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H ISTORY
OF
M ECHANICS
ou tline
1.1 Ancient thoughts
1.2 Renaissance—heliocentric theory
1.3 Birth of modern science—Galileo
and Newton
2
4
1.4 Fundamental revision of spacetime—
Einstein
1.5 Mechanics and modern physics
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The study of motion of physical bodies is called mechanics. This subject dates back to ancient
times; many wise and curious people have wondered about the motion of heavenly bodies, often in
preference to terrestrial ones. The question of whether the Sun goes around the Earth, or the Earth
goes around the Sun created one of the most heated debates in science.
A study of history of mechanics reveals that Aristotle formulated one of the first theories of
mechanics.1 Aristotle’s theory of motion was rather philosophical and relied heavily on logic.
Many centuries later it was observed that Aristotle’s predictions did not match with experimental
(or empirical) observations. Around the sixteen century, Galileo, Newton and other physicists
formulated a completely new theory of mechanics that forms the basis of modern science. In this
new theory, observations and experiments play a very important role as they are used to verify or
reject a physical theory.
At the turn of the twentieth century it was found that Newton’s laws of motion do not work
for the particles moving with speeds close to that of light, and for microscopic particles such as
electrons. To understand these phenomena, two new theories were proposed: theory of relativity
and quantum mechanics. The present book will focus on Newton’s laws, and will briefly discuss the
modern theories.
1 Theory
of mechanics by other ancient civilisations such as the Chinese, Indians, Mayans and Egyptians have not
been studied very widely. To name some of the work from these civilisations, an Indian philosophy school named
Nyaya Vaisheshika attempted to study the structure of matter. Also, several historians claim that the Aryabhatta
school developed a heliocentric planetary model before Copernicus. In this book, however, we will focus on western
history of science since it is well documented.
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In the following discussion, we will outline only major discoveries in mechanics—from the ancient
to the present time. The reader can refer to Ferris (1988), Spangenburg and Moser (1993) and
Wikipedia for further discussions on these topics.
While studying the history of mechanics, it becomes evident that mechanics was the front-runner
in founding modern scientific ideas. Let us start our discussion on the history of mechanics.
1.1 Ancient Thoughts
In this section we we will focus on Greek thoughts since they are well documented (see footnote
on page 1). The Greeks discovered that the Earth is round, and estimated the radius of the Earth
fairly accurately. They also attempted to measure the Earth–Moon and Earth–Sun distances. They
came up with many interesting ideas regarding the planets, the stars and the universe. Due to lack
of time and space, we cannot describe all these ideas here. In the following paragraphs we will only
discuss the ideas of Eudoxus, Aristotle and Ptolemy.
Many Greek thinkers worried about the movement of stars and planets. The most prominent
among them was Eudoxus (410–355 bc, or 408–347 bc). He belonged to Plato’s academy in Athens.
Deeply influenced by Plato’s appeal for perfect geometrical forms (sphere, regular solids and so on),
Eudoxus proposed that the universe is composed of 27 concentric spheres surrounding the Earth
(Fig. 1.1). Each sphere rotates with a constant velocity. The axes of some of the spheres were
postulated to be inclined to each other. The stars were carried by the outermost sphere. Each
planet moves in four spheres (20 for 5 planets), and the Sun and the Moon move in three spheres
each. By adjusting the rates of rotation and the inclination of the axes of the spheres, Euxodus
could explain the movement of heavenly bodies reasonably satisfactorily. In this model, the Earth
is at the centre of the universe, hence, is called the geocentric model of the universe.
The most complex motion in the sky is that of the outer planets. They exhibit a phenomenon
called retrograde motion during which a planet moves westward, then eastward and again westward (Fig. 1.2). Note that each planet must be observed at a fixed time in the night. Eudoxus
explained this phenomena by introducing epicycles [Fig. 1.3(a)]. The planet moves on a smaller
sphere that itself rotates around a larger sphere. The apparent motion of the planet from the Earth
in the epicycle model is depicted in Fig. 1.3(b). Note that a single epicycle produced the trajectory
shown in Fig. 1.3(b); several epicycles are required to generate the retrograde motion observed in
the sky.
Aristotle (384–322 bc) introduced more spheres to take the count up to 55. In spite of that, the
results from more refined observations performed later did not match with the Aristotelian model.
To keep the geocentric picture intact, more complex set of spheres were introduced. Claudius
Ptolemy (85–165 ad) constructed the most refined geocentric model that lasted for nearly 1400
years. As we will describe later, the heliocentric picture, much simpler than the geocentric one, can
explain the retrograde motion. Unfortunately, prevalent prejudices typically hinder thinkers from
attempting simple alternate ideas.
Aristotle set out to explain why the heavenly spheres move. He postulated that the prime
mover rotates the spheres containing the heavenly bodies. The planets, the Moon and the Sun are
regulated by several spheres. For Aristotle, the heavenly cosmos was imperishable and perfect,
and the heavenly spheres moved in circular orbits. Aristotle and earlier Greek philosophers chose
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Heavens
Sa
M
V
Fire
Air
Water
Hg
S
E
Ma
J
Fig. 1.1 Eudoxus’ and Aristotle’s model of the cosmos. The symbols E, M, Hg, V, S, Ma, J,
Sa represent the Earth, Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn.
Fig. 1.2 Observed motion of Mars and Saturn. These planets move westward,
eastward and westward. This phenomenon is called retrograde motion.
circular orbits over other trajectories due to the perfect symmetry of a circle. Aristotle argued that
unlike heavenly bodies, everything on the Earth was changeable and corrupt. The motion on the
Earth is linear, not circular.
To explain the motion in the terrestrial environment, Aristotle argued that each object is made
up of four elements—Earth, Water, Air and Fire—in various proportions.2 Each object moves in
such a way that it returns to its natural state, that is, to one of the spheres on the Earth [Fig. 1.1
(sphere of earth, water, air or fire)]. Solid objects like stones would tend to fall on the Earth since it
is their natural place. Air and fire are above the ground, hence, objects composed mainly of air or
fire would go up towards their natural places. A consequence of this theory is that heavier bodies
containing more earth will fall to the Earth faster than lighter ones.
Aristotle’s arguments for the vertical motion on the Earth are logically consistent, but they do
not work for projectile motion. He postulated a rather awkward argument to explain projectile
2 Indian
philosophy Nyaya Vaisheshika invokes five elements, called panchmahabhutas: Earth, Water, Air, Fire
and Akasha. In this philosophy, Akasah or ether is supposed to eternal, all-pervading and imperceptible to the senses.
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1
y
Earth
0
Planet
−1
Epicycle
−2
−2
(a)
−1
0
x
1
2
(b)
Fig. 1.3 (a) The retrograde motion of an outer planet explained using epicycles. The planet
moves on a sphere that itself rotates around a sphere whose centre is the Earth. (b) The apparent
motion of the planet in Earth’s frame of reference.
motion. He theorised that air rushes into the vacuum created by the projectile’s trail and pushes
the projectile. Compare the complexity of Aristotelian theory of motion with the relatively simple
Newton’s laws of motion.
According to Aristotle, an unforced object comes to rest when it reaches its natural place. Hence
a block sliding on a horizontal plank will come to rest unless forced, since rest is the natural state
of the block.3 Aristotle also gave a theory for the origin of force that can be illustrated using an
example. For projectile motion to occur, the thrower’s arm provides force to the projectile; the
thrower to his/her arm; and so on, forming an infinite chain of causes. Aristotle argued that there
must be an unmoved mover, something which can initiate motion without itself being set in motion.
This view was preserved by the medieval Church during the Dark Ages, and it became a ruling
paradigm.
In hindsight, Aristotle’s theory of the cosmos and causation were constructed by pure logic
without resorting to proper observational and experimental tests, hence, they were metaphysical and
philosophical. Around two thousand years later, many inconsistencies were found in the Aristotelian
theory, the resolution of which led to the birth of modern science.
1.2 Renaissance—Heliocentric Theory
1.2.1 Nicolaus Copernicus
In medieval times, astronomers performed very accurate observations of planetary orbits
including their retrograde motion, which could not be explained by Ptolemy’s system. Copernicus
3 Contrast this argument with Newton’s first law of motion according to which no force is needed to sustain a
constant velocity of a moving body.
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(1473–1543), a Polish astronomer, discovered that the retrograde motion of the planets (Mars and
Saturn) could be explained quite easily if one assumes that the planets go around a stationary Sun.
Copernicus’ idea is illustrated in Fig. 1.4; Earth and Mars orbit the Sun in a circular orbit. The
temporal positions of the Earth are indicated as points 1–7 in the inner orbit, and that of Mars
as the corresponding dots in the outer orbit. For an observer on the Earth, the position of the
Mars in the sky appears as points 1–7 on the right edge of the figure. Clearly, Mars appears to
move westward (left) at points 1, 2 and 3; eastward (right) at points 4 and 5; and again westward
at points 6, 7 and beyond. This is how one can explain the retrograde motion of the Mars in the
heliocentric model. Note that the retrograde motion takes place during the phase when the Sun,
Earth and Mars are collinear. In Example 1.1, we will show this behaviour quantatively.
φ
M
7 6
E
y0
4
3
1
6
5
S
7
2
5
4
3
2
1
0
x
Fig. 1.4 In the heliocentric model, the Earth and Mars orbit around the Sun in the inner and
outer circles respectively. The apparent positions of Mars in the sky are shown in the right
edge of figure. The dots on the trajectories occur at t = t/TE = −1/4 + j/12, where TE
is the time period of Earth’s orbit and j = 0 : 6.
EXAMPLE 1.1 In the heliocentric picture, describe the motion of Mars in the Earth’s sky.
Assume the orbits of Mars and Earth around the Sun to be circular.
SOLUTION In the heliocentric model depicted in Fig. 1.4, the inner and outer circles are
Earth’s and Mars’ orbits respectively. We assume that the Earth, Mars and Sun are collinear
at t = 0. In a coordinate system fixed to the Sun, the coordinates of Mars and Earth are (see
Fig. 1.4):
xMars = RM S cos(ωMars t); yMars = RM S sin(ωMars t)
(1)
xE = RES cos(ωE t); yE = RES sin(ωE t)
(2)
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where RM S and RES are the distances of Mars and Earth from the Sun respectively, and ωMars
and ωE are the angular velocities of Mars and Earth around the Sun respectively.
The relative coordinates of Mars with respect to the Earth are:
xr = xMars − xE = RM S cos (ωMars t) − RES cos (ωE t)
(3)
yr = yMars − yE = RM S sin (ωMars t) − RES sin (ωE t)
(4)
If we measure time in units of one terrestrial year, and the distance in units of RES , then
t = t TE = t xr =
r = RES r yr =
RM S
RES
RM S
RES
cos
sin
TE
TMars
TE
TMars
From observations, the ratio
2πt − cos (2πt )
(6)
2πt − sin(2πt )
(7)
TMars
RM S
= 1.52 and
= 1.88.
RES
TE
The line joining the Earth and Mars makes an angle φ with the x axis (see Fig. 1.4). In terms
of xr and yr ,
(5)
where t , r are the nondimensional time and distance respectively. Hence
2π
;
ωE
tan φ =
yr
xr
(8)
In Fig. 1.4, we mark the positions of the Earth and Mars at 7 equidistant points in the interval
t = −0.25 : 2.5. At t = 0, the Sun, Earth and Mars are collinear. Note that the orbital motion
of Mars is slower than that of the Earth. In Fig. 1.5(a), we exhibit the relative position of Mars
relative to Earth. An observer on the Earth however cannot measure (xr , yr ), but she can measure
the angular position φ of Mars. The angle φ(t), illustrated in Fig. 1.5(b), increases all times except
between the maximum and the minimum of φ(t) (for t ≈ −0.1 : 0.1). The increase of φ indicates a
westward motion for Mars, while its decrease corresponds to an eastward motion. Thus, the plot of
Fig. 1.5(b) reveals that retrograde motion occurs in the interval t ≈ −0.1 : 0.1, which translates to
approximately 0.2 terrestrial year or 70 days. Note that Mars retrogrades when the Sun, Earth and
Mars are collinear. More careful analysis of the above (for example, exact computation of maximum
and minimum) shows that Mars retrogrades for 72 days every 25.6 months.4
4 http://en.wikipedia.org/wiki/Apparent
retrograde motion
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1
1
2 3 4
7
5
6
−0.5
−2
−2
0
xr
(a)
2
−1.0
0.0
0.5
t
(b)
Fig. 1.5 (a) Plot of yr vs. xr for t = −0.25 : 2.5. (b) Plot of the angle φ = tan−1 (yr /xr ) as
a function of time. The retrograde motion takes place when the angle φ decreases (for t ≈ −0.1 : 0.1).
The sample points shown by dots in both the plots are same as those shown in Fig. 1.4.
1.2.2 Tycho Brahe
Tycho Brahe (Denmark, 1546–1601) can be considered to be one of the greatest astronomers
before the advent of telescopes. In an island near Copenhagen, Brahe built the ultimate astronomical
observatory of his day.5 Brahe observed that some new bright objects appeared and then
disappeared in the sky. This dispelled Aristotle’s idea that the heavens are unchanging. Later
these objects were identified to be comets and supernovae.
Brahe did not believe in Copernicus’ heliocentric picture. He thought of a composite model in
which the planets moved around the Sun, but the Sun itself moved around the Earth. To obtain
an observational support for his model, from the year 1576 to 1597, Brahe made very precise
astronomical observations of planets (up to 2 to 3 minutes of arc), which provided him with a huge
data bank. Brahe was a brilliant astronomer, but not a very good theorist. He could not analyse
his data; this task was done by his student Kepler after his death.
1.2.3 Johannes Kepler
Kepler (Germany, 1571–1630) inherited Tycho Brahe’s data and analysed them with great care. The
analysis of the Mars data itself took around a decade. He summarised his findings in the following
three laws of planetary motion:
1. The orbit of a planet lies in a fixed plane containing the Sun. The planet moves in the plane
in an elliptical orbit with the Sun as one of its foci.
2. The radius vector from the Sun to the planet sweeps out equal areas in equal amounts of time.
5 Tycho
Brahe used sophisticated sextants for his measurements that yielded 2 to 3 minutes of accuracy. Note that
Ulugh Beg (Samarkand, 1394–1449) and Maharaja Sawai Jai Singh (Jaipur, 1688–1743) also made observations of
stars and planets using non-telescopic instruments; the accuracy of their instruments was approximately 1 minute of
arc. Jai Singh’s observatories, called Jantar Mantar, located at Jaipur, Delhi, Mathura, Banaras and Ujjain are still
around.
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3. T 2 = kR3 , where T is the period of revolution, R is the semi-axis of the elliptical path of the
planet and k is a constant.
Recall that all earlier thinkers believed the planetary orbits to be circular. This prejudice came
from Plato’s fondness for symmetries in nature. Kepler claimed Mars’ orbit to be elliptical. This
claim was quite courageous because the eccentricity of Mars’ orbit [e = (rmax − rmin )/(rmax + rmin )]
is 0.093, which is very close to zero (eccentricity of a circle). The difference rmax − rmin is only 18%
of the average radius of Mars. Kepler strongly believed in Brahe’s data and his own calculations.
In Chapter 9, we will discuss Kepler’s laws with accurate planetary data. Meanwhile, you can refer
to Figs 9.10 and 9.14 for the confirmation of Kepler’s laws. Note that Mars’ orbit appears quite
close to a circle.
Kepler’s laws played a very important role in the development of mechanics. Newton used
Kepler’s laws to arrive at the law of universal gravitation.
1.3 Birth of Modern Science—Galileo and Newton
Around the fifteenth century, the Renaissance brought in many new ideas in Europe. During a
span of around 300 years, modern science was born due to the contributions of many thinkers like
Descartes, Galileo, Newton, Laplace, Hooke and so on. Here we will focus on the contributions of
Galileo and Newton.
1.3.1 Galileo Galilei
Galileo Galilei (Italy, 1564–1642) is considered to be the father of modern science. He placed
firm emphasis on observations. For example, he rolled balls down an inclined plane and measured
the time elapsed using a water clock (similar to an hour glass). He recognised this motion to be
equivalent to free fall, and observed that all the balls, independent of their constitution, took the
same time to descend a given height. This observation clearly disproved Aristotle’s idea that heavier
bodies fall faster than lighter ones.6
Galileo made many more important observations. For the first time he showed that the time
period of a pendulum is independent of its amplitude, which led to the invention of clocks. He also
got hold of a then-recently invented instrument called the telescope and started observing the sky.
He discovered the moons of Jupiter. The moons revolving around Jupiter rather than the Earth
provide further credence to Copernicus’ heliocentric theory.
One of the most important contribution of Galileo is the law of inertia according to which a body
retains its velocity unless a force acts on the body. This law contradicts Aristotle’s theory that an
object slows down and stops unless some force acts on it. Galileo’s law of inertia is valid only for
6 Galileo also constructed a beautiful logical argument to prove that all masses fall with the same acceleration.
Tie up a heavy body with a light body. By Aristotle’s theory that heavier bodies fall faster than lighter ones, the
composite body should fall faster than both its constituents, as it is heavier than both. We can also argue that the
lighter body should impede the motion of the heavier body (like for a combination of a fast runner and a slow runner),
hence the speed of the composite body should be an average of the speeds of its constituents. Hence a contradiction!
Thus Galileo concluded that under gravity, all the bodies fall at the same rate, a hypothesis that does not suffer from
the above contradictions.
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a free particle, which is an abstraction. Galileo also gave a law of relativity according to which the
laws of nature are the same in all reference frames moving with constant velocity.
Galileo provided a new paradigm for science in which the laws are constructed and refuted
on the basis of observations. Einstein said the following in his homage to Galileo, ‘Pure logical
thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from
experience and ends in it.... Because Galileo saw this, and particularly because he drummed it into
the scientific world, he is the father of modern physics—indeed, of modern science altogether.’
For a brief discussion on the present paradigm of physics, refer to Appendix A.
1.3.2 Isaac Newton
Isaac Newton (England, 1642–1727) was born the year Galileo died. Newton synthesised the ideas
of Copernicus, Kepler and Galileo, and produced compact and abstract laws of motion, and the
law of universal gravitation. These theories had an enormous impact on science. Newton was very
strong in mathematics; he is the first physicist who wrote the laws of physics in precise mathematical
forms. Note that Newton is also the discoverer of calculus.7
You are aware of Newton’s laws of motion from your school days. So we will not delve into them
except to remind you that Newton’s laws are universal, that is, they are applicable to all systems
except those that move with speeds comparable to that of light.
There is a story that Newton discovered the idea of gravitation when an apple fell on his head.
This is a story for children and definitely untrue. Newton used Kepler’s laws to discover the law
of universal gravitation. Huygens (Netherlands, 1629–1695) first derived an expression for the
acceleration of an object going around in a circular path. Using Huygens’ formula, the acceleration
of a planet going around the Sun is:
a = ω2 R =
4π 2
R
T2
(1.3.2)
where k is a constant. Substitution of Eq. (1.3.2) in in Eq. (1.3.1) yields:
4π 2 1
a=
k R2
(1.3.1)
where ω is the frequency of the circular motion, and R is the distance of the planet from the Sun.
According to Kepler’s third law,
T 2 = kR3
(1.3.3)
From Newton’s second law of motion, the force acting on the planet is:
4π 2 1
F = ma = m
k R2
7 Leibnitz
(1646–1716) also discovered calculus independent of Newton.
(1.3.4)
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which is inversely proportional to R2 . So Newton concludes that the Sun attracts planets by a
gravitational force that varies as 1/R2 .
Newton felt that the same force must be making the Moon go around the Earth. He also noted
that the Moon falls towards the Earth all the time while still following a circular path. Newton
cleverly used Kepler’s third law for the Earth–Moon system. The acceleration of the Moon towards
the Earth is:
aM =
4π 2 REM
2
TM
(1.3.5)
where REM is the distance between the Earth and the Moon, and TM is the time period of Moon’s
orbit around the Earth.
After this, Newton used Eq. (1.3.3) to deduce the acceleration of an object on the surface of the
Earth:
aE = aM
REM
RE
2
4π 2
= 2
TM
REM
RE
3
RE
(1.3.6)
where RE is the radius of the Earth. Substitution of TM = 27 days 7 hours and 43 minutes (present
data, not that of Newton’s time), REM /RE = 60.1, and RE = 6400 km yields:
aE = 9.84 m/s2
(1.3.7)
Newton measured the acceleration due to gravity by rolling various masses on an inclined plane
and found the experimental value to be in general agreement with the above theoretical prediction.
Using this result, Newton concluded that the forces acting between the Sun and planets, the Earth
and the Moon, and the Earth and objects on the Earth have the same origin—gravitational force.
A remark is in order here. Newton proved the inverse square law of gravitational force for elliptical
orbits. The proof is geometrical and somewhat involved.
From the above calculations, it is also clear that the acceleration due to gravity on the surface
of the Earth is independent of the mass of the object. Newton verified this theoretical observation
by measuring the acceleration of objects made of various materials on an inclined plane. Thus
Aristotle’s theory of motion was proven to be incorrect.
This is the genius of Newton. He used Kepler’s third law of planetary motion and mathematical
arguments to deduce the universal gravitational force between objects at disparate scales. The
universality of gravitation indicates that the laws of physics on the Earth and in the heavens are
the same, contrary to Aristotle’s belief. Newton deduced the universal law of gravity according to
which any two masses m1 and m2 separated by a distance r exert an attractive force on each other
whose magnitude is:
f =G
m1 m2
r2
where G is Newton’s constant of gravitation. The direction of the gravitational force is towards
each other.
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Although Newton constructed general abstract laws in the language of mathematics, these
laws are based on observations. Newton made various predictions based on his laws, for example,
the orbits of comets, tidal effects, the oblate nature of the Earth, perturbations of planetary orbits
from the ellipse due to other astronomical objects, and so on. Many of these predictions could be
tested using observations. This method, started by Newton and Galileo, is the method of modern
science. Newton’s ideas had far-reaching consequences in the field of science. For his wide-ranging
contributions, many people consider Newton to be the greatest scientist of all time.
Newton wondered very deeply about space and time; he conjectured them to be absolute, that
is, the same for all observers. Many of Newton’s predictions based on absolute space and absolute
time proved to be correct. But problems surfaced when the speed of a particle is close to that of
light. That’s where Einstein comes to the rescue.
1.4 Fundamental Revision of Spacetime—Einstein
At the turn of the twentieth century, an experiment by Michelson and Morley showed a very
puzzling result. According to the experiment, the speed of light is the same irrespective of the
relative motion between the observer and the source. In addition, the Maxwell’s equations indicated
that the electromagnetic waves move with a constant speed independent of the motion of the source
or the observer. These results contradict the law of addition of velocities given by Newton and
Galileo. Many scientists tried to explain the constancy of the speed of light using various tricks
within Newton’s framework. However, iconoclast Einstein (Germany and USA, 1879–1955) had
another idea. He noticed that the constancy of the speed of light and the postulate of absolute
space and absolute time are contradictory. So he abandoned the concepts of absolute space and
time. He proposed a new theory called the special theory of relativity in which the time difference
between any two events could be different for different observers.
The spacetime structure of special relativity differs completely from that of Newton. The time
difference between two events could be measured differently in two inertial frames; the length of
a stick is also different in two inertial frames. The theory of relativity has many other startling
predictions, for example, it is possible to convert mass to energy and vice versa. All the predictions
of relativity have been found to be consistent with experiments, and hence it is considered to be a
correct theory. For slow moving particles, the kinematics and dynamics of the special theory are
consistent with Newton’s theory of motion.
Einstein went a step further and discovered an equivalence between accelerating frames and
gravity. Using this idea, he proposed a new theory of gravity that is more general than
Newton’s theory of gravity. For example, Einstein predicted that light bends near a star, but no such
conclusion can be drawn from Newton’s theory of gravity. Einstein’s theory of gravity is called the
general theory of relativity. Einstein’s theory of relativity has had a revolutionary impact in physics.
Figure 1.6 showcases some of the important thinkers and scientists who have made far-reaching
contributions to the field of mechanics.
1.5 Mechanics and Modern Physics
As discussed in the previous sections, modern science essentially starts with the ideas of Galileo
and Newton on mechanics. New areas like fluid mechanics, elasticity and wave theory were directly
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Fig. 1.6 Scientists who have made leading contributions to the field of mechanics. [Source: Wikipedia]
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inspired by Newton’s theory. Newton’s laws also form the basis for engineering mechanics, which
is used for designing machines, robots, cars, aeroplanes and so on. Engineers use theory of statics,
dynamics, fluid mechanics, elasticity for their product design and manufacturing.
Electrodynamics and statistical mechanics took birth around the mid-nineteenth century. Two
revolutionary theories, relativity and quantum mechanics, were born in the twentieth century when
Newton’s theory was observed to fail at high speeds and at small length scales. These theories
together form modern physics. See Fig. 1.7 for an illustration of different branches in physics.
Classical physics
Quantum mechanics
Quantum statistical
mechanics
Classical
mechanics
Electrodynamics
Statistical
mechanics
Condensed
matter physics
Particle
nuclear
physics
Newtonian
mechanics
Relativity
Nonlinear
dynamics/
chaos theory
Soft matter
physics
Rigid body
statics
Continum
physics
Rigid body
dynamics
Fig. 1.7 Different areas of physics.
Quantum mechanics differs fundamentally from Newton’s and Einstein’s conception of dynamics.
For example, in quantum mechanics we cannot simultaneously specify the position and velocity of a
particle precisely—unlike in Newton’s formalism. The ideas of force and trajectory are abandoned
in quantum mechanics.
In a major part of the twentieth century, the study of elementary particles was one of the main
areas of research. A successful theory of elementary particles called quantum field theory was
discovered by marrying electrodynamics, relativity and quantum mechanics. The quantum field
theory, which is still incomplete, is presently the best known physical theory of nature. One of the
major unsolved problems in quantum field theory is the quantum theory of gravity. Quantum field
theory is also used to study properties of matter like metals, semiconductors and so on.
Another connected area of research which has come up in recent times is nonlinear dynamics
and chaos. We explain this field using an example. Newton solved the two-body problem exactly—
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bound particles under gravitational potential have elliptical orbits. Way back in 1888, Poincaré
showed that no such solution exists for a three-body problem. In fact, if two test particles under
the influence of two heavy masses start from two nearby initial conditions, the future for these two
particles could be very different. This kind of sensitivity to initial conditions is called chaos. A
similar feature has been found in our atmosphere—this property makes the long-term prediction of
weather impossible. Chaos theory and related fields are active areas of physics research.
Presently, it appears that we know all the basic equations needed to describe the physical
universe. For example, physicists are quite confident that quantum chromodynamics describes
the physics of the nucleus; Navier–Stokes equation describes turbulent flows; certain quantum
mechanical equations describe the physics of superconductors and other complex materials. The
major difficulty in physics, however, is how to solve these equations. Physicists have been trying
very hard to solve them using analytical methods and powerful computers, yet complete success has
eluded them so far. Maybe, we need to change the whole paradigm! Only the future can tell!
Exercises
1. Galileo rolled various balls through a groove in an inclined plane. He found that the time
taken by balls of different materials and different radii were approximately the same. Are
these results consistent with Newton’s laws of motion? Would a hollow ball and a solid ball
of same radii fall with the same acceleration?
2. Describe the motion of Saturn in the sky assuming the orbits of the Earth and Saturn around
the Sun to be circular. The time periods of Earth’s and Saturn’s orbit are 365.25 and 10756.2
terrestrial days respectively. The distances of the Earth and Saturn from the Sun are 1.5×1011
and 14.3 × 1011 metres respectively. Do retrograde motion take place for Saturn? If yes, for
how long, and after what interval?
3. Analyse the retrograde motion for other solar planets. The data can be obtained from
Appendix G.
4. When will the next retrograde motion take place for Mars and Saturn? Current coordinates
of the planets can be obtained from the Internet.
5. Historically, how were the following measurements made? Suggest improvements using modern
gadgets.
(a)
(b)
(c)
(d)
(e)
(f)
Radius of the Earth
Distance of the Moon from the Earth
Distance of the Sun from the Earth
Distance of planets from the Sun
Mass of the Earth
Mass of the Sun
Projects
1. Repeat Galileo’s experiment to show that all objects fall with the same acceleration on the
surface of the Earth (Exercise 1).
2. Repeat the above experiment to compute the acceleration due to gravity.
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3.
4.
5.
6.
Construct a Sun dial.
Locate the positions of planets in the sky and observe their motion.
Observe the motion of stars, solar planets and the Moon.
How did Kepler compute the relative distance between Mars and Earth?
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