The Book of Physics: Volume 1 E-Book

Simone Malacrida

• Herausgeber: BookRix
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2023

Beschreibung

In this book, the great history of physics discoveries is traced, starting from the scientific revolution of Galileo and Newton to the physics of today and the near future. The understanding of physics is approached both from a theoretical point of view, expounding the definitions of each particular field and the assumptions underlying each theory, and on a practical level, going on to solve more than 350 exercises related to physics problems of all sorts. The approach to physics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science. The entire book is divided into five distinct sections: classical physics, the scientific revolutions that took place in the early twentieth century, physics of the microcosm, physics of the macrocosm, and finally current problems that are the starting point for the physics of the future. The paper stands as an all-encompassing work concerning physics, leaving out no aspect of the many facets it can take on.

Leseprobe

Simone Malacrida

The Book of Physics: Volume 1

BookRix GmbH & Co. KG81371 Munich

“The Book of Physics: Volume 1”

“The Book of Physics: Volume 1”

SIMONE MALACRIDA

In this book, the great history of physics discoveries is traced, starting from the scientific revolution of Galileo and Newton to the physics of today and the near future.

The understanding of physics is approached both from a theoretical point of view, expounding the definitions of each particular field and the assumptions underlying each theory, and on a practical level, going on to solve more than 350 exercises related to physics problems of all sorts.

The approach to physics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science.

The entire book is divided into five distinct sections: classical physics, the scientific revolutions that took place in the early twentieth century, physics of the microcosm, physics of the macrocosm, and finally current problems that are the starting point for the physics of the future.

The paper stands as an all-encompassing work concerning physics, leaving out no aspect of the many facets it can take on.

ANALYTICAL INDEX

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INTRODUCTION

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PART ONE: CLASSICAL PHYSICS

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1 – THE SCIENTIFIC METHOD

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2 – MEASURING SYSTEMS

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3 – CLASSICAL MECHANICS: KINEMATICS

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4 – CLASSICAL MECHANICS: DYNAMICS AND STATICS

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5 – CLASSICAL MECHANICS: THEORY OF GRAVITATION

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6 – FLUID THEORY AND FLUID DYNAMICS

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7 - OPTICS

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8 – WAVES AND OSCILLATORY PHENOMENA

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9 – THERMODYNAMICS AND HEAT TRANSMISSION

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10 – STATISTICAL PHYSICS

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11 - ELECTROMAGNETISM

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12 – CRISIS OF CLASSICAL PHYSICS

INTRODUCTION

INTRODUCTION

This book was born from the need to reconcile, in a single text, all the physical theories studied to date, complete with their theoretical and experimental framework.

There is no doubt that physics, as we understand it today, originated from the introduction of the scientific method, first on a philosophical level, then on an experimental and practical level.

When the scientific method entered the practice of reasoning on which to base assumptions and deductions, there was an enormous leap in quality compared to all previous knowledge.

We can say that all the discoveries and applications that took place in the past with respect to that event are actually the result of semi-empirical approaches and not exactly of science as we understand it today.

That point of no return was such as to determine a historical watershed, in the same way as we are used to considering events of the caliber of the French Revolution, the fall of the Roman Empire or the discovery of America.

Since that time, scientific investigation has had an impressive acceleration, ranging in every field of knowledge and has impressed on society, in terms of applications and daily consequences, a decidedly different imprint than in the past, coming to create those conditions and those prerequisites necessary for the Industrial Revolution, which occurred only less than two centuries after those first scientific stirrings.

A first caesura of this path occurs with the end of the nineteenth century and with the acknowledgment that, in the range of knowledge in all sectors, such contradictions had been reached that the previous theoretical schemes had to be completely revised.

From that period, historically known as the crisis of classical physics, came the two revolutionary theories of the twentieth century which are the basis of contemporary physics, the one we use today to describe Nature and what surrounds us.

In this period of time, which lasted a good two centuries, physics has managed to scientifically explore various disciplines such as mechanics in all its forms (static, dynamic and kinematic), astronomy, the theory of gravitation, optics, the phenomena and oscillatory ones, fluid dynamics, thermodynamics, heat transmission, statistics applied to physics, electric and magnetic phenomena.

As can be seen from this small list, the elaboration of theories that predict and explain the experimental results has been so pervasive as to have left nothing unexplored, with the limitations that the equipment of the time could have (it is obvious that it was completely beyond place to think of probing the characteristics of the atom and of the atomic nucleus, not having at one's disposal the suitable material means to detect the essential experimental data).

What has just been described is dealt with in the first part of this book which coincides with the treatment of classical physics.

The second part of the book is inspired by the great revolutions of the early twentieth century, namely quantum physics and special relativity.

They have played such an extraordinary role in the development of physics that it was decided to devote an entire part to them.

The third part of the book deals with the physics of the microcosm, i.e. the physics that develops on a molecular, atomic, nuclear and fundamental particle scale.

We will see how far scientific investigation has gone and what the problems of these developments are today.

The fourth part instead, as a counterpart, deals with the physics of the macrocosm and has the theory of general relativity as its founding stone.

It is everything related to astronomy, astrophysics and cosmology.

Also in this case the recent results of these theories will be tangible.

The fifth and last part has the most difficult task compared to the others.

In fact, if on the one hand the theory of relativity has generated speculations on the macrocosm and quantum physics those on the microcosm, there is numerous evidence of their possible (and desirable) meeting in a single theory.

The last part of the book deals with this peculiar aspect.

The book is divided into chapters, each of which can very well be treated independently of the previous and subsequent ones (in fact, in the literature there are numerous writings relating precisely to each of the chapters exposed).

However, there is a logical correlation in the order of the chapters, a sort of progressive knowledge towards what was previously unknown.

The attentive reader will realize this and will be able to follow this leitmotif which is none other than the re-proposition of the history of physics.

A note should be made about the execution of the exercises.

It is true that in the first part, the one dedicated to classical physics, exercises carried out at the high school level are presented (precisely because in high school one begins to study those specific sectors of physics), but it is equally true that the theoretical formalism is, almost early on, focused on college-level mathematics that assumes knowledge of advanced mathematical analysis, advanced geometry, and other mathematical disciplines.

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What is the point of studying physics?

Let's try to give a brief explanation (entirely personal, of course).

We cannot hide the fact that the interpretation of physical laws, if pushed to the maximum level, can only lead to speculative questions typical of philosophy, especially when dealing with the infinitely large (as in the case of cosmology) or the infinitely small (as in particle physics).

Physical laws, precisely because they have the peculiarity of explaining nature, the universe and everything that surrounds us, must not only be in agreement with the experimental data, but constitute a theoretical model for the simulation of reality itself.

Their structure and interpretation therefore influence the way of describing reality, as already happened with the advent of relativism and indeterminism at the beginning of the twentieth century.

The physical laws are written with a symbolism that is mathematical. The great "strength" of mathematics lies in at least three distinct points.

First of all, thanks to it it is possible to describe reality in scientific terms, that is by foreseeing some results even before having the real experience.

Predicting results also means predicting the uncertainties, errors and statistics that necessarily arise when the ideal of theory is brought into the most extreme practice.

Second, mathematics is a language that has unique properties.

It is artificial, as built by human beings.

There are other artificial languages, such as the Morse alphabet; but the great difference of mathematics is that it is an artificial language that describes nature and its physical, chemical and biological properties.

This makes it superior to any other possible language, as we speak the same language as the Universe and its laws.

At this juncture, each of us can bring in our own ideologies or beliefs, whether secular or religious.

Many thinkers have highlighted how God is a great mathematician and how mathematics is the preferred language to communicate with this superior entity.

The last property of mathematics is that it is a universal language.

In mathematical terms, the Tower of Babel could not exist.

Every human being who has some rudiments of mathematics knows very well what is meant by some specific symbols, while translators and dictionaries are needed to understand each other with written words or oral speeches.

We know very well that language is the basis of all knowledge.

The human being learns, in the first years of life, a series of basic information for the development of intelligence, precisely through language.

The human brain is distinguished precisely by this specific peculiarity of articulating a series of complex languages and this has given us all the well-known advantages over any other species of the animal kingdom.

Language is also one of the presuppositions of philosophical, speculative and scientific knowledge and Gadamer has highlighted this, unequivocally and definitively.

But there is a third property of mathematics which is far more important.

In addition to being an artificial and universal language that describes nature, mathematics is properly problem solving , therefore it is concreteness made science, as man has always aimed at solving problems that grip him, just take a look at what has been discussed in this paper about the overcoming of physical theories.

The texture of reality is therefore marked by physical laws that underlie mathematical equations and which, over time, tend to generalize more and more on the wave of new discoveries and inconsistencies of old theories.

Today we are faced with one of those fundamental steps.

On the one hand we know that there are problems of congruence of the two main theories (general relativity and quantum field theory), on the other we have not yet defined a new theoretical canvas that overcomes these obscure points towards a wider knowledge.

As always, it is a constant challenge and, in some way, eternally inherent in human nature.

This characteristic is part of an eternal race towards a better description of what surrounds us and a better understanding of all existing phenomena, in the wake of a derivation from the myth of Ulysses, which embodies man's eternal propensity towards knowledge.

PARTE ONE: CLASSICAL PHYSICS

PARTE ONE: CLASSICAL PHYSICS

1

THE SCIENTIFIC METHOD

THE SCIENTIFIC METHOD

Introduction

The beginning of modern physics coincides with the formulation and application of the scientific method, which took place in a systematic way in the early seventeenth century above all by Galileo and with decisive contributions by the philosophers Bacon and Descartes.

This logical and philosophical structure became the basis for the construction of scientific knowledge in the following centuries and for the first mathematical approach through the introduction of analysis by Newton and Leibnitz in the second half of the seventeenth century.

Before Galileo, knowledge had progressed above all through empirical attempts or purely metaphysical reasoning, relying on logical constructs such as the syllogism or the principle of authority. There were therefore no scientists as we understand them today and the closest thing to our concept of science was given by the scholars of natural philosophy.

A forerunner of the scientific method was Leonardo da Vinci who, about a century before Galileo, understood the fundamental importance of real experimentation and mathematical demonstration, without however arriving at the definition of a system and a method.

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The vision of Galileo Galilei

Galileo started from some fundamental assumptions, which are still valid today, among which:

1) Nature responds to mathematical criteria

2) To establish the laws of physics it is necessary to carry out experiments

3) Logical hypotheses and mathematical theories must be in agreement with experiments

Therefore Galileo abandoned the empty search for the primary essences and qualities that had characterized knowledge so much before the seventeenth century and set quantitative facts, measurable and verifiable through experiments and expressible through the language of mathematics, as the cornerstone of science.

One of the key points is given by the reproducibility of the experiments: under suitable conditions and hypotheses to be prepared, a certain experience must be able to be repeated in every place giving the same results and therefore confirming (or denying) the mathematical theory formulated to explain this experiment.

In particular cases where it is not possible to carry out a real experiment, Galileo introduces the concept of thought experiment.

By applying the same mathematical and quantitative criteria in the formulation of the hypotheses, the thought experiment has the same validity as the one actually performed. In this way Galileo understood how the Copernican revolution of heliocentrism (the Sun placed at the center of the Solar System and not Earth as medieval claimed instead Scolastica referring to the authority of Aristotle) was correct and how Kepler's laws were correct at an astronomical level.

The scientific method is therefore the way in which science increases the knowledge of Nature and the Universe.

The characteristics of such knowledge are that of being objective, reliable and verifiable.

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Inductive method

The scientific method consists of two large macro-sectors.

On the one hand we have the collection of empirical evidence through experiments that must be brought back to a common theoretical logic, on the other we have the hypotheses and theories that must be in agreement with the experimental reality.

This dualism somehow reflects the ancient division of logical reasoning between the inductive method and the deductive method. While Galileo made particular use of the second, Bacon and Newton were frequent users of the first.

Let us briefly see the characteristics of these two different approaches to science and the scientific method and their implications in physical and philosophical terms.

The inductive method was the real driving force of modern physics and only went into crisis many centuries later, when it was clear that the theories formulated were in clear conflict with each other and with the experimental data.

The twentieth century led to a great transformation not only in the theories elaborated, but also in the approach to science, in the philosophical and logical explanation as well as in the method used.

The inductive method starts from empirical observation and ends in the formalization of a theory, carrying out a series of intermediate steps.

Observation identifies the characteristics of the physical phenomenon and measures them with reproducible methods while the subsequent experiment programmed by the observer allows these characteristics to be detected.

After that it is necessary to prepare an analysis of the correlation between the measurements, manipulating the experimental data in order to extract from them the greatest possible content of information.

This correlation is the first step towards the definition of a physical model which must be an abstraction of the real functioning given by the empirical results.

It must be said that the same experiments can lead to different physical models and the goodness of one model, compared to another, is given by the degree of precision with which the experimental data are explained.

The physical model is, in turn, formalized following a mathematical approach for the definition of a mathematical model which contains a series of equations, whose solutions must coincide with the experimental data.

At the end of the cognitive cycle of the inductive method there is the formulation of the theory which, based on the mathematical model, generalizes the physical model and explains the correlation between the measurements and the experimental data.

By applying the inductive method, new knowledge is generated both by abstracting from the particular to the universal as now done, and by subjecting the theory to experimental verification and overcoming it, with the same scheme, if an observation identifies characteristics not in agreement with what is predicted by the theory itself.

This mental scheme was the one applied by Bacon and Newton and which enjoyed considerable success for centuries.

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Deductive method

On the other hand Galileo showed himself closer to the deductive method, also called experimental.

The basic idea of the deductive method is that the theory is built at the beginning and not at the end of the cognitive process, as instead happens in the inductive method.

The deductive method starts from the construction of a mathematical theory which determines a physical model from which hypotheses can be formulated; such hypotheses must predict something experimentally measurable.

By carrying out an appropriate experiment, it is observed whether the event foreseen by the theory, and therefore by the hypothesis, occurs or not.

There are two ways of interpreting the verification between experimental observation and theoretical prediction.

For many centuries, it was agreed that the necessary criterion was that of verifiability.

With this criterion it was deduced that, if the coincidence between the prediction of the theory and the experimental reality does not occur, the theory is denied and therefore a new theoretical approach must be formulated. If, on the other hand, the theory agrees with the experimental data, it is correct.

This was the approach given by Galileo himself.

The second way consists instead in the so-called falsificationism, i.e. starting from the assumption that a theory can never be confirmed, but only refuted.

If there is a coincidence between the theoretical forecast and the experimental data, it can simply be concluded that the theory has not been denied and can be accepted on a provisional basis.

This approach derives mainly from Popper's studies during the 20th century.

The deductive method had great impetus after the logical criticisms carried out by Russell in the early twentieth century against the inductive method.

In the meantime, theories and ways of thinking closer to the deductive method developed, such as the studies on relativity carried out by Einstein, and concepts such as probabilism and indeterminism were introduced which sanctioned the definitive decline of induction.

Finally, the enunciation of Godel's incompleteness theorems gave the final blow to that logical scheme, leaving the only way of deduction open.

Popper's studies then made sure that falsificationism was taken as an assumption of today's science.

In particular, Russell posed as a capital point the logical inconsistency of induction which, on the basis of individual cases, abstracted a universal law.

Many contemporary studies tend to confirm this thesis, above all after the evident intrinsic incompleteness of every theory or logical scheme (demonstrated by Godel in the 1920s).

In fact, to make the inductive system valid, it would take an infinite number of empirical cases to confirm it, which would not generate any new knowledge.

Conversely, based only on a limited number of experimental cases, every inductive theory is, in reality, only a conjecture.

As proof of Popper's falsificationism, it must be said that the function of experiments is a refutation, as already observed by Einstein regarding physical theories and the connection with the deductive and experimental method.

Every physical theory can be called scientific if and only if it is expressed in a form that can be criticized and falsified in objective terms. From this point of view, Popper criticized many pseudo-scientific theories such as historicism, psychology, materialism and metaphysics, dismantling, among other things, studies by eminent philosophers such as Marx, Freud, Hegel and Kant.

Applications in physics

Going back to the origins of the scientific method and to Galileo, the first applications of this criterion were stated in 1638, in the scientific treatise "Mathematical speeches and demonstrations around two new sciences pertaining to mechanics and local motions ".

This treatise was the dawn of modern physics and that date can be taken as a dividing line between a pre-scientific era and a scientific era.

In that treatise, Galileo generalized the experiments and theories studied in previous years regarding motion on an inclined plane and falling bodies, arriving to correctly describe the laws of statics, leverage and dynamics, especially of motion naturally accelerated, of the uniformly accelerated one and of the oscillatory motion of the pendulum.

Furthermore, Galileo conceived the existence of the vacuum as a state in which there was no resistance of materials and in which motion was possible, arriving correctly to conclude that bodies, having different masses and shapes, fall with equal speed in the vacuum, as opposed to to all the theories of the time.

Always with this approach, Galileo overturned Aristotle's point of view on the principle of inertia through an ideal experiment, i.e. imagining the limiting case of a body moving on a horizontal plane without friction.

In this case, for Galileo, the body remains in its state of motion without any term of space and time, simply for a principle of conservation of energy.

All this knowledge formed the necessary background for the formulation of the laws of Newtonian mechanics in the second half of the seventeenth century, even if there was a need for a new mathematical formulation, that of mathematical analysis, not yet ready in Galileo's time.

Three other aspects of Galileo's scientific method were important for the continuation of modern physics.

The first aspect concerns the astronomical discoveries deriving from the acceptance of the theories of Copernicus and Kepler. Galileo was the first to build a telescope and to scientifically probe celestial objects, such as planets and satellites.

The second aspect is the concept of infinity and its measurement, which will be very useful in mathematical analysis.

The last question concerns the so-called Galilean principle of relativity.

Galileo was the first to scientifically ask himself the question of the validity of physical laws, especially of mechanics, and of the role of different observers in different reference systems.

Galileo started from the hypothesis that the laws of mechanics are always the same for inertial reference systems, ie reference systems that satisfy the principle of inertia. Simply put, such frames of reference are not accelerated.

These reference systems can be expressed through the formalism of the Cartesian axes in three dimensions (with Cartesian coordinates) and by adopting the rules of Euclidean geometry.

The observer present in the reference system is integral with the reference system, therefore it does not have its own motion, but only that of the system.

The first point that Galileo highlighted is that of the simultaneity of the experiment.

Two observers placed in different inertial frames of reference must perform the same experiment at the same instant in order to have an identical result. Therefore they will have to exchange information to synchronize this experiment. Galileo tried to measure the speed of light and deduced that it was so high compared to daily practice, as to make the time necessary for the exchange of information irrelevant.

The first conclusion of Galilean relativity was that time remained the same in the passage from one inertial system to another.

Since the two reference systems have different speeds, Galileo posed the problem of how to carry out a transformation of the speeds, passing from one system to another.

By applying Euclidean geometry together with Cartesian coordinates, he vectorically composed the velocities according to the well-known law of the parallelogram. This law, already known by Leonardo, now found an explanation in the Galilean theory of relativity.

Ultimately, given two inertial systems, the passage of space-time coordinates from one system to another according to Galilean relativity is given by:

Where v is the relative speed between the two systems, composed according to the parallelogram rule.

With these scientific assumptions and with the method developed by Galileo, there were the real foundations for starting the path of modern physics, starting right from the mechanical concepts.

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MEASURING SYSTEMS

MEASURING SYSTEMS

International system: fundamental units

The International System of Measurement (known as SI or MKS System) is a measurement system that is based on the metric system and introduces seven fundamental units for physics.

1) For lengths, the metre, symbol m, is defined as the distance traveled by light in a vacuum in the time of 1/299'792'458 seconds.

2) For the masses, the kilogram, symbol Kg, is defined as the mass of the internationally recognized prototype.

3) For the time, the second, symbol s, is defined as the duration of 9'192'631'770 periods of the radiation corresponding to the transition between two hyperfine levels (from F=4 to F=3 for MF=0) of the state fundamental element of the cesium-133 atom.

4) For the temperature, the Kelvin, symbol K, is defined as 1/273.16 of the thermodynamic temperature of the triple point of water.

5) For the intensity of electric current, the Ampere, symbol A, is defined as the electric current flowing between two linear and parallel conductors, placed in a vacuum at a distance of one meter and producing a force equal to 0.0000002 newton per meter in length.

6) For the amount of substance, the mole, symbol mol, is defined as the amount of substance of a system that contains a number of entities equal to the number of atoms present in 12 grammicarbon-12.

7) For luminous intensity, the candela, symbol cd, is defined as the intensity of a source that emits monochromatic radiation at a frequency of 540 THz with an intensity equal to 1/683 watt per steradian.

International system: derived units

All the others can be derived from the seven fundamental units.

We mention only the main ones:

Frequency

hertz

Hz

Power

newtons

No

Pressure

pascal

Pa

Power

joule

j

Power

watt

w

Electric charge

coulomb

c

Electric potential

volt

v

Electric capacity

farad

f

Electrical resistance

ohm

Ω

Electric conductance

siemens

St

Magnetic flux

weber

Wb

Magnetic flux density

tesla

T

Inductance

henry

h

luminous flux

lumens

lm

Lighting

lux

lx

Radioactive activity

becquerel

Bq

Radioactive dose absorbed

gray

Gy

Equivalent radioactive dose

sievert

St

Recall that in physics, plane angles are always measured in radians and solid angles in steradians.

Also, exponential notations and the use of commas for significant figures are valid.

Prefixes

Being a metric system, the following prefixes apply:

you decide

d

centi

c

milli

m

micro

dwarf

no

pic

p

femto

f

deed

to

zepto

z

yocto

y

deca

from

hecto

h

kilo

k

mega

m

gig

g

tera

T

peta

P

exa

AND

zetta

Z

yotta

Y

Recall that in computer science there are also base 2 prefixes.

Other commonly used units

Despite the attempt at standardization implemented by the International System, there are some units that lend themselves more to being used, both for common use and for scientific convenience.

We can make the following equivalences:

TIME

Since time is measured in seconds, 1 minute corresponds to 60 seconds, 1 hour to 60 minutes and 1 day to 24 hours.

SPACE

Since lengths are measured in metres, an angstrom equals one-tenth of a nanometer and a nautical mile equals 1' 852 metri.

Since areas are measured in square meters, a hectare equals 10' 000 metri quadratiand a barn equals 100 square femtometres.

Given that volumes are measured in cubic metres, a liter is equivalent to a cubic decimetre.

MASS

Since masses are measured in kilograms, a quintal equals 100 Kgand a ton equals 1' 000 Kg.

SPEED'

Since speeds are measured in meters per second, one knot equals the speed of one nautical mile per hour.

PRESSURES

Since pressures are measured in Pascals, one bar equals 100,000 Pa, one millibar equals 100 Pa, and one millimeter of mercury (mmHg) equals 133.322 Pa.

POWER

Since energies are measured in Joules, one calorie equals 4.1868 J and one kilocalorie equals 4168.8 J.

The following units of measurement are also considered valid:

electronvolt (symbol eV): energy equal to

atomic mass unit (symbol u): mass equal to

astronomical unit (ua): length equal to

In astronomy, the light year equal to 63'284 au and the parsec equal to 206'625 au are also considered.

CGS system

The CGS system is a derivation of the international system, in which the basic units of kilogram and meter are replaced by those of gram and centimeter.

Other units of measurement of this system that find applications in physics are:

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erg: defined as that energy equal to

dyne: defined as that force equal to

poise: defined as that viscosity equal to 0.1Pa*s

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British imperial system

It is a system that is not based on the metric system.

We have:

LENGTH:

One inch equals 2,54 cm.

A thousandth of an inch is the mil.

One fortieth of an inch is the line.

One hand equals 4 pollici.

A span equals 9 pollici.

One foot equals 12 pollici.

One elbow equals 18 pollici.

One yard equals 3 piedi.

One arm equals 2 yards.

One pole equals 5.5 yards.

One chain equals 11 fathoms.

One stage equals 10 chains.

One terrestrial mile equals 8 stadia or 1609,344 metri.

MASS:

One ounce equals 28,35 grammi.

An eighth of an ounce is called a dram.

A pound is 16 once.

For areas and volumes, these quantities are used squared and cubed.

We emphasize that one acre equals 0,4046 ettari.

For volumes for liquids, the following equivalences apply.

One fluid ounce equals 28.4 ml.

A pint equals 20 onceliquids.

One gallon equals eight pints.

TEMPERATURES:

The Fahrenheit scale is used.

To switch from Fahrenheit scale to Celsius scale, the following rule is used:

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US customary system

The only differences with respect to the British imperial system consist in the use of the Rankine scale for temperatures with the following equivalence:

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And a different classification of volumes for liquids:

One fluid ounce equals 29.57 ml.

A pint equals 16 onceliquids.

One gallon equals eight pints.

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