Showing posts with label PHYSICS. Show all posts
Showing posts with label PHYSICS. Show all posts

Monday, 18 January 2021

COMPLEX NUMBERS , DEMOIVRE'S THEOREM & EULER'S FORM

       COMPLEX    NUMBERS

Any  number of the form  a+bi is known as  complex  number where a & b are real numbers & i=  √-1 , a is called real part & b is called imaginary part.

           A complex number is generally denoted by z. If a is 0 then 

      z is purely imaginary & if b is 0 then z is purely real.

NOTE:

  1) i  = √-1  

  2) i² = -1

  3) i³ = i² i =-1.i =-i

  4) i⁴ =i³.i =-i.i =-i² =-(-1) =1

  5) i⁵ = i⁴ .i =1.i =i

  6) i⁶ =i⁵.i = i.i = i² = -1

 7) i⁷ =i⁶ .i =-1. i =-i

 8 ) i⁸ =i⁷.i =-i.i =- i² = -(-1)=1

 9) i⁹ =  i⁸.i = 1.i =i

10)i¹º =i⁹.i =i.i =i² = -1

  &  so on .


Read blog on Line & it's Equation


                ALJEBRA OF COMPLEX NUMBERS


  1)  ADDITION : When two complex numbers 

   are added their sum is also a complex number.

   eg.

  if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁+ z₂ is given by

       z₁+ z₂ = (a₁+b₁i ) + (a₂+b₂i ) 

                  = (a₁+a₂) + (b₁+b₂)i

2)   SUBTRACTION : When two complex 

   numbers are subtracted their  difference is 

    also a complex number .

   eg.

    if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁-z₂ is given by

       z₁- z₂ = (a₁+b₁i ) - (a₂+b₂i ) 

                 = (a₁-a₂) + (b₁-b₂)i

2) MULTIPLICATION : When two complex numbers 

    are multiplied their multilication is also a complex number .

   eg.

 if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁. z₂ is given by

       z₁. z₂ = (a₁+b₁i ) . (a₂+b₂i ) 

                 = (a₁a₂-b₁b₂) + ( a₁b₂+b₂a₁)i


                 EQUALITY OF COMPLEX NUMBERS 

Two complex numbers are said to be equal if

1)Their real parts are equal

2)Their imaginary parts are also equal.

 eg. If z₁=a+bi  & z₂=2+3i 

             z₁=z₂ if a+bi  =2+3i i.e a=2 & b=3


         CONJUGATE OF A COMPLEX NUMBER


Two complex numbers which are differ only in the sign of their imaginary parts are said to be conjugates of each other.

If z is complex number then it's conjugate is denoted by z .ie. z bar.

if 2+3i is given complex number the it's conjugate is given by 2-3i.

          

                 z                              conjugate  z

          2+3i                                             2-3i.

               2-3i                               2+3i

             -2+3i                              -2-3i

             -2-3i                                -2+3i

              -3i                                     +3i

             +3i                                      -3i


NOTE : 1)SUM &product of two complex conjugates are real.

               2)difference of two complex conjugates are imaginary.


Read the blog on Pascal Triangle & FIBONACCI NUMBERS


                       MODULUS & ARGUMENTS

If z=a+bi is a complex number then modulus of z is given by r=√a²+b² .

Argument or amplitude of z is given by  𝛩=tan⊣(y/x) 

i.e tan inverse of y/x


           POLAR FORM OF A COMPLEX NUMBER

IF z=a+bi is a complex number then it's 

   polar form is given by z=r(cos𝛩+ i sin 𝛩)


EXPONENTIAL FORM OF A COMPLEX NUMBER

1)   eⁱӨ = e raise to iӨ =cos Ө+i sinӨ

2) e⁻ⁱө  =e raise to -iө  =cos Ө-i sinӨ

3) eⁱӨ +e⁻ⁱө =2cosө

4) eⁱӨ - e⁻ⁱө =2isinө


  DE MOIVRE'S THEOREM

For any rational number n the value or one of the value of 
(cos Ө+i sinӨ)ⁿ= (cos nӨ+i sin nӨ)

e.g. 1) if (cos Ө+i sinӨ)⁴ = (cos 4Ө+i sin4Ө)

       2) if (cos Ө-i sinӨ)⁴ = (cos 4Ө-i sin4Ө)

      3)  if (cos Ө+i sinӨ)⁻⁴ = (cos 4Ө-i sin4Ө)

      4)  if (cos Ө-i sinӨ)⁻⁴ = (cos 4Ө+i sin4Ө)


EULER'S FORM 

1) if z₁=a₁+b₁i  & z₂=a₂+b₂i then r=√a²+b² .

&     z₁
__________=  ( r₁/r₂) eⁱ(ө₁-ө₂)      
     z₂                

2) z₁ . z₂  = (r₁.r₂) eⁱ(ө₁+ө₂)  

i.e    (r₁.r₂). e raise to i.(ө₁+ө₂)  

 This is known as Euler's form



Tuesday, 22 September 2020

UNITS AND DIMENSIONS OF PHYSICAL QUANTITIES IN PHYSICS ...

                                                    UNITS  AND DIMENSIONS 


      SR.NO.                   PHYSICAL  QUANTITIES                 UNIT                                DIMENSION

       1                               DISTANCE                                       m , cm                                 M   L   T

       2                               DISPLACEMENT                           m  , cm

       3                               SPEED                                            m/sec , cm/sec

       4                               VELOCITY                                  m/sec , cm/se

       5                                ACCELERATION                        m/sec² , cm/sec²

       6                                MASS                                             kg  .gm

       7                                TIME                                               sec

       8                                FORCE                                       Newton ,dyne

       9                                WORK                                         joule , erg

      10                               POWER                                             Watt , j/sec

      11                              AREA                                                  m² , cm²

      12                              VOLUME                                            m³ ,  cm³ 

      13                              LENGHT                                                m , cm

      14                             DENSITY                                          kg/m³ ,  gm/cm³

      15                             MOMENTUM                              kg m/sec  , gm cm/sec

      16                             PRESSURE                                  Newton/m² . dyne/cm²

      17                             TEMPRATURE                          kelvin

      18                              TORQUE                                   Newton m , dyne cm

      19                              IMPULSE                                 Newton sec

      20                              CURRENT                               Ampere

      21                              CHARGE                                 coulomb

     22                             KINETIC ENERGY                    joule , erg

     23                             POTENTIAL ENERGY               joule ,erg

     24                             SURFACE TENSION                   N/m . dyne/cm

     25                             SURFACE  ENERGY                     joule

     26         RADIUS OF GYRATION                                   m , cm

     27         FREQUENCY                                                      Hertz  , cps

    28          ENERGY                                                                         joule

    29          ELECTRIC POTENTIAL                                             volt

    30          ELECTRIC  RESISTANCE                                      ohm

    31         ELECTRIC  CONDUCTANCE                             /ohm , Siemen

    32         ELECTRIC      CAPACITANCE                              farad

    33         MAGNETIC     FLUX                                                   Weber

   34          INDUCTANCE                                                                Henry

   35          MAGNETIC  FLUX   DENSITY                             Wb/m²

   36          ILLUMINATION                                                  lux

   37         LUMINOUS  FLUX                                                  cd 

   38         SPECIFIC  HEAT                                           J/Kkg

   39         SPECIFIC    VOLUME                              meter cube/kg

  40          STEFAN'S  CONSTANCE

  41          STRESS

 42           STRAIN

 43           TEMPERATURE   GRADIANT

 44           VELOCITY    GRADIANT

 45           UNIVERSAL GRAVITATIONAL   CONSTANT

 46           UNIVERSAL GAS   CONSTANT 

 47           ANGLE

 48           SOLID ANGLE

 49           WEIGHT

 50           RADIOACTIVITY

 51           INTENSITY OF LIGHT

 52          QUANTITY  OF  SUBSTANCE

 53          CURRENT  DENSITY

 54          MAGNETIC  INDUCTION

 55          ANGULAR    DISPLACEMENT

 56          ANGULAR  VELOCITY

 57          ANGULAR  ACCELERATION

 58          ANGULAR MOMENTUM

 59          CENTRIPETAL  FORCE

 60          FARADAY  CONSTANT

 61          HEAT

 62         HEAT    CAPACITY

 63         MAGNETIC MOMENT

 64         MAGNETIC DIPOLE MOMENT

  65        MAGNETIC SUSCEPTIBILITY

  66        COEFFICIENT  OF  VISCOSITY

 67         WAVE LENGHT

 68         WAVE NUMBER

 69         THERMAL   CONDUCTIVITY

 70        ELECTRICAL   CONDUCTIVITY

 71       ELECTRICAL   DIPOLE MOMENT

 72        ELECTRIC  FIELD

 73        ELECTRIC  FLUX                                 Nm²/C

 74       PERMITIVITY  OF FREE  SPACE            H/m

 75      PERMEABILITY OF SPACE              

 76         SPRING  CONSTANT                           N/m

77        MODULUS   OF  ELASTICITY             N/m²

78            YOUNG,S   MODULUS                      N/m²

79             BULK  MODULUS                           N/m²

80               RIGIDITY  MODULUS                     Newton/square meter

81              CONDUCTANCE                                /ohm

82               CONDUCTIVITY                                /ohm meter

83            RESITIVITY                                               Ohm meter

84         COEFFICIENT OF   FRICTION               NO UNIT

85          EQUIVALENT  CONDUCTANCE            /ohm m²

86        MOLAR    CONDUCTANCE                      /ohm .m² . /mole







Thursday, 30 January 2020

PHYSICS & PHYSICAL QUANTITIES IN PHYSICS .



                          PHYSICS  

PHYSICS  is Derived  from  a  Greek  Word  "FUSIS " which means Nature i.e. Physics is Study of nature AND natural phenomenon .

      PHYSICS  have many branches . Some of these are as follows.

1)Classical Mechanics
2)Wave Mechanics
3)Heat and Thermodyanamics

4)Optics
5)Electricity, Magnetism & Electronics
6)Modern Physics

7)Relativity
8)Solid state physics
9)Nuclear Physics
10)Astrophysics
11)Quantum Mechanics etc.

    Up to 12th standard mainly we study above first 6 branches. Normally all boards in India they teach Classical Mechanics,Wave Mechanics & Heat & Thermodyanamics as PAPER 1 and
Optics , Electricity magnetism and Electronics & Modern Physics as PAPER 2 .

Physical quantity  : Physical quantities are characteristic or property of an objects which can be measured or calculated by the use of other measurements .

There are three types of physical quantities.
1)Fundamental quantities
2)Derived quantities
3)Suplementary Units

1)Fundamental quantities : Those physical quantities which does not depend on any other physical quantities for their measurements , they can be expressed by themselves are know as fundamental quantities .

Their corresponding units are known as fundamental units.

There are seven fundamental quantities .Their units are known as fundamental units or base units.They are ...

a)Length (l) : It's unit is metre (m) or cm.
b)Mass (m) : It's unit is Kg or gm .
c)Time (t) : It is measured in second (s)
d)Temperature (T) : It is measured in Kelvin(K).
e)Current (I) : It is measured in Ampere (A).
f)Amount of substance : It is measured in mol .
g) Intensity of light (I) :It is measured in Candela                                                                           (Cd) .


2) Derived Quantities : Those physical quantities which depend on Fundamental quantities for their measurements , are known as derived quantities . Examples of derived quantities are  Area , velocity, acceleration , force , density, volume , Pressure , momentum  etc.

3) Supplementary Units : There are two supplementary units .
1)Plane angle :It is measured in degree or radian .
2)Solid angle  : It is measured in steradian .




We should know some basic defination and units of physical quantities which are used in physics.

1) Motion : It is defined as change in the position of the object with respect to some observer . Means change in the position of the object should be seen by the observer , then only it is motion . If change in the position of the object is not seen by the observer then it is not a motion for that observer.

There are different types of motion . But for simplicity we will catagorise as  :-
a)Linear motion or translational motion 
b)Angular motion or circular motion

1) Linear Motion ( straight line motion ) or translational motion  : Motion of the particle along the straight line .

2) Circular Motion or Angular motion (Rotational motion)  : Motion of the particle along the circumference of circle .

3)Uniform circular motion (UCM)  : Motion of the particle along the circumference of the circle with uniform speed is called as uniform circular motion.

4)Displacement (S): It is the shortest distance between starting and end point of motion . It is also defined as change in position of the object in a specific direction. it is denoted by S . It is a vector quantitty . It is measured in Meter (m) or centimeter (cm).

5)Distance (S) or(x) : It is a actual path lenght i.e. it is actual distance covered by the body. it is a scalar quantity . It is measured in meter (m) or centimeter (cm).

6)Velocity (V) : It is defined as change in displacement per unit time or it is also defined as time rate of displacement. It is denoted by U or V . It is a vector quantity. It is measured in meter per second(m/s) or centimeter per second (cm/s) .

Velocity = Displacement/time

7)Speed (V) : It is defined as change in distance per unit time or it is time rate of distance . Normally it is denoted by V. It is a scalar quantity . It is measured in meter per second (m/s) or centimeter per second (cm/s) .

Speed= Distance /time

8)Acceleration (a) : It is defined as change in velocity per unit time or it time rate of velocity. It is denoted by  'a '. It is a vector quantity . It is measured in meter per second square (m/s²)or centimeter per second square (cm/s²).

acceleration = Velocity/time  or V/t

9)Mass (m) : It is actual matter contained in the body . It is denoted by 'm' . It is scalar quantity. It is measured in kilogram(kg) or gram(g) .

10) Weight (W) : It total gravitational force acting on the body.It is denoted by W. It measured in kilogram meter per second square (Kg m/s²) or gram centimeter per second square (gm cm/s²).
            Weight =mg






11)Density (d) : It is the ratio of mass to volume or it is defined as mass per unit volume of an object.
It is measured in kg/m³ or gm/cm³ .

Density =Mass/ volume 

12)Force (F) :It is defined as that physical quantity which changes or tries to change the position of the object . It is a vector quantity . It is measured in Newton (N) or Dyne (D) .
1N =10⁵ Dyne .

Force  =ma 

13)Work (W) :It is a Product of force and displacement . It is Vector quantity. It is one kind of Energy . It is measured in Joule (J) or erg . It is also measured in N m or dyne cm .
1J =10⁷ Erg

Work =F.S 

14) Power  (P) : It is Capacity to perform work . It Measured in Joule / second (J/s) or Erg per second (Erg/s) .
Power =Work/time =W/t =FS /t=F (S/t )=FV =mav  .

15)Energy (E) : It is work. It is meaured in J or Erg .

16)Kinetic Energy (K.E.) :  It is energy posses by the body on account of it's motion. It is meaured in J or Erg .
K.E. =1/2mv².

17) Potential Energy (P.E.) : It energy posses by the body on account of it's Configuration or position. It is also measured in J or Erg .

P.E. =mgh 

18)Pressure (P) : It is a force acting per unit area . It meaured in Newton per meter square (N/m²) or dyne per centimeter square (Dyne/cm² ).

Pressure =Force/Area =F/A 

19)Torque (T) : It is a rotational force which produces circular motion in the body .It is nmeasured in Newton meter (Nm) or dyne cm.

Torque  = r F 

20 )Couple  : It is also a rotational force ,which produces circular motion in the body. It is also defined as two equal and opposite forces acting along same line but in opposite direction. It is also measured in Nm or dyne cm.

Couple =rF 


21) Impulse (J) : It is a large force generated for a short interval of time . It is measured in Newton meter (Nm) or dyne cm .

Impulse = F t  

22) Momentum (M) : It is a product of mass and velocity . It is measured in Kg m/s or gm cm/s .

Momentum =mv 

23 )Angular momentum  (L) : It is a product of  moment of inertia and angular velocty . It is measured in kg m².rad/sec .

L =Iw 

24)Moment of inertia (I) : It is product of mass and square of it's distance from axis of rotation . It is anologous to mass in linear motion . It is measured in Kg m² or gm cm²  .

I =MR²  

25)Surface tension (T) : It is a force acting per unit length .It is measured in Newton per meter (N/m) or dyne /cm .

Surface Tension = F/l  

26)Radius vector (r) : It is a vector drawn from centre to any point on the circumference of the circle .

27)Angular displacement (0) :It is a angle trace by the radius vector at the centre of the circle . It is measured in radian (rad) .



28)Angular velocity (w) : It is a angular displacement per unit time . It is measured in rad/sec.

Angular Velocity = angular displacement/time 

29) Angular acceleration (a)  : It is a ratio of angular velocity to time . It is measured in rad/sec² .
  Angular acceleration =angular velocity /time

30)Time period (T) : It is a time taken by the particle to complete one revolution. It measured in second (sec) .

31) Centripetal force (F) : It is force acting on a particle performing  uniform circular motion  along the radius towards the centre of the circle . 
It is measured in Newton (N) or dyne .

Centripetal force  = mv²/r =mrw² =ma

32) Centrifugal force (F) : It is a force acting on the particle performing UCM along the radius but away from centre. It is measured in Newton(N) or Dyne 

Centrifugal force  =  - mv²/r  =-mrw² =-ma 


33)Stress : Force acting per unit area is knownas stress. It's unit is N/m² i.e. Newton per meter square.

  Strees  =F/A 

34) Strain : It is the ration of change in dimension to original dimension .
It has no unit.

Strain = Change in dimension/original dimension

35)Angle of contact :The angle between the tangents drawn to the free surface of the liquid  and surface of the solid at point of contact , measured within the liquid.






36 )Surface Energy : It is amount of energy rquired per unit area to  increase the size of the surface of the liquid.

           Surface Energy = TA
 where T =surface tension    , A = area 









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