COORDINATE SYSTEM

&

FRAMES OF REFERENCE

BSc – I (UNIT III)

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Contents

1) Coordinate System

Cartesian, Plane Polar and Spherical Polar 3

2) Position Vector

In Two dimensional Cartesian Coordinate System 4

In Three dimensional Cartesian Coordinate System 4

3) Unit Vector

In Plane Polar Cartesian Coordinate System 5

In Spherical Polar Cartesian Coordinate System 6

4) Position Vector

In Two dimensional Cartesian Coordinate System 7

In Three dimensional Cartesian Coordinate System 7

In Plane Polar Cartesian Coordinate System 8

In Spherical Polar Cartesian Coordinate System 9

5) Relationship between Cartesian and Plane Polar Coordinates

Velocity 11

Acceleration 11

6) Relationship between Cartesian and Spherical Polar Coordinates

Velocity 12

7) Frames of Reference

Inertial Frame 13

Non – Inertial Frame 13

8) Pseudo Force 13

9) Coriolis & Centrifugal Force

Uniformly Rotating Frame (Explanation of Centrifugal & Coriolis Force 14

Effect of Centrifugal Force on the value of Acceleration due to gravity on Earth 17

Direction of 18

Components of Coriolis Force due to rotation of the Earth on a particle projected horizontally 18

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In classical mechanics, space is considered to have three dimensions. The

three dimensions are represented by three mutually perpendicular axes

& . The fixed point at which these three axes meet is called the origin. A

point in space is represented by three coordinates & which are

distance of the point from the origin along the axes & .

In a two dimensional case, the motion takes place in a plane. The two coordinates for a point in this

system are given by,

a) The radial distance of the point from the origin. It is denoted by . It is

always positive and can take any value between zero and infinity.

b) The angle which the radius vector makes with the positive direction of

-axis. It is denoted by . It is always measured in anti-clockwise

direction with respect to the positive direction of -axis, , and can

take any value between & .

If & are the cartesian coordinates of the point P, then

&

Hence, we get

&

The spherical polar coordinate system provides a convenient method of representing the coordinates of a

point on the surface of a sphere. The three coordinates for a point P in this system are,

a) The radial distance of the point from the origin. It is denoted by . It is always positive and can

take any value between zero and infinity.

b) The angle that the radius vector makes with the positive direction of -axis. It is called the co-

latitude angle or zenith angle and is denoted by . The angle can take any value between & .

c) The angle with which the vertical plane containing , known as the azimuthal plane, makes with the

plane. It is called the azimuthal angle and is denoted by . It is the angle, measured anti

clockwise, that line makes with the positive direction of -axis,

, and can take any value between & .

If & are the cartesian coordinates of the point P, and the

projection of in the plane is , then

Also, we get

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Again,

And,

Consider a particle in the plane having Cartesian coordinates & at any time, then

Position vector of the point

Using the properties of vectors, we get

Where, & are the unit vectors in the direction of and respectively.

Also,

The magnitude of the position vector is given as,

If is the unit vector in direction of , then .

Consider a particle in the space having Cartesian coordinates & at any time, then

Position vector of the point

From , draw perpendicular on the plane meeting the plane at . Draw parallel to axis

meeting the axis at and parallel to axis meeting the axis at . The distances & are

the & coordinates of point .

Using the properties of vectors, we get

Where, & are the unit vectors in the direction of and

respectively.

Also,

The magnitude of the position vector is given as,

If is the unit vector in direction of , then .

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The two unit vectors in two dimensional plane polar coordinates are

a) The radial unit vector along the direction of increasing radius vector .

b) The unit vector along the direction of increasing .

The directions of and are perpendicular to each other.

If is a point having Cartesian coordinates and polar

coordinates , then .

The unit vector will be along and the unit vector will be

perpendicular to .

Now,

The unit vector is given as,

Let us consider a point having polar coordinates . Then

is given as

Therefore vector is given as

But and . Hence, we get

The vector points in the direction of increasing . The magnitude of the vector is given as,

Hence, the unit vector is given as

We have,

&

Hence, we get

Now taking scalar product of the unit vectors we get

Hence, the two unit vectors are perpendicular to each other. This implies that the plane polar coordinate

system is orthogonal.

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The three unit vectors in spherical polar coordinates are

a) The radial unit vector along the direction of increasing radius vector .

b) The unit vector along the direction of increasing .

c) The unit vector along the direction of increasing .

The directions of and are perpendicular to each other.

If is a point having Cartesian coordinates and

spherical polar coordinates , then .

The unit vector will be along and the unit vectors

and will be perpendicular to .

Now,

The unit vector is given as,

Now since unit vector is perpendicular to unit vector and leads by .

Therefore, we get

Now, we draw perpendicular on plane and let

The unit vector will be perpendicular to unit vector and lead it by .

We have,

Hence,

Therefore , we get

We have,

Hence, we get

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Now taking scalar product of the unit vectors we get

Hence, the three unit vectors are mutually perpendicular to each other. This implies that the spherical

polar coordinate system is orthogonal.

The velocity of a particle is defined as the time rate of change of the position vector.

Hence,

Now, . Hence, velocity is given by,

The magnitude of the velocity is given by,

Acceleration is defined as the time rate of change of velocity. Hence, acceleration in two dimensional

Cartesian coordinate system is given as,

The magnitude of the acceleration is given by,

Where, and are the and components of acceleration respectively.

The velocity of a particle is defined as the time rate of change of the position vector.

Hence,

Now, . Hence, velocity is given by,

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The magnitude of the velocity is given by,

Acceleration is defined as the time rate of change of velocity. Hence, acceleration in three dimensional

Cartesian coordinate system is given as,

The magnitude of the acceleration is given by,

Where, and are the and components of acceleration respectively.

The velocity of a particle is defined as the time rate of change of the position vector.

Hence,

Now, . Hence, velocity is given by,

But,

,

and

Hence,

Components of velocity: The two components of the velocity are,

a) The quantity is called the radial component of velocity and is due to change in magnitude of

, remaining constant.

b) The quantity is called the transverse component of velocity and is due to change in

magnitude of , remaining constant.

The magnitude of radial velocity is, and that of transverse velocity is . If is a

constant , the angular velocity then .

The magnitude of the velocity is given by,

Acceleration is defined as the time rate of change of velocity. Hence, acceleration in Plane polar

coordinate system is given as,

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But

and

. Hence, we get

Components of acceleration: The two components of the acceleration are,

a) The quantity is called the radial component of acceleration and its direction is along

. It consists of two parts,

i. The quantity gives the acceleration due to change in magnitude of . It has a positive sign and

is directed away from the centre.

ii. The quantity gives the centripetal acceleration due to change in . It has a negative sign

and is directed towards the centre. If is a constant , the angular velocity then .

b) The quantity is called the transverse component of acceleration and its direction

is along . It consists of two parts,

i. The quantity gives the acceleration due to change in magnitude of .

ii. The quantity arises due to interaction of linear and angular velocities due to changes in

and respectively. This is similar to Coriolis acceleration.

The magnitude of the acceleration is given by,

The velocity of a particle is defined as the time rate of change of the position vector.

Hence,

Now, . Hence, velocity is given by,

But,

Hence,

Components of velocity: The three components of the velocity are,

a) The quantity is called the radial component of velocity and is along the direction of .

b) The quantity is called the angular velocity and is along the direction of .

c) The quantity is along the direction of .

The magnitude of radial velocity is, , that of angular velocity is and that of

. The magnitude of the velocity is given by,

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i. If a particle travels only in a straight line it has only radial velocity, .

ii. If it moves along a circular path with a uniform velocity, it has only angular velocity .

iii. If the particle moves along a curved path in a plane it has both the components along and

along at right angles to .

iv. When the particle moves in a curved path in space, it has in addition the third component in

the plane perpendicular to the plane in which and lie. is the component of in the

plane and gives the angular velocity due to rate change of in the direction

perpendicular to the vector .

Acceleration is defined as the time rate of change of velocity. Hence, acceleration in Spherical polar

coordinate system is given as,

But,

But,

Hence, we get

Where,

The magnitude of the acceleration is given by,

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If and are the Cartesian coordinates of the point P having and as the coordinates in plane polar

coordinate system, then

and

Differentiating and with respect to time , we get

Multiplying by and by and adding, we get

Multiplying by and by and subtracting from , we get

If and are the Cartesian coordinates of the point P having and as the coordinates in plane polar

coordinate system, then

and

Differentiating and with respect to time , we get

and

Differentiating velocities, and with respect to time , we get

Multiplying by and by and adding, we get

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Multiplying by and by and subtracting from , we get

If and are the Cartesian coordinates of the point P having and as the coordinates in spherical

polar coordinate system, then

, and

Differentiating and with respect to time , we get

Multiplying by , by and by , we get

On adding, we get

Again we have,

Now,

Multiplying both sides by , we get

Also multiplying by gives,

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Subtracting and , we get

On multiplying by and by , we get

Subtracting from , we get

But . Hence, we get

A reference frame in which Newton’s first law of motion, i.e. law of inertia, holds good is known as the

inertial frame of reference. In an inertial frame of reference, a body continues to be in rest or in uniform

motion until an external force acts on it. All the Newton’s laws of motion hold good in an inertial frame.

All frames of reference moving with a constant velocity with respect to an inertial frame are also inertial

frames of reference. They can hence be called as the unaccelerated frames of reference. The best

approximation of an inertial frame is intergalactic space.

When a frame of reference is accelerated relative to an inertial frame, the form of basic physical laws

changes. Such frames of reference are called as Non – inertial frames of reference. One of the best

examples of a non – inertial frame of reference is the Earth.

Let us consider a non – inertial reference frame be moving with an acceleration relative to an

inertial frame . Now let us consider a particle of mass in which no external force is acting on the

reference frame . Then the acceleration of the particle in the frame is zero but the particle will appear to

have an acceleration relative to frame . Thus for an observer in frame, a force will

appear to be acting on the particle whereas for an observer in the frame , there is no force acting on the

particle.

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Let initially the frame and coincide with each other at time . The frame starts from rest

and moves away with an acceleration . Then displacement of the origin in time is given as

. If is a particle moving in space and and are its position vectors at time in the

frames and respectively, then and .

Now,

.

Differentiating the equation twice with respect to , we get

Where is the acceleration in the inertial frame and is the acceleration in the non – inertial frame .

If is the mass of the particle then,

When no real force acts on the particle in the inertial frame, . Hence, we get

Thus, we find that in the frame , the particle appears to experience a force even when no force is

acting on it in the inertial frame . This force arising due to motion of frame with respect to the frame

is called as Fictitious force, Apparent force or Pseudo force.

Let be an inertial frame of reference represented by coordinates and be a frame sharing the

common origin with and represented by coordinates which is rotating with angular velocity

with respect to frame . Frame is a non-inertial frame of reference.

Let the position vector of a point in the inertial frame be represented by,

Where are the coordinates of the point and are the unit vectors along the and

directions respectively in the inertial frame

The components of the position vector in the rotating non-inertial frame is represented by,

Where are the coordinates of the point and are the unit vectors along the and

directions respectively in the inertial frame

As the frame is rotating, the directions of and axes are continuously changing. The directions

of unit vectors are hence also changing. Hence, although constant in magnitude keep on

changing direction. Hence,

are not zero.

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But unit vectors in the inertial frame are constant in magnitude as well as direction. Hence,

are all zero.

Differentiating equation with respect to time , we get

Now, let us denote velocity in frame by and that in

frame by . Hence,

and

.

As the linear velocity of a particle is given by,

, where, is the angular velocity. Hence, we have

Hence, we get

Hence, the velocity of the particle in the inertial frame is the vector sum of its velocity in rotating non-

inertial frame and the linear velocity which arises due to rotation, the value of which at any instant

depends upon the value of at that instant.

The relation

holds good for other similar vectors and may be expressed as an

operator equation,

Applying this operator to the velocity vector , we get

Substituting the value of from equation in the RHS of the above equation, we get

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Where, is the acceleration in the inertial frame and is the acceleration in the rotating non – inertial

frame.

The term is known as the Coriolis acceleration. It appears only when the particle moves in a

frame which is rotating.

The term is the centripetal acceleration.

The term

is the acceleration due to change in angular velocity, . If the angular velocity is

constant then the term disappears. Hence, we get

According to Newton’s second law of motion, the effective force on the particle of mass in the

rotating frame is given as . Hence, equation implies,

Where, is the fictitious or the pseudo force acting in the non – inertial frame and is given by,

As, is a constant,

. Hence, the last term vanishes. Thus, we have

The Coriolis force is given by . It is defined as the fictitious

force which acts on a particle when it is in motion relative to a rotating frame of reference. It acts only

when the particle in the non-inertial rotating frame is not at rest or the frame itself is not at rest.

The direction of Coriolis force is always perpendicular to and i.e. the angular velocity of the rotating

frame and the velocity of the particle relative to the rotating frame. The negative sign indicates that

Coriolis force is in the direction opposite to that given by the right hand screw rule. If the angle between

and be and is the unit vector perpendicular to both and .

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If , . This is when the angular velocity vector of the rotating frame and the linear velocity

of the particle relative to the rotating frame are parallel.

If , i.e. when and are perpendicular to each other, Coriolis force is maximum.

The centrifugal force is given by . It is defined as the fictitious

force which acts on a particle at rest relative to a rotating frame of reference. It is numerically equal to

the centripetal force but is oppositely directed i.e. outward away from the axis of rotation.

We take a frame fixed on earth to be a rotating frame of reference rotating with an constant angular

velocity . Let the particle be at rest. Let is the actual acceleration due to gravity and be the

acceleration due to gravity in the rotating frame. We can express as,

Since, the particle is at rest, and hence second term becomes zero.

Also, since the angular velocity is constant,

and hence last term becomes zero. Hence, we get

Let us take angular velocity along the axis. The actual value of at a point in the latitude will

hence act along the direction towards the centre of the earth . The components of will be,

Let us draw perpendicular on axis and let where is the radius of the circle in which the

particle rotates. If be the radius of earth then, .

The observed value of acceleration due to gravity at will be hence,

Magnitude of is as,

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Taking radius of earth to be, , we get

Hence the value of acceleration due to gravity at different latitudes is,

At equator, . Hence,

At the poles, . Hence, . Therefore decrease in the value of due to rotation is zero. The

observed difference between the value of at the poles and the equator is about . This is

due to the fact that the earth is flattened at the poles.

At latitude , . Hence, .

Hence, if the earth stops rotating, the value of at the equator will increase by and that

at latitude will increase by . The effect of the centrifugal force due to rotation of

earth on the acceleration due to gravity is maximum at the equator and minimum at the poles.

The direction of will not be exactly along the vertical but

will be deviated from the vertical by an angle . The value of

can be found out by applying the law of sines to the triangle

in which various sides represent the acceleration vectors in

magnitude and direction. Hence, we get

As is small, . Hence, we get

Now we have, and . Hence, we get

.

The deflection at latitude will be,

Hence, a plumb line at will not point in the true vertical direction but along which is displaced

from the vertical by a small angle equal to .

Let be a point on the surface of the earth at latitude . Consider a rectangular Cartesian system with its

origin at point . The axis of this system is taken vertically upward at , radially outward from the

centre of the earth. The plane is the horizontal plane containing the point . The axis points

towards the East and axis points towards North. The earth rotates from West to East and the positive

direction of axis is taken towards the East. The angular velocity vector lies in the plane in a

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direction parallel to the polar axis about which the earth rotates. The vector has no component in the

East – West direction i.e. axis. Hence, can be represented as, .

Let a particle be projected from with velocity, . As the particle has been projected

vertically, the vertical component of its velocity , and its horizontal components are .

The Coriolis force is given as, . Hence, we get

The magnitude of Coriolis force is,

The co – latitude can be defined as

. Hence, the above equation can be written as,

The magnitude of horizontal component is given by,

The horizontal component of Coriolis force has maximum value at the poles and minimum at

equator .

The magnitude of horizontal component is given by,

At the equator if the body has an initial velocity , it will appear to be lifted upwards.

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