May 31, 2018 Thursday

Bedtime Story 

Initial Angular Momentum was Imparted during Its Origins

Last night I had left you with the accretion process that often takes place in the interstellar dust due to gravity which is truly a run-away process just like a ball set rolling down the slope.

Once it starts, there is no stopping it – initiation of accretion only further accelerates the process as more and more dust matter get closer to each other significantly increasing the effect of gravitational force.

Ever increasing mass in the center of the action eventually leads to the collapse, and as the collapse continues things get interesting because the law of conservation of angular momentum begins to display its effect.

This specific conservation law dictates that the spinning of a body will accelerate as the spinning body collapses within itself.

There are several real-life events, the most classical one that you can watch on several YouTube videos is the increase in the rate of spin of a skater when he or she pulls in his/her arms closer to the body.

The increase in spin keeps the angular momentum constant as now the radius (arms) of the body (figure skater) has decreased from before (when the arms were out stretched).  

What holds true for stars also holds true for planet earth.

Earth rotates today and we have days and nights because it is the same spin that was imparted to it during its formation that keeps on going today thanks to the law of conservation of angular momentum.

Because our planet is largely spinning and revolving in vacuum, there is a very little torque acting on it, at least as far as our miserable existence is concerned, it can almost be discarded.

Of course, certain amount of external torques do have acted on our planet over so many billions of years and continue to do so in the form of collisions from meteorites and asteroids.

In fact, not just meteorites and asteroids but it is believed that our planet had undergone multiple collisions and bombardments even with objects that were as large as planets.

In fact, it is believed that one such major blow that is fancifully known as the “giant-impact” or the “Big Splash” took place when a protoplanet fancifully named as Theia struck our planet just 0.1 billion years after the solar system had formed; Which amounts to almost 4.4 to 4.45 billion years back.

Our Solar System itself formed 4.6 billion years ago, once again thanks to the gravity leading to the gravitational collapse (which is different from accretion) of a small part of a giant molecular cloud.   

Soon after our solar system had formed something seriously gigantic that is believed to be a protoplanet called Theia struck our planet at an oblique angle when our planet was nearly fully formed.

It is believed that Theia’s iron core would have sunk and merged with the young Earth’s iron core and similarly a large part of its mantle would have got assimilated with our mantle.

We shall continue with the violent journey of the birth of our planet in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 30, 2018 Wednesday

Bedtime Story 

Some Important Conservation Laws

We were talking about conservation laws in physics and how they form the bed rock of the description of nature. 

Some of the main conservation laws of classical physics are:

Conservation of mass-energy

Conservation of linear momentum

Conservation of angular momentum

Conservation of electric charge

I am sure you must be all vaguely familiar with some of these laws if not very well versed with all of them.

These are only some of the conservative laws and are both exact and absolute, which means as claimed earlier, that they apply to all the processes in the nature.

Even more interesting than this truth is the Noether’s theorem which is a very interesting result that was arrived after studying these various laws of conservation.

Before going into this theorem, let me tell one very important implication of the conservation of angular momentum.

Today no one questions the fact that days and nights are caused by earth’s rotation even though it is a fairly new knowledge considering the amount of time sapiens as species have lived through not knowing this.

But I find it very odd that very few people raise the question that why does earth spin or rotate about its axis, and goes on to do so endlessly, at least endlessly as far as we as species are concerned.

It is today also fairly well accepted fact that there surely was a Big Bang in which a primordial cloud was formed consisting principally of the simplest elements of the periodic table, hydrogen and helium which I had discussed in some of bedtime stories of last year.

What happened before that is a question for research and today some interesting theories have come up with alternatives to what could have happened before.

Heavier elements as we know are “cooked up” inside massive supernovae and spitted out all over when these super bombs explode.

All these interstellar dust along with the omnipresent gravitational force allows that magical process known as accretion to set in.

Accretion is the process that is responsible for the formation of almost all astronomical objects ranging from galaxies to planets.

Strangely enough, it is the least taught natural process that gave rise to our existence taught in High Schools.

What also accompanies this accretion process is the imparting of relatively low degree of angular momentum to this slowly gathering cloud of interstellar dust matter because of its heterogeneous nature.

As time goes by and the process of accretion continues, the central part of the nebula undergoes fast compression and a forms a hot hydrostatic core which is actually the seed of what will in future become a star.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 29, 2018 Tuesday

Bedtime Story 

Solutions to n-body Problem Needs Explanation

Last night I had left you with a promise to list out the solutions to the n-body problem (whose specific case is the two-body problem) but I realize that it is not that a facile task.

Ground work has to be laid before the solutions can be proposed.

Some terms that are listed out below are familiar to most such as mass, position, vector but there are some other terms that needs more detailed consideration.

The masses of the two bodies are represented by m1 and m2.

Position of one body relative to the other is given by the vector r.

The distance between the two is represented by r, all in the standard SI units of course.

One of the key concepts in the two body problem that needs to be introduced is the specific relative angular momentum represented by small letter h with an arrow on top but I will simply use the small letter h here.  

This is a mass-specific angular momentum and angular momentum as you would recall is the rotational equivalent of linear momentum and is conserved quantity.

When we say it is a conserved quantity it means that the total mass-specific angular momentum of the two bodies does not change unless acted upon by an external torque.

Torque is similar to force but is used when we apply the idea of force to change of angular momentum.

Now whenever you come across the words conservation and law together, treat them with great reverence, only in physics of course; everywhere else these terms would most likely be not only misused but thoroughly abused as well.

There are far more charlatans and spiritualists chanting out there nonsensical mumbo jumbos sexed up with borrowed lexicon of physics and mathematics (more from physics usually as it is the most fundamental of natural sciences) than there are genuine physicists and mathematicians on our dear 4.6 billion years old planet.     

Conservation laws of physics are fundamental is describing the real world or the nature as Feynman use to call it; even though mathematics is the queen of sciences and widely used in physics mathematical equations do not always describe reality and yet they may be either true or logically derivable.

It is the physics that is by far, the best in describing reality though it still does not answer the question ‘Why’ but rather ‘How’.

One can safely say that all of the physics rests on these conservation laws and it is these laws that determine what processes are permissible in nature and what are not.

Once when I say that conservation laws are fundamental to physics, it necessary implies that they have broader and wider implications to other sciences such as chemistry, biology, geology, astronomy and it is endless.

Science cannot have it that some laws are applicable at one place and defied at other; this is the fundamental philosophy of unity of science because the different subjects of science are merely examining the same nature at different levels of organization.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 28, 2018 Monday

Bedtime Story 

Solving the Two-Equations Yields us the Barycenter and its Vector

Last night we were left with two equations which had converted the single two-body problem into two one-body problems.

Now, whenever you have two such equations most of us know that they can be played around with.

The two can be added, can be subtracted and it’s all a fair game.

Let us see what we get if we add them.

m₁x₁ + m₂x₂ = (m₁ + m₂)R = F₁₂ + F₂₁ = 0

If you are wondering, like me, that how did F₁₂ + F₂₁ turned out to be zero, you need not fret long.

This has come from the Newton’s third law which states that when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The beauty is that the addition of the two equations resulted in the generation of a new concept that is known as the center of mass or more accurately the barycenter.

Barycenter refers to the center of mass of two or more bodies that are orbiting around each other.

Now if you decide to subtract the Equation 2 from that of 1, you will end up in getting the equation that will describe the vector between the masses and how it changes with time.

So if capital R is the position of the barycenter than the small r represents its vector and on solving the equations one would get r = x₁ – x₂.

Thus the two-body problem is solved as we have got the position of the barycenter and its vector for all times t. 

Now this was a simple two-body problem that gets more complicated when the two point bodies are interacting with each other only due to the gravitational force.

Then it becomes a more specific case of two-body problem known as gravitational two-body problem.

When smart human apes decide to solve problems that are found in nature, they tend to simplify things and make certain assumptions (sometimes too many) such as

(i) the two bodies do not collide with each other,

(ii) do not pass through each other’s atmosphere and further

(iii) do not interact with each other through any other known force.

Now please keep in mind that we are merely considering just two point bodies interacting with each other through one force of gravity, and yet even then its solution does not come easy.

Just to exemplify the difficulties of such n-bodies problem, I shall simply enlist the solutions (yes plural for it has multiple solutions for varying conditions of two-bodies even though they are loaded with assumptions that very much simplify the solutions).

First let us go through the parameters that would be required to arrive at the solutions.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 27, 2018 Sunday

Bedtime Story 

Breaking Down the Two-Body Problem

In fact, the two-body problem when reduced two independent one-body problems, the scenario becomes even more familiar in the form of Newton’s second law.

In case you have forgotten, it would be worth going through this law once again which now may seem very primitive (after general relativity) but still serves us wonderfully well in our Middle World.

Middle World quite curiously is a term coined by the evolutionary biologist Richard Dawkins, introduced first in his 2005 TED talk held at Oxford titled: “Queerer Than we can Suppose: the Strangeness of Science”.

Middle World is the habitat that we humans live in, lying between the microscopic world of quarks and subatomic particles and the cosmic world of giant stars and galaxies.

Newton’s second law states that the rate of change of momentum of body is directly proportional to the force applied on it.

Moreover, the change in the momentum takes place in the direction of force applied.

Mathematically, it is represented as:

F = dP/dt = d(mv)/dt

Now the two-body problem seeks to determine the position of two point particles at various times that only interact with each other.

In the real world or rather the real universe, such two-bodies are often encountered in the form of satellite orbiting a planet, or a planet or orbiting a star or even a star orbiting around another star when the system is known as binary star.

So let us see how a two-body problem can be broken down into two independent one-body problems.

So let us say that there are two point bodies whose vector positions are given by x₁ and x₂ and there masses are represented with m₁ and m₂.  

So mathematically, the problem is reduced to determining the trajectories x₁(t) and x₂(2) for all times t given the initial positions x₁(0) and x₂(0) and the initial velocities v₁(0) and v₂(0).

In this case scenario, the force acting on the first body with mass 1 is only from mass 2 and hence we can label it as F₁₂.

Analogously, the force acting on body 2 of mass 2 is only from the first body of mass 1 and hence we can label the force as F₂₁.

This data is enough to apply Newton’s second law on to this bodies which will give us the following two equations:

F₁₂ (x₁, x₂) = m_1 1  

F₂₁ (x₂, x₁) = m_2 2  

I hope you noticed the two dots on top of the letter x.

They have an interesting significance.

The xs with 2 dots on top represents the second derivative with respect to time, which in other words means they represent acceleration vectors.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 26, 2018 Saturday

Bedtime Story 

According to Ada, Analytical Engine was Capable of Tackling Three-Body Problem

If you recall, which you may very well not, that Ada Lovelace had mentioned in one of her Notes that the fabrication of the beautiful woven portrait of Jacquard necessitated the usage of 24,000 punched cards.

But then she goes on to add that such large number of punched cards can be dramatically reduced by using the technique of looping which she then introduces it.

Rearranging operations and using intelligent loops, she says, would significantly reduce the number of punched cards, requiring only 3 in operations which at the first instance would seem to need 330.

She goes on to say that Analytical Engine would be capable of carrying out immensely complex computations with far greater accuracy which are currently not computable by the available mechanical calculators or if possible, then with great room for errors.

One of the complex problems that she mentions where Analytical Engine would prove to be extremely useful is the famous three-body problem. 

Though you may all know vaguely what this three-body problem refers to, let me explain to you in greater detail what it means.

This is going to be one hell of a diversion from our bedtime stories of logic, mathematics and computer science but it’s a story so wonderful that not putting it down on paper on pen or rather the word processor would be a serious crime.

To the physicists, the three-body problem is merely a special case of n-body problem and the simplest case of the n-body problem is the two-body problem which has been comprehensively solved.

The origin of n-body problem rests on astronomy; motion of celestial bodies and of course, the third most important factor that pervades the entire universe – the gravity.

This problem was perhaps addressed very indirectly by the Italian explorer, navigator and cartographer Amerigo Vespucci on whose name the Americas came to be named after.

He followed the path of his even greater predecessor Christopher Columbus and reached central and South America, just a decade after him, only critical difference being that unlike Columbus in 1492, he did not mistake the Americas for Asia.

In 1502 he demonstrated, using his knowledge of the position of the Moon that Brazil and West Indies where he had set his feet upon, could not be Asia’s eastern outskirts and had to be some landmass hitherto unknown to Europeans.

To him astronomy and trigonometry were mere tools to serve his primary purpose of exploration.

It was Galileo Galilei first, more but more importantly Isaac Newton who being far more concerned about the motion of planets rather than his own location on this planet Earth, directly took on this fundamental problem of classical mechanics.

Though the three-body problem that I introduced here may be alien to you, we all have been introduced to its simpler version – the two-body problem - in our high schools when we were introduced to the classical Laws of Motion and gravity.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 25, 2018 Friday

Bedtime Story 

Recap of Notes 

Some I had to go back and re-read my own bedtime stories on Ada Lovelace Notes because of its extreme difficult nature and recurrent use of tables which are essentially algorithm in the language of the Engine.  

I have realized after reading my own bedtime stories on the Notes how difficult it must have been for the readers to get a broad sense what Ada Lovelace was writing about, specially to the uninitiated and the uninterested. 

So let me try to sum it all up in a more bedtime story-like fashion though yet again I must make it clear that in spite of my attempt the subject is inherently difficult and there is a limit to how much one can simplify certain technical facts.

To begin with, we saw Ada comparing the Analytical Engine with the Difference Engine and clearly describing the former to be superior to the later.

While the Difference Engine is capable of computing values of 6^th degree polynomial, the Analytical Engine can carry sequence of operations.

Moreover she considers the Analytical Engine as the “material and mechanical representative of analysis” which at that time no mechanical calculating devise could claim for itself.

This is a supreme kind of attribution that Ada devotes to a machine, something that no other mechanical machine previous to it had ever deserved.

She then explains how this marvel would carry out its analysis; using the punched cards as its data feeding mechanism.

Just look how eloquently she describes the punched card mechanism: “the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves.”

From then on, she becomes less poetic and more technical taking examples of specific type of computations and how they will be carried out using Operation-cards which would determine the operations to be performed and Variable-card which define where the intermediary and final values would be located. 

She speaks of “cycles” and “cycles of cycles” which Mon Ami in today’s parlance would recognize them as loops and nested loops.

We saw earlier how she even provided us with mathematical notation for this idea with the equations (6), (7) and (8) of the Note E.

(6.)

where p stands for the variable; (+ 1)^p for the function of the variable, that is, for Φp; and the limits are from 1 to p, or from 0 to p-1, each increment being equal to unity. Similarly, (4.) would be,—

(7.)

the limits of n being from 1 to n, or from 0 to n-1,

(8.)

or

This indeed is very modern and was very far-sighted of this young mathematician.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

May 24, 2018 Thursday

Bedtime Story 

How Ada was Reintroduced

I shall not go much into Alan Turing or his work right now as he will follow us later but the point that has to be stressed is that of the few people on this planet then who were aware of the works of Babbage and Lovelace, Turing was definitely one of them.

Just an example to show you how Turing related humans to machines, I find nothing better than this simple quote of his from the paper ‘Intelligent Machinery: A Report by A. M. Turing’ (Summer 1948) – “A man provided with paper, pencil and rubber, and subject to strict discipline, is in effect a universal machine.”

Yet, even as decades rolled by from 50s, to 60s and to 70s, both Ada Lovelace and Charles Babbage remained little known and largely unrecognized,

It was only in the 1980 when a team led by Jean Ichbiah (It is a French name and the guy was a French computer scientist) and the French company Groupe Bull under a contract with the United States Department of Defense developed a structured, statically typed, imperative, wide-spectrum, and object-oriented high-level computer programming language and named it Ada that Lovelace finally began to be recognized as a pioneering figure in the field of computer science.

The language was named Ada very specifically after Augusta Ada, Countess of Lovelace or more commonly as is known to us as simply Ada Lovelace.

I am not an expert in computer programming languages and I do not know why Ada did not become very popular as a programming language even though for its safety-critical support features it is still being used for military applications, avionics and air traffics.

So even though the language itself did not itself become very widespread it certainly enhanced the awareness of both Ada Lovelace and Babbage in the consciousness of the masses.   

Suddenly their names began to appear on the popular media and even their biographies began to be read.

With their popularity, arose both interest and curiosity in the Difference Engine, so much so, that some computer scientists began to question that if the real Difference Engine were to be invented, would it really work?

A serious project was initiated which in the year of 2002 led to the construction of a full-fledged Difference Engine that was based on the exact same plan as devised by Babbage but for one alteration.

What was that one alteration I have no knowledge but to surprise of many, this Difference Engine indeed did work!

What came as even a bigger amazement or perhaps a coincidence that the cost of building the Difference Engine in this century turned out to be almost the same as requested by Babbage to the British Crown back in 1823, adjusted for inflation of course.

As for the Analytical Engine, it still remains elusive physically though conceptually perhaps it is alive and working in almost every home of a modern family even those struggling for their basic necessities in third-world countries.

It would shock both Charles Babbage (and Ada Lovelace too) that the money for the Engine that he was refused by the Crown today is accessible to even those families who cannot afford their own homes or even clean water.

Now, isn’t that something insanely and bigotedly astounding if you care to think of it with a broader historic perspective!

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg