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.
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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:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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