SCALAR & VECTOR

Most of our physical quantities are linearly depends its variables. I not sure to explain what is mean by linear dependence, suppose we have a physical quantity R is the function of variables x, y and z if they linearly depends R can be written as,

                            R = c₁x + c₂y + c₃z  where c₁, c₃, and c₄ are constants or values. The sign “+” is not only ordinary addition also a symbol of linear operator; avoiding this confusion R can be written as the matrix form

.R = [c₁  c₂  c₃]   [x

                              y

                              z]                         

                                                                                                                                                                 In physics most of the quantities can be written as the linear combination of other physical quantities either fundamental or non fundamental quantities. But in our undergraduate levels, we deals with majority the linear combination of constants or direction or both, these are respectively called scalar and vector.

Vector is the quantity has both magnitude and direction this is we learned, but please tells like that, vector which is the quantity gives the direction, for example velocity of water gives the direction of water flows. Not, velocity of fluid is in the direction of motion. And don’t distinguish scalar and vector in the scale of magnitude and direction. They are differs in the order of writing the linear combination, scalar is zero order matrix that is no specific indication, vector is the first order matrix that is only one indices is needed. Let a_i is the vector and ‘i’ is the indices it has values, i = 1,2,3,…….etc.. Due to the single indices vector is either row only or column only matrix. A_ij  is not a vector because it has two indices ‘i’ and ‘j’.

From above discussion you ask that what is the significance of indices, this is why the mathematics and physics differ. The  scalar and vector are the mathematical concepts, but the indices gives the physical significance, here indices in vector gives the space coordinates we know that each coordinates in a coordinate system is independent so only one value is possible In each coordinate, this why the vector has only one indices.Fundamental levels we can discuss higher order (more than one indice) matrix or tensors. 

GRAVITATIONAL MOTION OF TWO BODIES

                Here I shows the advance of theoretical calculation over experimental calculation using a simple event.
All you know the experiment of Galileo, to show that all bodies of different masses will fall simultaneously without depends the mass.
          But it doesn't true the heavier mass reach at ground first. This argument cannot prove using experiment.
Consider two  bodies of mass M & m separated withe distance of 'd'. According to the Newton's gravitational law two bodies approaching to their center of mass of system.

The acceleration of mass M,                                              

             g₁ = (1/M) [G (Mm)/r²];

The acceleration of mass m,

             g₂ =- (1/m) [G (Mm)/r²];

the resultant acceleration is g = g₁-g₂ = {(1/m)+(1/M)} [G (mM)/r²]
                                         g = [G (m+M)/r²]

From the above equation you can find the gravitational acceleration depends the masses of the body.
consider two masses m, m'. If the mass of earth is M then,
                                        g = [G (m+M)/r²];
                                  g’  = [G (m’+M)/r²) 

 ^{if  m'> m then g'>g, ie. m' attain faster than m.}

^{This give that acceleration of two bodies depends their mass.} 

^{Then how we can say Galileo's experiment is true.}

                                  

DERIVATION OF STOKE’S EQUATION IN VISCOUS FLUID

I mean that the equation is,

F = 6πƞrv; where, ƞ is the coefficient of viscosity, r is the radius of sphere which flows through the fluid, v is the terminal velocity and F is the force acted on the fluid for the viscous drag.

I am sorry to convey that I couldn’t find the proper derivation of Stoke’s equation. But here I wish to say about the magical power of our mind, for past one month I searched for the derivation of Stoke’s equation, but I couldn’t find it anywhere at last I could derive it by myself. But it is not correct

fBut it helps you to find the proper derivation.

We have the Newton’s equation for viscosity,

F = ƞA (dv/dx) --------------------------- (1)

‘A’ is the area of fluid displaced and (dv/dx) velocity gradient or it is written as   “div v” here (dv/dr), since the quantities varies radial.

In this figure we can see that only half of the sphere is push the fluid, therefore A = 2πr² because, this figure 1 misunderstands you actually the fluid motion is considering from bottom to half of the sphere, but go on half to top the flow will be turbulent due to downward motion of sphere, so we cannot use Newton’s equation on the another half face.

                                                                                        

We have the equation for continuity

(dβ/dt) + V (grad β) + β (div v) = 0; OR

(dβ/dt) + v(dβ/dr) + β(dv/dr) = 0;

Where β is the density of fluid, here β is independent of time because fluid undergoes terminal velocity that is no force for  compressing the liquid. Therefore the equation be

(dv/dr) = - {v/β}(dβ/dr)

Therefore equation (1) becomes

F = -ƞ2πr²{v/β}(dβ/dr)-----------------------------------------(2)

Let ‘M’ be the mass flows out by the sphere, And using Archimedes principle, the volume displaced  is equal to the volume of sphere , So β = (3M/4πr³)

And    (dβ/dr) = -(9M/4πr⁴)

These two equations are substitute in equation (2) we get the Stoke’s equation

I.e.  F = 6πƞrv

This is my derivation, you can discuss about this derivation with me. 

GRAVITY ON STARS

         In this section I write a quiet interesting thing, that doesn't related to star formation. You know that gravity is the weakest force than other three fundamental force, but it is the great important and strong in the universe. I think gravity is the only force an important role in star formation, from the nebulae not form a single star, it forms a group of stars. But the distance between the stars are gradually increases. Why didn't they far away from each other there is only force is exist that is gravity(by strong), but gravity has attractive nature then why this happen, which force does this work.
          Actually i don't know why it is happen, because i am primitive in classical mechanics. But i think that the force behind this mystery is gravity. Yes it has a face of repulsion, don't mistaken it has not, but it is the responsible for this expansion. Are you heard about the the great "Three body problem", from its solution, it says that in three mutually gravitationally interacting body if it has total positive energy then all three masses can move away from each other, or can escape.Here there is many of bodies therefore the total energy is always positive that will cause to this expansion. Sorry i couldn't have any mathematical expression, of course i don't know the solution of three body problem the above idea is taken from "Classical Mechanics by H.Goldestein", but i believe that this is the reason. If you can solve the many body problem.
      

NEWTON'S EQUATION IN GRAVITATION

     All students are familiar with the equation of Newton’s equation for gravitation, but a few of them are completely understood about this equation. I can prove it, just look that the equation of gravitation.

The gravitation force,                                                              

                                                F = GM₁M₂

                                                           R²

Where M₁, M2 are the mass of two bodies and R is the distance from the centre of mass (c.m) one body to another.

I take two rings 1,2 of different size(1>2) and mass M₁ and M₂ respectively. First I placed them into same axial line shown in fig.1

                                                             

In this figure, we can calculate the force of attraction using Newton’s equation which is same in above equation. But, the real problem is I decrease the distance R up to zero, that is now the two rings are in same plane and coincided its centre. Therefore according to Newton’s equation the force of attraction to be infinite or strong force.

Is it happening yes of course it is not. Then where is the mistake if you said it is on Newton’s equation surely you are a fool because Newton is not a fool, then where is it?
   If you couldn't find, I sorry because it is not in the part of this equation, But you should understand it because it give the completeness of Newtons law of gravitation. 
There is an important equation in Newton's gravitation that is
       F = M₂ (-dV/dr)

Where V is the gravitational potential produced by the ring 1 on ring 2.The term (dV/dr) also known as gradient. You must know that the newton's equation only valid at varying gravitational potential with respect to 'r'. Inside of a body the gravitational potential to be constant.Here the radius of ring 1 is r' then gravitational potential to be,V = GM₁/r’²

        I only say that the gravitational potential is a constant why it is constant? this question is for you, if  you get share it!


Most of the teacher were ate this section, they are only try study and teach for exams. Every fool can study theorems and law, but only student has applied and analysis. So be a student.    

   

                                              "Make sure that, it is true"

                       

What an idea?

In our society there are so many scientists, they are self educated and free thinkers. But some of them are think so ridiculous( I am one of them). Because they do not bothered intense of science. There are so many people around the world, believes Eisenstein's theory of relativity is false{ And I also amazing how they study special theory of relativity}. So please don't makes foolish, Study and the research are the only way to find facts.

    Here I  writes one of my mistakes,

" Our world is in the edge of energy crisis, so it is important  to find the new source of energy. For this purpose I suggest a method to overcome this problem. Our earth have an magnetic field it is an static field so we can produce energy from this magnetic field by electromagnetic induction [http://www.education.kerala.gov.in/englishmedium/physicseng/chapter3.pdf. ]

But it is only work at varying field, for this purpose our all vehicles carrying coils( of 10000 turns) if it moves it changes magnetic field and produce electric energy and it can store battery.To calculate this energy by the equation (E = Blv) E is induced emf, B is earth's magnetic fields, l is the length of the coil increases by number of turns, and v is the velocity of the car). I know the induced emf is so small, but is on charging so after complete travelling there is a significant amount of energy. And it is also works I can charge 9V battery as full.

Eureka!!!!, I had an invention next Noble prize is for me (I think)."   

But, I couldn't get the Noble prize, because there is the biggest blunder, that energy couldn't helps to decrease the energy crisis sometimes it increase the energy crisis. Because of the two great ideas in physics First law of thermodynamics and Second law of thermodynamics[http://en.wikipedia.org/wiki/Laws_of_thermodynamics]. The magnetic field doesn't do work done, Therefore the energy converted (According to first law) into electric energy by electromagnetic induction is the energy used to ride the car. And also according to Second law of thermodynamics We cannot completely convert the energy which used to drive car to electric energy there is some loss. Therefor we consume more energy to produce small quantity of energy, then you say is the better way ???.

But I use this energy for saving my electricity bill. 

So keep study the physics and know the beautifulness of our world.