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4 Newton and Beyond
However, the gift of augury is not necessarily accorded even to the great, and as
Science Fiction writer Arthur C. Clarke aptly remarked in the citation that opens
this chapter, many a would-be scientiﬁc prophet has lived to regret rash statements.
Shortly after Kelvin’s unfortunate prophecy, the launch of the 20th Century
heralded the development of the theories of Relativity and Quantum Mechanics,
both of which turned classical physics on its head and led to what is now termed
“Modern Physics.” However these ideas did not emerge, fully formed, like the
goddess Aphrodite rising from the sea. Rather, they developed from the solid
foundations laid by the scientiﬁc greats of the past, and it is the purpose of the next
three chapters to explore these past achievements. In later chapters we will discover
how drastic modiﬁcations were forced on these nineteenth century theories by
developments in the so-called Golden Age of Physics, which took place in the ﬁrst
half of the twentieth century.
4.2
Newton Stands on the Shoulders of Giants
“If I have seen further, it is by standing on the shoulders of giants”. So wrote Isaac
Newton in a letter [2] to his arch-rival, Robert Hooke in 1676. He was most likely
paraphrasing Bernard of Chartres, a 12th century scholar, who compared us (the
moderns) with dwarves riding on the shoulders of giants (the Ancients), and suggested that it is for this reason alone that we are able to see further than our
antecedents. The expression served also as a sarcastic rejection by Newton of the
notion that he had exploited Hooke’s work, for Hooke was a sickly man, and
certainly not a giant. To set the scene for what is to come in this chapter, a quick
review of the work of these “giants” is appropriate at this point.
In 1638, Galileo published his ﬁnal book, Discourses and Mathematical
Demonstrations Relating to Two New Sciences. In it he describes the motion of
falling bodies, projectiles, the concept of the relative motion of two bodies, and
some fundamental ideas about the pitch of vibrating strings, friction and inﬁnity.
Following after Galileo, the end of the 17th Century saw a blossoming of studies
into the motion of objects, both earthbound and celestial. Christiaan Huygens, John
Wallis and Gottfried Leibniz introduced concepts such as the Conservation of
Momentum and the Conservation of Energy, which were discussed in Chap. 2.
Earlier in the century, between 1609 and 1619, based on the meticulous
observations of Tycho Brahe in Prague, Johannes Kepler deduced his three laws of
planetary motion.1 Kepler’s laws are speciﬁc and apply to a very narrow ﬁeld, but
they marked a turning point in the application of science to a celestial problem, and
a victory for the heliocentric model of the solar system propounded by Copernicus.
These laws may be stated as follows:
1
A recent historical novel, Kepler, by Mann-Booker prize winning novelist, John Banville,
describes the life of Kepler and sets the atmosphere of the period in which his work took place.
4.2 Newton Stands on the Shoulders of Giants
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Law 1 the orbit of every planet is an ellipse with the Sun at one of the two foci;
Law 2 a line joining a planet and the Sun sweeps out equal areas during equal
intervals of time;
Law 3 the square of the orbital period of a planet is proportional to the cube of the
semi-major axis of its orbit.2
Following on from the discoveries of his predecessors, Newton began a series of
investigations that culminated in 1687 with the publication of his masterpiece:
Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural
Philosophy). In this book, a pinnacle of rational scientiﬁc thought, Newton introduced his three laws of motion and the principle of gravitation, which explain
celestial motion. They are still today the starting point for most courses in physics,
and essential knowledge for anybody desirous of even a cursory understanding of
the physical world about them. Although we have discussed some of these principles in Chap. 2, for clarity in what follows we reiterate them here.
The ﬁrst law is usually written: every body continues in its state of rest, or of
uniform motion in a straight line, unless it is acted upon by an external force. With
this law, Newton is clarifying the work of Galileo and laying to rest forever
Aristotle’s idea that objects come to rest when the driving force is removed from
them.
The second law gives us a quantitative description of what happens when an
external force is indeed applied to any object: the object is then accelerated along
the line of the applied force and the magnitude of the acceleration is inversely
proportional to the mass of the object. As we have seen in Chap. 2, this law is
expressed as F = ma, where F is the applied force, m is the mass of the object and
a is the acceleration of the object. F and a are vectors. The ﬁrst law is actually a
special case of the second law. If we choose F = 0 in the second law we see that the
acceleration of the object is zero, and the object’s state of motion is therefore
unchanged. If at rest, it stays at rest; if in motion it continues along with the same
velocity.
The ﬁrst and second laws accord with our everyday experience in that it is hard
to set heavy objects moving (inertia), but once they are moving it is hard to stop
them (momentum). Aristotle’s principle might be expressed as F = mv, which
would indicate that if we remove the external force, the object stops immediately.
Newton realised that the bringing to rest of a freely moving object was the result of
the external forces of friction and air resistance, not the removal of the driving force.
If we could place a moving object into an environment where friction and air
resistance are removed, the object would continue its motion forever. The recent
2
The orbital period is the time taken for a planet to complete one orbit of the sun; the major axis of
an ellipse is the longest diameter which is a straight line running from a point on the ellipse
through the centre and both foci to a point on the opposite side of the ellipse; a semi-major axis is
half of the major axis.
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4 Newton and Beyond
exit of the space probe Voyager 1 from the solar system thirty-ﬁve years after its
launch from Earth is a modern demonstration of the truth of Newton’s ideas.
Newton’s third law states that for every action, there is an equal and opposite
reaction. It is somewhat more subtle than the ﬁrst two laws. In one sense it is a
clariﬁcation of the requirement in Law 1 that the driving force is external to the
body being propelled. You can stand in a sailboat and push as hard as you like on
the mast but you will not propel the boat one millimetre forwards because your feet
exert a rearward force on the bottom of the boat equal and opposite to the force
exerted by your push on the mast. You would be better off turning around and
blowing over the stern, in which case there would be a “reaction”, if ever so slight,
to the “action” of the force driving your breath. This is the principle behind the
rocket ship and the jet engine.3
The third law also explains the “kick” that occurs when a ﬁrearm is discharged.
In Chap. 2 we introduced the concept of conservation laws, which play an
important role in physics. It can be readily shown that the Laws of Conservation of
Momentum and Angular Momentum are a consequence of Newton’s third law, and
the Law of Conservation of Energy arises from Newton’s second law.
However, it turns out, as we will see in later chapters, that these conservation
laws are more fundamental than Newton’s laws and apply in domains such as
Atomic Physics where Newton’s laws are inapplicable.
Even in the everyday world there are instances where Newton’s laws are
inapplicable. An experimenter sitting on a carousel would notice that there are
effects due to the rotation which complicate the application of Newton’s laws, and
forces such as centrifugal force and the Coriolis force arise. When we are standing
outside the rotating carousel, we may dismiss these forces as “ﬁctitious” or
“pseudoforces” due to the rotary motion. However, we spend our life on a rotating
body (the earth) and the effects of centrifugal force (e.g. the equatorial bulging and
flattening of the earth at the poles) and the Coriolis force (hurricanes) are anything
but ﬁctitious. It is customary to add the caveat that Newton’s laws only apply in an
inertial frame of reference, but if a physicist is asked to deﬁne an inertial frame of
reference, the usual reply is that it is a frame of reference in which Newton’s laws
apply. One can think of a non-inertial frame of reference as being one that is
undergoing some form of acceleration.
Another attribute of Newton’s laws is that they are time-reversible. If one were
to view a movie running backwards in time, the same laws of physics would apply
as in the real world. It might look strange to see an object rising from the floor and
floating up onto a table. However, if you were to work through the equations of
motion with the ﬁnal velocity of the object just before impact reversed, you would
3
It is a popular misconception that the air blown out by a jet engine meets a resistance from the
atmospheric air behind the engine, and that as a consequence the engine is thrust forward in the
same manner that skaters pushing on a wall can thrust themselves away from it. This is incorrect.
A rocket ship works perfectly well in deep space where there is no atmosphere. The forward thrust
is a consequence of Newton’s third law.
4.2 Newton Stands on the Shoulders of Giants
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describe the events in the backwards-running ﬁlm accurately. We will see later that
time-reversibility is a requirement of physical laws for individual bodies.
4.3
Newton’s Law of Gravity
After expressing his three laws of motion, Newton proceeded in Principia to lay out
his theory of gravitation. Everybody is aware of the apocryphal story that the idea
of gravity came to Newton after he observed an apple fall from a tree. However,
Robert Hooke had a different opinion of the law’s origin and when Newton propounded his theory to the Royal Society in 1686, Hooke immediately claimed that
Newton had plagiarised the law from him.
Putting aside a controversy which still continues to this day, the law of gravity
can be stated mathematically (as we saw in Chap. 2) with the force F being such
that its modulus F is given by:
F¼G
m1 m2
r2
where m1 and m2 are the two masses, and r is their separation. The force F is
exerted on each mass, and its direction is towards the other mass,4 i.e. it is an
attractive force, tending to move the two masses towards each other. G is a constant
that is known as the Universal Gravitational Constant (see Chap. 2) and is one
of the most important constants of the universe. G will be discussed further in
Chaps. 7 and 13.
The law of gravity, combined with the three laws of motion, explained most of
the astronomy known at the time. Kepler’s laws can be deduced from Newton’s.5
The orbits of planets and their masses could be, and were, calculated. A web of
gravitational pulls between stars, planets, comets and other celestial objects holds
the universe together, and it is the same force that holds our feet on the ground.
The incredible scope of Newton’s laws is apparent, and is in stark contrast with
the limited nature of Kepler’s three laws. Mechanics, as we know it today, came
into existence with the publication of the Principia.
Although the importance of Newton’s Principia was quickly recognised, the
book was not without its critics. In England it was argued that Newton had removed
any place for God in the universe, and was promoting atheism. (It should be
remembered that Newton himself was devoutly religious and in later life gave up
4
This is an example of the third law.
As we discussed in Chap. 2, the aim of physics is to explain nature with a minimum of “laws”, in
the same way that mathematics is constructed from a minimum of axioms. Kepler’s laws can be
derived from Newton’s and are therefore not considered fundamental. In practice, they may still be
used if they are easier to apply in a particular example.
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4 Newton and Beyond
physics for theology). In France and Germany, Descartes and Leibniz objected that
Newton hadn’t explained the nature of gravity, which appeared as almost supernatural in his work.
In a second edition of the Principia published in 1713, more than a quarter of a
century after the original, Newton addressed some of these criticisms. He admitted
that he did not understand the nature of gravity, or how it exerted a pull on objects
across empty space. His reply was similar to that of Samuel Beckett who, when
asked to explain who Godot was in his absurdist classic, “Waiting for Godot”6 is
alleged to have replied: “if I knew the answer to that question I would have put it in
the play.” In a new addendum, Newton rejected the idea that the Principia promoted atheism, maintaining that the order that his laws found in the universe was a
sign of the presence of a creator, who was beyond human understanding.
The three hundred years that have elapsed since the Principia ﬁrst appeared have
bred a familiarity with the concept of gravity that has dulled our amazement at the
idea that a force can act at a distance through empty space. We now take gravity for
granted. As we will see in Chap. 7, Einstein proposed a different view of gravity
which sees its effects arising through the distortion of space-time by the presence of
a mass. Whether this concept is easier to grasp than the idea of action at a distance,
which so troubled Descartes and Leibniz, is a matter for the beholder.
In the Principia, Newton proposes his laws and deduces their consequences
using precise mathematical logic. He ﬁnds that these consequences agree with the
results of observations, in some cases providing an explanation for hitherto unexplained effects (e.g. tides and the orbits of comets). He does not speculate on the
nature of gravity. In adopting this approach he has taken the baton from Galileo in
laying down the methodology of the science of physics. Physics answers questions
beginning with how, when and where. Questions beginning with why are best left
to philosophers.
In the following chapters we will be introduced to many concepts which are
counter-intuitive and puzzling, and yet meet the test of experimental veriﬁcation,
and the challenge of Occam’s Razor. One wonders whether in three hundred years’
time, humankind will be as blasé to these 20th and 21st Century ideas as it is today
to gravity.
4.4
Let There Be Light
Another branch of physics in which Newton made a ground-breaking contribution
is the science of optics (or the study of the properties of light). The importance of
light to human vision has ensured that its nature has remained a subject of speculation and investigation throughout recorded history. The earliest Egyptian and
“Waiting for Godot” is a play about two tramps waiting for a third character, Godot, who never
shows up.
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