Newton's 2nd Law & Coriolis force - Lecture 23

Newton's 2nd Law of Mechanics
F = M * A
M = mass of an object , kg
A = absolute acceleration
F = total actual force acting on the object;
unit : kg * m / (s**2)
F and A are both vectors

reference frame ,
* Absolute acceleration of an object is measured
in a reference frame at rest :
Such coordinate is called ( Inertial Coordinate)
* Such V is absolute velocity & A is absolute acceleration.
* wind is NOT absolute velocity
It is not measured in an inertial coordinate
* wind is a measure of the velocity of air parcels relative to a weather station ;
i.e. relative to a location on a rotating earth ,
Hence wind is a velocity in a Rotating Coordinate
It is a relative velocity .
* A change of wind is a relative acceleration
* if A refers to the relative (instead of absolute) acceleration in the formula above,
F would refer to more than just the actual force
We need to consider the effects associated with the rotation of the earth.
Those effects are called "forces" &
we include them in "F"

Forces that affect wind :
(1) Coriolis force
(2) Centrifugal force
(3) Pressure gradient force
(4) Gravitational force
(5) Frictional force

How to analyse/understand the nature of wind measured on earth?
To begin with, we need to introduce a force called
Coriolis Force

F(Coriolis)
= 2 * V * W * sin f

where V = wind speed
W = rotation rate of the earth,
one revolution per day
f = latitude
The dependence of latitude appears because only the local vertical component of the earth's rotation affects the horizontal wind.

Direction of F(Coriolis) is 90 degree to the right of the wind vector in Northern Hemisphere.
e.g. F(Coriolis) is southward if the wind is westerly ;
Opposite in southern hemisphere .
F(Coriolis) only tends to change the direction of the wind & does not change the wind speed.

What is the physical nature of F(Coriolis) ?
We can get a deeper understanding of F(Coriolis) by
relating it to the basic notion of "conservation of angular momentum" .

Illustrated with a simple example :
Coriolis demo
* two players, A and B, standing at opposite sides of a merry-go-round that rotates in counterclockwise direction,
* A throws a ball straight towards B
* B cannot catch the ball since it goes to the right of B .
* B would conclude that the ball has been deflected
to the right presumably by a certain force.

* Note : A & B are on a rotating coordinate
The rotational movement of an object with respect to an axis has a property called "Angular Momentum"

absolute angular momentum
= (absolute tangential velocity)
* (distance from the axis of rotation)

"tangential velocity" is the component of the velocity
in the direction of the curve of rotation.

absolute tangential velocity of the ball
= relative tangential velocity of ball
+ tangential velocity due to merry-go-round

Newton's law of motion implies that,
if there is no force acting on the ball,
its absolute angular momentum would not change.

* Recall, A throws a ball radially towards B
* When the ball leaves A's hand,
the ball has zero relative tangential velocity;
it only has the velocity of the merry-go-round
* initial absolute tangential velocity
= the velocity of merry-go-round at A's position

* The original amount of absolute angular momentum in the ball does not change
after it leaves A's hand because no real force acts on it thereafter.
i.e. Conservation of absolute angular momentum
* As soon as the ball leaves A's hand,
its distance to the axis of the merry-go-round decreases.
* This means that its new angular momentum associated with
the rotation of the coordinate becomes smaller.
* For the absolute angular momentum to remain the same,
the ball's relative angular momentum must increase
to make up for the difference
* Consequentially, a nonzero relative tangential velocity of the ball would appear;
i.e. The ball begins to move tangentially from its initial radial path ;
hence deflecting to the right .
* To A & B there appears to be a force
(first recognised by a French engineer, Coriolis)
causing the deflection.

* a person standing on the ground is not on such a rotating coordinate;
he sees a ball going straight.
To him, there is no Coriolis force acting on the ball.
* Coriolis force also referred to as an apparent force;
Only apparent to observers on a rotating coordinate .

* rotation of the earth is very slow compared with that of the merry-go-round
Hence its effect on the ball is negligibly small,
compared to the effect of the rapidly rotating merry-go-round
as far as A & B are concerned

How might one describe what happens
to an air parcel moving in a curved trajectory ?
e.g. wind near a trough
Two equivalent points of view :

(1) One could say that
the parcel has a centripetal acceleration.
A(centripetal) = V**2 / R
where V= speed of parcel
Its direction is 90 degrees to the "inside of the trajectory", called inward normal
R = radius of curvature

(2) Alternatively, one could say that
there is a centrifugal force acting on the parcel.
The direction of this force is outward normal to the trajectory.
with a magnitude
F(centrifugal) = V **2 / R
Interpretation (2) is often used ,for convenience, e.g. when we discuss the balance of forces acting on air parcels later on.

question for thought :
Why doesn't the moon crash down on earth under the influence of the gravitational force of the earth ?


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