CLASSICAL MECHANICS| Acceleration of a particle| Coriolis force and Centripetal force

Acceleration of a particle with respect to a not rotating frame of reference

Let us consider an inertial frame of reference S (x, y, z) and an non inertial frame of reference S'(x’,y’,z’) where origin of both frames coincide and S’ is rotating with constant angular velocity with respect to S frame.

Let P(x, y, z) be the position Let of a particle at any instant t and P(x’,y’,z’) with respect to S’ frame.

Position vector of the particle —

with respect to s frame,

with respect to s’ frame,

Thus velocity of the particle (u) with respect to S frame,

corresponding hungry operator

velocity of this particle in S frame.

velocity of the particle is S’ frame .

———–3

Accelerations of the particle with respect to S frame,

Using hungry operator,

we get

Where

is the acceleration of the particle with respect to S’ frame.

The last term

is the acceleration due to change in w. Since w is constant

——————–4

acceleration of the particle with respect to S’ frame.

acceleration of the particle with respect to S frame.

————-Coriolis acceleration

—————Centrifugal acceleration

Define Coriolis force and Centripetal force

Multiplying equation 4 by m(mass of the particle) on both sides

Where F’ effective force acting on the particle in the rotating frame (S’)

F true force acting on the particle in the inertial frame (S)

F0 fictitious force acting on the particle in the non-inertial / rotating frame (S’)

Note : Again

i.e fictitious force = Coriolis force + Centripetal force

Definition of Coriolis force

Coriolis force is defined as the fictitious force which acts on the particle when it is in motion relative to a rotating frame of reference, according to the name of the inventor.

Coriolis force =

i.e. Coriolis force is

i)proportional to angular velocity w of the rotating frame

ii) proportional to the velocity u‘ of the particle relative to rotating frame.

The direction of the Coriolis force is always perpendicular to the

i) angular velocity (w) of the rotating frame and

ii) velocity of the particle (u‘) relative to rotating frame

N.B.- If

$i) θ = 0$

Coriolis force =0

$ii) θ = π / 2,$

Coriolis force=

( maximum)

Definition of centrifugal force

Centrifugal force is defined as the fictitious force which acts on a particle at rest relative to a rotating frame of reference.

Centrifugal force

Discuss the effect of Coriolis force on a particle moving on the surface of earth ( or Coriolis force due to earth’s rotation) :

Let ρ be a point on the earth surface at a latitude λ

Hence co-latitude of the point =

$ϕ = π / 2 - λ$

Consider a rectangular coordinate system with the origin at ρ , z-axis is taken vertically upwards at ρ.

X-Y plane is the horizontal plane containing the point p, the x- axis pointing towards the East and Y-axis pointing towards the East and Y- axis pointing towards the North.

The earth rotates from west to east. The angular velocity vector w lies in Y-Z plane in a direction parallel to the polar axis about which the earth rotates. Hence w has no component along east-west direction i.e. Wx = 0 .

when the particle is at rest

————————-1

Let a particle projected from p with a velocity u‘ such that

Coriolis force acting on a particle moving in a velocity u‘ in a reference frame rotating with an angular velocity is given by

as the particle has been projected horizontally, its vertical component

Magnitude of Coriolis force;

i) Horizontal component :

Horizontal component of Coriolis force =

Magnitude

N.B.- Horizontal component is maximum at λ= 90° i.e. at the poles

Horizontal component is minimum at λ= 0° i.e. at the equator.

ii)Vertical component :

Vertical component of Coriolis force

Magnitude

Difference of Coriolis force and centrifugal force

One thing they both have in common. Both of them are “fictitious” of “pseudo” forces. That is a technical term which, refers to an apparent force caused by treating an accelerating frame of reference as an inertial one.

1. Briefly, we can say that a centrifugal force appears to throw something outwards in a rotating frame (in reality, it is just the object attempting to follow the inertial path – the real force is the centripetal force which is accelerating the object towards the centre of that rotation).

The Coriolis force is that appears to cause the apparent rotational motion of something on the surface of a rotating frame of reference. Actually, it is just an inertial path that an object is taking as it appears to an observer on that surface. It can be viewed as the surface rotating just below the object. It’s what appears to drive the rotational motion of hurricane’s and other such storms.

2. The Centrifugal Force is proportional to the square of the rotation rate.

But, the Coriolis Force is proportional to the rotation rate.

3.The Centrifugal Force is related to the body’s distance from the rotating frame’s axis.

The Coriolis Force is proportionate with the velocity vector that is orthogonal to the rotation axis.