June 09, 2018 Saturday
Bedtime Story
The Tautochrone Problem
The tautochrone problem can be stated this
way: is there any curve for which the time taken by an object sliding without
friction in uniform gravity to its lowest point is independent of its starting
point.
The solution to the problem is a cycloid
and the tautochrone curve is the same as the brachistochrone curve for any
given starting point.
The time taken by a body to drop down is
fixed and is calculated by multiplying pi with the square root of the radius,
the radius being the radius of the circle that generates the cycloid.
This problem was first solved by the great
Dutch scientist Christian Huygens in his 1673 magnum opus and masterpiece
Horologium Oscillatorium: sive de motu pendulorum ad horologia aptato
demonstrations geometricae which is Latin for The Pendulum Clock: or
geometrical demonstrations concerning the motion of pendula as applied to
clocks.
Using pure geometry, Huygens had solved the
tautochrone problem as evident from these lines in Horologium Oscillatorium:
“On a cycloid whose axis is erected on the
perpendicular and whose vertex is located at the bottom, the times of descent,
in which a body arrives at the lowest point at the vertex after having departed
from any point on the cycloid, are equal to each other…”
Huygens was interested in this problem
because he noticed that his pendulum clocks followed circular paths and hence
were not isochronous; which meant that pendulum clocks would vary in their
times depending on how far the pendulum swung.
Lagrange attacked this problem using
analysis and worked on it for two years between the age of 18 and 20.
It was while working on this problem that
Lagrange discovered a method of maximising and minimising functionals in a way
similar to finding extrema of functions.
Oh his discoveries he corresponded with
Euler intensively in these two years; Euler was then working at the Berlin
Academy thanks to the offer made to him by Frederick II, the King of Prussia.
Lagrangian solution is not only original
but elegant.
Just remember, that unlike Newtonian mechanics
that relies heavily on algebra and thereby vectors, Lagrangian mathematics
works only on scalars.
Lagrange, in working at the solution, used
scalar quantities such as potential energy, kinetic energy and arc length.
This parameter arc length determines the
position of the particle from the lowest point and is represented by s(t) with
s being the arc length at time periods t.
The potential energy of the point is
directly related to this arc length as it will determine the height and thus we
can say the potential energy is proportional to the height y(s).
The kinetic energy will be proportional to
2 because kinetic
energy is proportional to square of the velocity.
Note that there is a dot over the letter s.
What does that signify?
We shall ponder over it 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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