Knowledge

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Knowledge And Deep Knowledge Gangan
Prathap CUSAT JNCASR Kochi 682 022 and
Bangalore 560064
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Knowledge

• Choose an engine (maximum deliverable thrust)
  for a rocket weighing 1000kg to have a maximum
  acceleration of 10 m/s2.

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Knowledge

• Choose an engine (maximum deliverable thrust)
  for a rocket weighing 1000 kg to have a maximum
  acceleration of 10 m/s2.
• F m a
• Therefore,
• Engine which can deliver a thrust of
• 10000 Newtons
• is required.
• This is all an engineer is required to know.

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Deep Knowledge

• But why
• F m a
  (Galileo)
• and not
• F m v (Aristotle)
• is the question a natural philosopher will ask.

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Deep Knowledge

• Laws of conservation of energy and momentum will
  lead to
• F m a
• and not
• F m v
• is what a natural philosopher will find.
• (Newton, Laplace, Lagrange, Hamilton )

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Deep Knowledge

• From Newton to Hamilton,
• the Least Action Principle is introduced.
• Why bring in Action?
• or a
• Least Action Principle?

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Deepest Knowledge

• Emmy Noether (1882-1935)
• brings in the connection between
• Symmetry and Conservation Laws
• Noethers Theorem was published in 1918 and
  largely ignored for nearly forty years

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Deepest Knowledge

• If the laws of motion are to be invariant
  (symmetry)
• under any arbitrary change of frame of reference
  (time, space),
• then the laws of conservation of energy and
  momentum will emerge
• from a definition of action and this will lead
  to such invariant laws of motion

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Deepest Knowledge

If a physical system behaves the same regardless
of how it is oriented in space, the physical laws
that govern it are rotationally symmetric from
this symmetry, Noether's theorem shows the
angular momentum of the system must be
conserved. Rather, the symmetry of the physical
laws governing the system is responsible for the
conservation law. If a physical experiment has
the same outcome regardless of place or time
(working the same, say, in Cleveland on Tuesday
and Samaria on Wednesday), then its laws are
symmetric under continuous translations in space
and time by Noether's theorem, these symmetries
account for the conservation laws of linear
momentum and energy within this system,
respectively. (from Wikipedia)
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THE LAW OF THE LEVER GIVE ME A
PLACE TO STAND AND I WILL MOVE THE EARTH A
remark of Archimedes quoted by Pappus of
Alexandria in his "Collection" (Synagoge, Book
VIII, c. AD 340 ed. Hultsch, Berlin 1878, p.
1060).
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• THE LAW OF THE LEVER
• The two wonders
• 1. The phenomenological or empirical law
• Ancient Greeks such as Aristotle knew about the
  principle of the lever (or "law of the lever")
  very early on in history, but they had trouble
  proving their theories.
• 2. The proof from 1st principles
• Archimedes, a Greek mathematician who lived from
  287-212 B.C., made a statement about when levers
  are in equilibrium.
• "The law states that a lever is in
  equilibrium when the product of the applied force
  and the distance from the from the point of
  application to the fulcrum equals the product of
  the resisting force and the distance from it's
  point of application to the fulcrum."

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L1
L2
F1
F2
F1xL1 F2xL2 How did Archimedes derive this
law?
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More on the law of the lever The concept of
moment about a point or the equilibrium of such
moments was not known till the time of Stevinus,
i.e. nearly 19 centuries later! Archimedes
was the first to use the principle of virtual
work.
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d1
d2
L1 L2
F1 F2 The
principle of virtual work d1xF1 d2xF2 From
Euclids geometry d1/L1 d2/L2 Therefore,
F1xL1
F2xL2
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Deep understanding of the law of the lever
F1
F2 d1 ?
L1 L2
d2 Kinematics Kinetics
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Why does this very interesting relationship
emerge? In kinematics, we have
considered pure deformation, without worrying
about forces. In kinetics, we have considered
forces at equilibrium, without considering the
deformation at all.
Yet, they are
inter-linked through a very interesting
relationship. Note that a purely verbal
language would have never been able to show this
aweinspiring form. Yet the language of
mathematics grasps the poetry of the relationship
so elegantly.
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THERE IS A story about two friends, who were
classmates in high school, talking about their
jobs. One of them became a statistician and was
working on population trends. He showed a reprint
to his former classmate. The reprint started, as
usual, with the Gaussian distribution and the
statistician explained to his former classmate
the meaning of the symbols for the actual
population, for the average population, and so
on. His classmate was a bit incredulous and was
not quite sure whether the statistician was
pulling his leg. "How can you know that?" was his
query. "And what is this symbol here?" "Oh," said
the statistician, "this is pi." "What is that?"
"The ratio of the circumference of the circle to
its diameter." "Well, now you are pushing your
joke too far," said the classmate, "surely the
population has nothing to do with the
circumference of the circle." The
Unreasonable Effectiveness of Mathematics in the
Natural Sciences
Eugene Wigner
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WHAT IS PHYSICS? The physicist is interested in
discovering the laws of inanimate nature. What
is the concept, "law of nature"? Schrodinger
has remarked, that it is a miracle that in spite
of the baffling complexity of the world, certain
regularities in the events could be discovered.
Being able to notice this and express this as an
empirical or phenomenological law is the 1st
wonder I talked about. One such regularity,
discovered by Stevinus/Galileo, is that two
rocks, dropped at the same time from the same
height, reach the ground at the same time. The
laws of nature are concerned with such
regularities.
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• Galileo's regularity is a prototype of a large
  class of regularities.
• It is a surprising regularity for three reasons
• It is surprising that it is true not only in
  Pisa, and in Galileo's time, it is true
  everywhere on the Earth, was always true, and
  will always be true. This property of the
  regularity is a recognized invariance property
  and, without invariance principles similar to
  those implied in the preceding generalization of
  Galileo's observation, physics would not be
  possible.
• 2. The regularity is independent of so many
  conditions which could have an effect on it. It
  is valid no matter whether it rains or not,
  whether the experiment is carried out in a room
  or from the Leaning Tower, no matter whether the
  person who drops the rocks is a man or a woman.
  It is valid even if the two rocks are dropped,
  simultaneously and from the same height, by two
  different people.

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3. The preceding two points, though highly
significant from the point of view of the
philosopher, are not the ones which surprised
Galileo most, nor do they contain a specific law
of nature. The law of nature is contained in the
statement that the length of time which it takes
for a heavy object to fall from a given height is
independent of the size, material, and shape of
the body which drops. In the framework of
Newton's second "law," this amounts to the
statement that the gravitational force which acts
on the falling body is proportional to its mass
but independent of the size, material, and shape
of the body which falls.
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The preceding discussion is intended to remind
us, first, that it is not at all natural that
"laws of nature" exist, much less that man is
able to discover them. There is a succession of
layers of "laws of nature," each layer containing
more general and more encompassing laws than the
previous one and its discovery constituting a
deeper penetration into the structure of the
universe than the layers recognized before.
However, the point which is most significant in
the present context is that all these laws of
nature contain, in even their remotest
consequences, only a small part of our knowledge
of the inanimate world. All the laws of nature
are conditional statements which permit a
prediction of some future events on the basis of
the knowledge of the present, except that some
aspects of the present state of the world, in
practice the overwhelming majority of the
determinants of the present state of the world,
are irrelevant from the point of view of the
prediction. The irrelevancy is meant in the sense
of the second point in the discussion of
Galileo's theorem.
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Tempus in quo aliquod spatium a
mobili conficitur latione ex quiete uniformiter
accelerata, est aequale tempori in quo idem
spatium 10 conficeretur ab eodem mobili motu
aequabili delato, cuius velocitatis gradus
subduplus sit ad summum et ultimum gradum
velocitatis prioris motus uniformiter accelerati.
The time in which any space is traversed by a
body starting from rest and uniformly accelerated
is equal to the time in which that same space
would be traversed by the same body moving at a
uniform speed whose value is the mean of the
highest speed and the speed just before
acceleration began. t s/v s/(v0
v1/2) or s v0 v1t/2
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Let us represent by the line AB the time in which
the space CD is traversed by a body which starts
from rest at C and is uniformly accelerated let
the final and highest value of the speed gained
during the interval AB be represented by the line
EB drawn at right angles to AB draw the line AE,
(Condition 2/00-th-00-dialog1) then all lines
drawn from equidistant points on AB and parallel
to BE will represent the increasing values of the
speed, beginning with the A. Let the point F
bisect the line EB draw FG parallel to BA, and
GA parallel to FB, thus forming a parallelogram
AGFB which will be equal in area to the triangle
AEB, since the side GF bisects the side AE at the
point I for if the parallel lines in the
triangle AEB are extended to GI, then the sum of
all the parallels contained in the quadrilateral
is equal to the sum of those contained in the
triangle AEB for those in the triangle IEF are
equal to those contained in the triangle GIA,
while those included in the trapezium AIFB are
common. Since each and every instant of time in
the time-interval AB has its corresponding point
on the line AB, from which points parallels drawn
in and limited by the triangle AEB represent to
increasing values of the growing velocity, and
since parallels contained within the rectangle
represent the values of a speed which is not
increasing, but constant, it appears, in like
manner, that the momenta (momenta) assumed by the
moving body may also be represented, in the case
of the accelerated motion, by the increasing
parallels of the triangle AEB, and, in the case
of the uniform motion, by the parallels of the
rectangle GB. For, what the momenta may lack in
the first part of the accelerated motion (the
deficiency of the momenta being represented by
the parallels of the triangle AGI) is made up by
the momenta represented by the parallels of the
triangle IEF. (Condition Aristot-space-prop)
Hence it is clear that equal spaces will be
traversed in equal times by two bodies, one of
which, starting from rest, moves with uniform
acceleration, while the momentum of the other,
moving with uniform speed, is one-half its
maximum momentum under accelerated motion. Q. E.
D. t s/v s/(v0 v1/2) or s
v0 v1t/2
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FROM CLASSICAL TO QUANTUM PHYSICS The
principal purpose of the preceding discussion is
to point out that the laws of nature are all
conditional statements and they relate only to a
very small part of our knowledge of the world.
Thus, classical mechanics, which is the best
known prototype of a physical theory, gives the
second derivatives of the positional coordinates
of all bodies, on the basis of the knowledge of
the positions, etc., of these bodies. It gives no
information on the existence, the present
positions, or velocities of these bodies. It
should be mentioned, for the sake of accuracy,
that we discovered about thirty years ago that
even the conditional statements cannot be
entirely precise that the conditional statements
are probability laws which enable us only to
place intelligent bets on future properties of
the inanimate world, based on the knowledge of
the present state.
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FROM PHYSICS TO ENGINEERING As regards the
present state of the world, such as the existence
of the earth on which we live and on which
Galileo's experiments were performed, the
existence of the sun and of all our surroundings,
the laws of nature are entirely silent. It is in
consonance with this, first, that the laws of
nature can be used to predict future events only
under exceptional circumstances - when all the
relevant determinants of the present state of the
world are known. It is also in consonance with
this that the construction of machines, the
functioning of which he can foresee, constitutes
the most spectacular accomplishment of the
physicist. In these machines, the physicist
creates a situation in which all the relevant
coordinates are known so that the behavior of the
machine can be predicted. Radars and nuclear
reactors are examples of such machines.
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LAYERS of KNOWLEDGE
Shallow How to do? Engineering, Technology Deep
Why you do it the way you do? Science Deepest
Metaphysics, Philosophy