June 08, 2018 Friday
Bedtime Story
Newton's Minimum Resistance Problem
In the words of Newton, the Minimum
Resistance Problem was stated as follows:
“If in a rare medium, consisting of equal
particles freely disposed at equal distances from each other, a globe and a
cylinder described on equal diameter move with equal velocities in the
direction of the axis of the cylinder, the resistance of the globe will be but
half as great of the cylinder.”
This is a problem which boils down to
finding a solid of revolution that experiences minimum resistance while moving
through a homogenous fluid with constant velocity in the direction of the axis
of revolution.
Solid of revolution is the solid figure
that is achieved when a plane curve rotates around some straight line.
A simple example is that if a block of cube
placed horizontally is made to rotate around a vertical axis passing through
the center of its two sides, it will attain the shape of a cylinder.
This cylindrical shape so obtained would be
the solid of revolution.
Newton’s proposition was that were a globe
and a cylinder of same diameter were to move with same velocity in the
direction of the axis of cylinder in a homogenous medium, then the resistance experienced
by the globe would be one half of that of the cylinder.
From this famous proposition is derived a
scholium (Greek for ‘interpretation’) which says that a curve when rotated
about its axis generates a solid that experiences lesser resistance than any other
solid having a fixed length and width.
This is the first problem ever, at least in
recent recorded history that was solved using the method of calculus of
variations.
Newton published the solution in 1687 in
his Principia without the derivation.
David Gregory, a contemporary Scottish
mathematician and a huge proponent of Principia approached Newton and sought
from him the derivation of the solution which Newton obliged following which
Gregory shared it with his pupil and the world.
This was almost a decade before the
calculus of variations was applied to the famous brachistochrone problem.
Brachistochrone problem, as you would
recall, came up when Johann Bernoulli proposed it in 1696 to test if Newton
could solve it by using calculus of variations method.
Johann was on the side of Leibniz camp, and
he was hoping that the failure of Newton would finally settle the
Leibniz-Newton controversy in favor of his own camp.
It never happened as both Newton and
Leibniz (besides three others) submitted their solutions and thus the
controversy till date remains unresolved.
Lagrange himself personally never worked on
the brachistochrone problem directly; instead he applied his mind to the
tautochrone problem also known as the isochrone problem but in nature very
similar to the brachistochrone problem.
We shall discuss this tautochrone problem
to which Lagrange had applied his fabulous mind 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:
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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