1. The Coriolis Force
QMUL Interview
7th July 2016

2. Feff = |F – mω x (ω x r) – 2m(ω x v)| = ma
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
Fcentrifugal
FCoriolis
Feff = |F – mω2r ur – 2mωvsinθuθ | = ma ^
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

3. 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
Only acts on objects moving in rotating frame
Points perpendicular to velocity, v, & ω vectors
Proportional to velocity perpendicular to axis
= 2mωvsinθuθ ^ v ω
FCoriolis

4. 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
Only acts on objects moving in rotating frame
Points perpendicular to velocity, v, & ω vectors
Proportional to velocity perpendicular to axis
= 2mωvsinθuθ ^
Rotating frame

5. Coriolis effect on Earth
In N hemisphere, earth rotates anti-clockwise
Horizontal component of the centrifugal (CF) and gravity (g) forces balance for object at rest
Object going E (in to paper) has a larger rotational velocity
Increases horizontal component of CF  deviated S (to right) due to net Coriolis force gh
CFh
CFv CF ω g gv ω g gv
CFv gh CF
Coriolis
Force
CFh
Zero at equator where CF is vertical and max at poles where it is horizontal

6. Coriolis effect on Earth
In N hemisphere, earth rotates anti-clockwise
Horizontal component of the centrifugal (CF) and gravity (g) forces balance for object at rest
Object going W (out of paper) has a smaller rotational velocity
Decreases horizontal component of CF  deviated N (to right) due to net Coriolis force gh
CFh
CFv CF ω g gv ω g gv
CFv gh CF
Coriolis
Force
CFh
Opposite in southern hemisphere as rotates clockwise  to left

7. Coriolis effect on Earth
In N hemisphere, earth rotates anti-clockwise
Horizontal component of the centrifugal (CF) and gravity (g) forces balance for object at rest
Object going N, also moves E seen in inertial frame due to earth’s rotation
CFh points out from centre of net rotation  direction has changed giving component to east (right) gh
CFh
CFv CF ω g gv ω
CFh
CFhNS
CFhEW gh
Coriolis
Force
Opposite in southern hemisphere as rotates clockwise  to left

8. Coriolis effect on Earth
In N hemisphere, earth rotates anti-clockwise
Horizontal component of the centrifugal (CF) and gravity (g) forces balance for object at rest
Object going S, also moves E seen in inertial frame due to earth’s rotation
CFh points out from centre of net rotation  direction has changed giving component to west (right) gh
CFh
CFv CF ω g gv ω
CFh
CFhNS
CFhWE gh
Coriolis
Force
Opposite in southern hemisphere as rotates clockwise  to left

9. 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 π/(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

10. 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 L H
Northern Hemisphere L H
Southern Hemisphere
 = Pressure gradient force
 = Coriolis force
Net result is the formation of cyclones that rotate anti-clockwise (clockwise) in northern (southern) hemisphere

11. 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 L H
Northern Hemisphere
Northern Hemisphere L H
Southern Hemisphere
Southern Hemisphere
Eye of storm when balance
Net result is the formation of cyclones that rotate anti-clockwise (clockwise) in northern (southern) hemisphere

12. 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

13. 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