gv
CFh
g
gv
CFh
Coriolis
Force
Zero at equator where CF is vertical and max at poles where it is horizontal 5
Coriolis effect on Earth
In N hemisphere, earth rotates
anti-clockwise
Object going W (out of paper)
has a smaller rotational velocity
Horizontal component of the
centrifugal (CF) and gravity (g)
forces balance for object at rest
ω
gh
Decreases horizontal component
of CF  deviated N (to right) due
to net Coriolis force
ω
CFv
gh
CFv
CF
CF
g
gv
CFh
g
gv
Opposite in southern hemisphere as rotates clockwise  to left
CFh
Coriolis
Force
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Coriolis effect on Earth
In N hemisphere, earth rotates
anti-clockwise
Object going N, also moves E seen in
inertial frame due to earth’s rotation
Horizontal component of the
centrifugal (CF) and gravity (g)
forces balance for object at rest
ω
gh
CFv
CF
g
gv
CFh
CFh points out from centre of net
rotation  direction has changed
giving component to east (right)
Coriolis
Force
CFh CF
hNS
ω
CFhEW
gh
Opposite in southern hemisphere as rotates clockwise  to left
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Coriolis effect on Earth
In N hemisphere, earth rotates
anti-clockwise
Object going S, also moves E seen in
inertial frame due to earth’s rotation
Horizontal component of the
centrifugal (CF) and gravity (g)
forces balance for object at rest
ω
gh
CFh points out from centre of net
rotation  direction has changed
giving component to west (right)
CFhNS
CFv
CFhWE Coriolis
Force
CF
g
gv
CFh
CFh
ω
gh
Opposite in southern hemisphere as rotates clockwise  to left
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The myth: water down a plug
People believe that the Coriolis force causes water to spin the opposite way
down the plug in northern/southern hemispheres
This is a myth: it can spin either way and the Coriolis force is way to weak to
affect it under normal situations
Consider unit mass (m) of water
moving at v=0.1m/s in a sink of
radius r=0.5m at θ=78o north
FCorilois = 2 m v ω sinθ
= 2 x 1kg x 0.1m/s x
2π/(24*60*60) rad/s x
sin(78o)
≈ 1.4x10-5 N
Fcent = mv2/r
= 1kg x (0.1 m/s)2 / 0.5 m
≈ 2x10-2 N
= 1400 x FCoriolis
v = 0.1 m/s
r = 0.5 m
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Cyclones: a real Corilois effect
Air wants to move straight from high to low pressure due to gradient force
But, as we have shown, the Coriolis force acts to deviate it to the right (left) in
northern (southern) hemisphere causing it to spiral
Northern Hemisphere
H
Southern Hemisphere
H
L
L
 = Pressure gradient force
 = Coriolis force
Net result is the formation of cyclones that rotate anti-clockwise (clockwise)
in northern (southern) hemisphere
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Cyclones: a real Corilois effect
Air wants to move straight from high to low pressure due to gradient force
But, as we have shown, the Coriolis force acts to deviate it to the right (left) in
northern (southern) hemisphere causing it to spiral
Northern Hemisphere
H
Southern Hemisphere
H
L
L
Net result is the formation of cyclones that rotate anti-clockwise (clockwise)
in northern (southern) hemisphere
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Summary
The Coriolis force is an apparent force, which explains the deviation of
moving objects in a rotating frame of reference
It deviates objects to the right (left) for anti-clockwise (clockwise) rotation
The deviation vanishes at the equator and reaches its maximum at the poles
It leads to the formation of cyclones which spin in opposite directions in the
northern and southern hemispheres
It is not responsible for water rotating in opposite directions as it
goes down the plug hole since it is much too weak on this scale
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Derivation
Velocity in inertial frame = v in rotating frame + v due to rotation
ω = constant angular velocity
r = position in rotating frame
Since nothing special about r, take it out to get an operator
Calculate acceleration in inertial frame
Apply to Newton’s 2nd law: FI = maI
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