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ABSTRACT

In the text below Coriolis force will be derived solely relying on the mathematical vectors’ identities without need to visualize or conceive any physical abstraction at all, just successive application of mathematics will be used to derive this mechanical effect. This application of strict mathematical identities is suitable for students to understand the effect and the physical and mathematical reasons for its occurrence in real world, outside of abstract mathematical formulas. There is expectation in Classical Mechanics that all formula can be derived relaying only on two basic formulas and , therefore Coriolis force is not an exception at all, as it is conceivably shown in the text below. This effect is very important for understanding of turbulence and Turbo Effect in fluid dynamics.

STUDENT FRIENDLY DERIVATION OF CORIOLIS FORCE

For the correct derivation of the Coriolis force is necessary to be found following formula for the time derivations of the absolute value of a vector:

(1)

And also a formula of time derivation of the corresponding ort vector should be found:

Coriolis velocity is the velocity of approaching to axis of rotation by its definition:

(3)

And:

It is important here to be noticed that v_{c} is not the velocity in the respect to
the rotating platform as it is frequently citied in the available literature
from the domain of the Classical Mechanics, but it is velocity in the respect
of axis of rotation. The following figure depicts a groove with a grey ball
moving inside it with the constant speed v_{0} on the rotating bluish
disk. The groove contains green, purple and orange sections of the path and the
Coriolis force appears only on trajectory in the purple sections of the groove
because only in this section relative velocity to the respect of the center of
rotation is not trivial, i.e. different than zero as it is case with green
sections:

Fig. 1

Otherwise the Coriolis force would be twice stronger than the Centrifugal force on the greenish
parts of the trajectory and would act in the opposite direction than the
centrifugal force which is duly impossible - on these sections only centrifugal
force acts without suitable circumstances for appearance of Coriolis force.
Therefore Coriolis velocity is
undoubtedly the velocity of the body in respect to the axis of rotation only.
Most misunderstanding of the Coriolis’ effect was based on the false grasp of
the Coriolis’ velocity which
is somehow wrongly defined in the most of the textbooks of Classical Mechanics
as the velocity in respect to the rotational disk’s plane.

According (2) is:

According (2) too:

According (3) is:

(7)

=>

(8)

It is also , so following formula is obtained:

(9)

=>

Or:

(11)

The first term denotes force caused by acceleration of the body in respect to the axe of
rotation; the second term denotes force cause by angular acceleration of the
platform, the third term denotes centrifugal force that appears when body runs
on arc with constant distant from the axe of rotation (greenish section on above
figure) and finally the fourth term denotes precious Coriolis force which
appears only when the body has steady velocity on the line perpendicular to the
axe of rotation. So, v_{C} is the velocity of going away from the axe
of rotation and not any velocity of the body in respect to the platform.

Above formula clearly states that there is no Coriolis force on the greenish parts of the
trajectory depicted on the Fig. 1 because the Coriolis velocity is equal
to zero due to the constant radius of these paths making the term dedicated to
the Coriolis force trivial.

According both (5) and (6) the work of Coriolis force is:

(12)

Derivation of the work done by Coriolis force is:

(13)

=>

(14)

=>

(15)

=>

(16)

=>

(17)

=>

(18)

=>

(19)

=>

(20)

=>

(21)

=>

(22)

=>

(23)

=>

(24)

=>

(25)

=>

(26)

As angular velocity is always perpendicular to the radius vector, i.e. , we have:

(27)

Therefore Coriolis force is not reactive or conservative one as frequently considered in
literature, therefore it is real force fully capable to perform active work.

It is pertinent place to remind us that conservative or passive forces are the ones producing reactive
power (usually forces perpendicular to the trajectory) and the active forces
are fully capable to produce real work as they are not perpendicular to the
trajectory.

Active power is scalar
defined as:

(28)

Reactive power is vector defined as:

(29)

This is pure evidence that this force is not cause of the declination of missiles or pendulums’ rotations due to the globe’s rotation, quite contrary the pendulum or the bullet declines from the globe plane due to absence of the Coriolis force that exists on the globe’s surface. Any pendulum is some sort of vibrating gyroscope and the gyroscopic effect causes pendulums to roate. Gyroscopic effect and Coriolis’ effect are quite different ones in their very essences; Coriolis Effect is planar or 2D effect while gyroscopic effect is duly spatial or 3D effect.

Author:

Dipl.El.-Ing. Andrija Radović

andrijaradovic@hotmail.com

http://www.andrijar.com

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Author:

Dipl.El.-Ing. Andrija S. Radović

Tel: +381 64 1404075