PH101: PHYSICS1

Lecture 3

�

�

�

�

��

II. Cylindrical coordinate system �, �, �

How to specify a point ‘P’ in space ?

�, � Coordinate of the foot

of the point in XY plane

�-Height from the XY plane

�, �, � coordinates system is known as cylindrical coordinate system

Why the name cylindrical?

Point ‘P’ is the intersection of three surfaces: A cylindrical surface � � �������; A half plane containing �-axis with �=constant and a plane

�=constant.

�, �, �

O

P

Coordinate transformation: Cartesian to

cylindrical

�

�

�

�

��

��, �, ��

� � � ����� � � ����

� � �

Transformation equation is very similar to polar coordinate

with additional �-coordinate.

Reverse transformation

� � �� � �����

� � � �!��

�� � �

��, �, ��

Note: Instead of �, � many books use notation �", #).

Unit vectors in cylindrical coordinate system

�

�

�

�

��

�$

�$

�% �$, �% and �$ are unit vectors

along increasing direction of

coordinates �, �and �.

Polar coordinate unit vectors (�$, �%) +

additional unit vector in the � 'direction.

�$ = ���� �(+ ���� �(�% = - ���� �(+ ���� �(

�$ � �$

and * are orthogonal but their directions depend on location.

Position, Velocity, Acceleration, Newton’s law

in cylindrical coordinate system

+, � - � -. � �-�

/0 � 1 � � ��

2,=(3 ' -4� � �2-- � 3 �. � �3�

6, � 67 � 68. � 69�� :[(3 ' -4� � �2-- � 3 �. � �3�]

Position vector

Velocity

Acceleration

Newton’s law

Vector components are very similar to polar coordinate+

� 'component

=

>

?

��, �, ��

��

@

III. Spherical polar coordinate system

� → Radial distance from origin

� →Angle of radial vector with �-axis.

@ →Angle between X-axis and the

projection of radial vector in XY plane

�, �, @ is known as spherical polar coordinate

Note that point �, , #� is at the intersection of three surfaces

A sphere where � �Constant

A half plane containing �-axis and @= constant

A cone about �=axis with �=constant.Be careful, notations are different.

� and � are not planer coordinate.

O

P

Connection of spherical polar with cartesian

=

>

?

��, �, ��

��

@

�����

� � � sin� cos@

� � � sin � sin@

� ����

Transformation relations

� � � ���� ���@� � � ���� ���@

� � � ����

Hence

� � �� � �� � �����

� � � �!��� � ��

���

�@ � � �!�

�

�

Unit vectors in spherical polar

Position vector

, � GHI JKG # L$ � GHI GHI # M$ � cos �

=

>

?

��, �, ����

@

, . and #$are, respectively

perpendicular to � JKIGN. � JKIGN. # � JKIGN.

�$ ��

��$ =GHI JKG # L$ � GHI GHI # M$ � cos �

�$

@( is the unit vector perpendicular to

# �constant plane (��PQ��R�, I,e, perpendicular to unit vectors and �(

Thus #$ ��$S�$

�$S�$

=!T$ UVW 8UVW XYZ$ UVW 8 [\U X

UVW �

#$ � 'L$ sin# � M$ cos #

@(

�% �@(S�$

@(S�$

� L$ cos # cos � M$ sin# cos ' � sin

�%

�$ ��

�� �( ���@���� � �( ���@���� � �$ ����

�% �]�$

]�� �( ���@���� � �( ���@ ���� ' �$ ����

@( � '�( ���@ � �( ���@ ������%�^_Q��R_�Q����@!�

� � ��a�� ��@�( � � �a�� �a�@�( � ����� �$

Unit vectors in spherical polar

Partial differential of unit vectors in spherical

polar

]�$

]��

]

]��( �a�� ��@ � �( �a�� �a�@ � �$ ���

� �( ���� ���@ � �( ���� ���@ ' �$ ���� � �%

]�$

]@�

]

]@�( �a�� ��@ � �( �a�� �a�@ � �$ ���

� '�( ���� ���@ � �( ���� ���@ � ����@(

]�%

]��

]

]��( ��� ��@ � �( ��� �a�@ ' �$ �a��

� '�( ���� ���@ ' �( ���� ���@ '�$ ���� � '�$

Unit vectors in spherical polar coordinate are function of � and @ only.

]�%

]@�

]

]@�( ��� ��@ � �( ��� �a�@ ' �$ �a��

� '�( ��� sin@ � �( cos� ���@ � 'cos� �$

Additionally, you may verify:

Velocity in spherical polar coordinate

+, �b,

bN�

b

bN

�b

bN �

b

bN

� - � c

c

b

bN�c

c#

b#

bN� - � -* � sin #- #$

Chain rule

Acceleration

2,� 3 ' -4 ' #- 4GHI4 � �3 ' 2--

' #- 4 sin cos �. � #3 sin � 2-#- cos � 2-#- sin #$

You must try to prove this

� � ��a�� ��@�( � � �a�� �a�@�( � ����� �$

+, � - � -. � sin #- #$

Velocity –to remember!

����� � b�$ � b�% � GHI b# @(

+, ������

������b

∆N �

b

∆N% �

sin b#

∆N#$

e � �- �$ � ��- �% � � ����@- @(

=

>

?

��

@

f��$

�f��%

�����

� ����f@ @(

Elementary

displacement in

arbitrary direction

�� in �t

Done!

Well, We are done with the necessary

mathematical concepts!

Ok, Now in to Physics!Ok, Now in to Physics!Ok, Now in to Physics!Ok, Now in to Physics!