Hyderabad Institute of Arts, Science & Technology, Hyderabad
ENGG
ECON 103

Fall 2016
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Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
human ear 11 pm Hydrogen atom size 30 pm Size of small
bacterium 0.2 m Wavelength of visible light 0.4 to 0.8 m
Radius of sharp razor blade 5 m Point: diameter of smallest
object visible with naked eye 20 m Diameter of human hair
(thin to thick) 30 to 80
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
pieces. We therefore avoid fractals * Lewis Fray Richardson
(18811953), English physicist and psychologist. * Most of
these curves are selfsimilar, i.e., they follow scaling laws
similar to the abovementioned. The term fractal is due to the
mathematicia
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
accurate than often imagined, especially when trained. For time
periods between Ref. 34 a few tenths of a second, as necessary
for music, and a few minutes, humans can achieve timing
accuracies of a few per cent. Motion Mountain The Adventure
of Physics c
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
new situation shows a valid time? What happens for clocks that
also have a third hand for seconds? Challenge 85 s How
many minutes does the Earth rotate in one minute? What
is the highest speed achieved by throwing (with and without a
racket)? What was Ch
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
mind that the size of an object or of a piece of empty space is a
tricky quantity and that we need to be careful whenever we
talk about it. In summary, we avoid strangely shaped volumes
and curves in the following, and we take special care when we
talk ab
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
we can define sequences of positions in precisely three
independent ways. Indeed, the inner ear of (practically) all
vertebrates has three semicircular canals that sense the bodys
acceleration in the three dimensions of space, as shown in
Figure 21.* Simi
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
mentioned. Today this generalized concept of area, technically a
finitely additive isometrically invariant measure, is called a
Banach measure in his honour. Mathematicians sum up this
discussion by saying that since in two dimensions there is a
Banach me
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
length must be supplemented to avoid this freedom and to
prevent the square from being deformed? The extra line
interlinks must be in the same plane as the square and each
one may only be pegged to others at Challenge 128 s the
endpoints. Motion Mountain
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
influential mathematician. * The proof of the result does not
need much mathematics; it is explained beautifully by Ian
Stewart in Paradox of the spheres, New Scientist, 14 January
1995, pp. 2831. The proof is based on the axiom of choice, Vol.
III, page
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
you can check this at home. In this experiment, whatever the
spring load of the cannon, the two bodies will always collide in
midair (if the table is high enough), thus proving the assertion.
In other words, a flying cannon ball is not accelerated in the
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
definition? Obviously, * Mathematicians say that a socalled
Lebesgue measure is sufficient in physics. This countably
additive isometrically invariant measure provides the most
general way to define a volume. * Another famous exception,
unrelated to atom
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
200 s Forsythia europaea, F. suspensa, F. viridissima, F.
spectabilis Flower petal oscillation, discovered by Van Gooch in
2002 5.1 ks Motion Mountain The Adventure of Physics
copyright Christoph Schiller June 1990October 2016 free
pdf file available at w
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
display. Measure the time , in seconds, between the moment
that you see the rocket explode in the sky and the moment you
hear the explosion, measure the (plane) angle pronounced
alpha of the ball formed by the firework with your hand. The
diameter is 6 m
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
and are called coordinates. They specify and order the location
of a point in space. (For Page 79 the precise definition of
Euclidean spaces, see below.) What is described here in just half
a page actually took 2000 years to be worked out, mainly
because
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
modelling works during most of the adventures that follows.
However, this common model of space and time cannot be
confirmed by experiment. For example, no experiment can
check distances larger than 1025 m or smaller than 1025 m;
the continuum model is li
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
height is a conical glass half full? rubber band F I G U R E 31 Can
the snail reach the horse once it starts galloping away?
Challenge 74 e Can you place five equal coins so that each one
touches the other four? Is the stacking of two layers of three
coin
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
never asked, not even in Challenge 90 s other languages. Why?
Challenge 91 s Is there a smallest time interval in nature?
A smallest distance? Given that you know what
straightness is, how would you characterize or define the
Challenge 92 s curvature of
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
take positions in an apparently continuous * Hermann Weyl
(18851955) was one of the most important mathematicians of
his time, as well as an important theoretical physicist. He was
one of the last universalists in both fields, a contributor to
quantum the
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
us consider general shapes and general cuts in three
dimensions, not just the nice ones mentioned so far. We then
stumble on the famous BanachTarski theorem (or Ref. 42
paradox). In 1924, Stefan Banach and Alfred Tarski* proved
that it is possible to cut
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
limit to the attainable precision? Page 55 Fractals in three
dimensions bear many surprises. Let us generalize Figure 25 to
three dimensions. Take a regular tetrahedron; then glue on
every one of its triangular faces a smaller regular tetrahedron,
so that
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
makers are experts in producing motion that is as regular as
possible. We will Page 177 discover some of their tricks below.
We will also explore, later on, the limits for the preVol. V, page
45 cision of clocks. Is there a perfect clock in nature? Do clo
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
space just listed has to be checked. And again, careful
observations will show that each property is an approximation.
In simpler and more drastic words, all of them are wrong. This
confirms Weyls statement at the beginning of this section. In
fact, his s
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
(math * For a definition of uncountability, see page 281 in
Volume III. * Note that saying that space has three dimensions
implies that space is continuous; the mathematician and
philosopher Luitzen Brouwer (b. 1881 Overschie, d. 1966
Blaricum) showed th
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
Adventure of Physics copyright Christoph Schiller June 1990
October 2016 free pdf file available at
www.motionmountain.net 3 how to describe motion
kinematics 75 collision F I G U R E 44 Two ways to test that the
time of free fall does not depend on hori
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
large ones.We will discuss Page 54 size in more detail shortly.
How do we deduce space from observations? During childhood,
humans (and most higher animals) learn to bring together the
various perceptions of space, namely the visual, the tactile, the
audi
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
taken to simulate the inevitable measurement errors. Stating
that motion is the change of position with time is neither an
explanation nor a definition, since both the concepts of time
and position are deduced from motion itself. It is only a
description
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
vectors behave like arrows, vectors can be added and they can
be multiplied by numbers. For example, stretching an arrow =
(, , ) by a number corresponds, in component notation,
to the vector = (, , ). In precise, mathematical
language, a vector is an ele
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
pdf file available at www.motionmountain.net 2 from motion
measurement to continuity 63 Challenge 78 s infinitely
stretchable. Can the snail reach the horse? For a
mathematician, 1 km is the same as 1000 m. For a physicist the
two are different! Indeed, f
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
thumb at the end of your extended arm? Challenge 110 e
Angular size is not an intuitive quantity; it requires
measurement instruments. A famous example, shown in Figure
36, illustrates the difficulty of estimating angles. Both the Sun
and the Moon seem la