Physics
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Title: Physics
Author: Willis E. Tower
Thomas D. Cope
Charles H. Smith
Charles M. Turton
Release date: July 9, 2012 [eBook #40175]
Most recently updated: October 23, 2024
Language: English
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*** START OF THE PROJECT GUTENBERG EBOOK PHYSICS ***
PHYSICS
TOWER, SMITH, TURTON,
AND
COPE
[Illustration: (_See p.441_)
THREE-COLOR PRINTING
_Y._ Yellow impression; negative made through a blue-violet filter. _R._
Crimson impression; negative made through a green filter. _RY._ Crimson
on yellow. _B._ Blue impression; negative made through a red filter.
_YRB._ Yellow, crimson, and blue combined; the final product. (Courtesy
of Phototype Engraving Co., Philadelphia.)]
PHYSICS
BY
WILLIS E. TOWER, M. SCI. (Univ. of Illinois)
HEAD OF THE DEPARTMENT OF PHYSICS, ENGLEWOOD
HIGH SCHOOL, CHICAGO
CHARLES H. SMITH, M. E. (Cornell)
HEAD OF THE DEPARTMENT OF PHYSICS AND ASSISTANT
PRINCIPAL, HYDE PARK SCHOOL, CHICAGO
CHARLES M. TURTON, A. M. (Syracuse)
HEAD OF THE DEPARTMENT OF PHYSICS, BOWEN
HIGH SCHOOL, CHICAGO
IN COLLABORATION WITH
THOMAS D. COPE, Ph.D. (Pennsylvania)
ASSISTANT PROFESSOR OF PHYSICS, UNIVERSITY
OF PENNSYLVANIA
BASED UPON
PRINCIPLES OF PHYSICS
BY
TOWER, SMITH and TURTON
WITH 7 PLATES AND 448 OTHER ILLUSTRATIONS
PHILADELPHIA
P. BLAKISTON'S SON & CO.
1012 WALNUT STREET
COPYRIGHT, 1920, BY P. BLAKISTON'S SON & CO.
PREFACE
In the preparation of this text, the _pupil_, his experience, needs, and
interests have been constantly kept in mind. The order of topics,
illustrations, and problems have been selected with the purpose of
leading the _pupil_ into a clear understanding of the physical phenomena
continually taking place about him.
The recommendations and conclusions reached by the "New Movement in the
Teaching of Physics" have been incorporated into the book as a whole.
These conclusions indicate that the most efficient teaching in physics
involves a departure from the quantitative, mathematical methods of
presentation that were in general use a dozen or more years ago, toward
a method better adapted to the capabilities, interests, and requirements
of the young people in our physics classes.
The older methods are effective with a portion of the student body which
has the greater mathematical ability and training, but they discourage a
large majority of the pupils who are not gifted or prepared for severe
mathematical analysis. For this reason, many of the more difficult
mathematical demonstrations often given in physics texts are omitted.
Most of the problems involve only the units employed in practical
every-day measurements.
The portions of Mechanics that are ordinarily so difficult for the
average pupil are not taken up until he has covered considerable ground
with which he is more or less familiar and not until he has become
somewhat accustomed to the methods of study and the technical terms of
the subject.
The pupil comes to the study of physics with a great number of
experiences and impressions of physical phenomena continually occurring
about him. In recognition of this fact, it has been thought best to
consider first the explanation of common things well known to all
pupils, such as the diffusion of gases, evaporation of liquids,
expansion of bodies when heated, and capillary action. Since the
molecular theory of matter is now supported by so many conclusive
evidences, we have not hesitated to make free use of it in the early
chapters. The applications of this theory are extremely helpful in
explaining every-day phenomena. Our experience shows that beginners in
physics understand and apply this theory without difficulty.
The illustrations and drawings have been selected from a pedagogical
rather than a spectacular point of view. Practically all of them are
new. The problems and exercises have been selected for the distinct
purpose of illustrating the principles taught in the text and for their
practical applications.
Many direct applications to common every-day experiences are given in
order to connect the subject matter with the home environment and daily
observation of physical phenomena. Some phenomena are mentioned without
detailed explanation as it is felt that the presentation of these
subjects in this manner is better for this grade of student than a
complete analysis.
Some of the special features of the text may be briefly summarized as
follows:
(A) _Simplicity of presentation_ is emphasized. The methods of attack,
the illustrations and examples employed in developing the subjects are
particularly adapted to beginners in physics.
(B) The text is divided into some _seventy-seven sections_, each
containing material enough for one recitation.
(C) Each of these sections is summarized by a list of _important topics_
which point out to the pupil the principles and subject matter requiring
most careful attention. The lists of important topics are also of
assistance to the teacher in assigning recitations.
(D) The _problems and practical exercises_ emphasize physical principles
as distinguished from mathematical training. A list of exercises is
placed at the end of the several sections. They are in sufficient number
to permit testing at many points and of a choice of problems by
teachers.
The authors wish to express their appreciation for suggestions and
helpful criticisms to many who have read the text in manuscript or
proof. Especially to Professor A. P. Carman of the University of
Illinois and his associate, Professor F. R. Watson, who have gone
carefully over the whole text; and to Mr. Chas. M. Brunson, Scott High
School, Toledo, Ohio, Mr. Frank E. Goodell, North High School, Des
Moines, Iowa, and to Mr. Walter R. Ahrens, Englewood High School,
Chicago, for assistance in reading the proofs. Also to Mr. W. H.
Collins, Jr., Bowen High School, Chicago, who supervised the preparation
of drawings for the diagrams and figures; and to many firms and
individuals that have courteously furnished material for illustrations.
WILLIS E. TOWER.
CHARLES H. SMITH.
CHARLES M. TURTON.
ON THE STUDY OF PHYSICS
When a pupil begins the study of Physics he has in his possession many
bits of knowledge which are fundamental in the science. He has learned
to throw a ball and can tell how a thrown ball moves. He has drawn out
nails with a claw hammer. He has seen wood float and iron sink. He has
sucked liquids up through straws. In his mother's kitchen, he sees water
as ice, liquid, and steam. On a wintry day he reads the temperature on a
thermometer. He sees sparks fly from car wheels when the brakes are
applied. He has played with a horseshoe magnet, and has found the north
by means of a compass. The telephone, the electric light and the motor
he sees, and perhaps uses, many times a day. He dresses before a mirror,
focuses his camera, watches the images at a moving picture show, and
admires the colors of the rainbow. He has cast stones into water to
watch the ripples spread, has shouted to hear the echo, and perhaps
plays some musical instrument. These, and a thousand other things, are
known to the intelligent and normal boy or girl who has reached the age
at which the study of Physics is properly begun.
To a great extent even the terms used in the science are familiar to the
beginner. He speaks of the horse-power of an engine, reads
kilowatt-hours from the meter in the cellar, and may know that
illuminating gas costs one dollar per thousand "cubic feet." "Ampere"
and "volt" are words he frequently hears and sees.
When he takes up the study of Physics, the attitude of the student
toward these familiar things and words must undergo a change. Casual
information about them must be changed to sound knowledge, purposely
acquired. Hazy notions about the meanings of words must be replaced by
exact definitions. Bits of knowledge must be built into a structure in
which each fact finds its proper place in relation to the others.
The only agent which can accomplish these changes is the student
himself. He must consciously and purposely seek the truth and must
reflect upon it until he sees it in its relation to other truth. Upon
him, and upon him alone, rests the final responsibility for the success
or failure of his study.
But the student is not without assistance. In his teacher he finds a
guide to stimulate, to direct, and to aid his efforts, and a critic to
point out wherein his efforts have failed and wherein they have
succeeded. Weights, measures, and other apparatus are furnished to
enable him to answer for himself questions which have arisen in his
studies.
In addition to these the student has his text book, his teacher for his
hours of private study. A good text book is an inspiring teacher in
print. It directs attention to things familiar to the student through
long experience, and inspires him to make a closer scrutiny of them. It
invites him to observe, to analyze, to compare, to discover likenesses
and differences in behavior. It questions him at every turn. Its ever
repeated challenge reads, "Weigh and consider." It furnishes him needed
information that he cannot otherwise acquire. It satisfies his desire to
know, "By whom, where, when, and how was this first discovered?"
The student of Physics must never forget that he is studying not pages
of text but the behavior and properties of iron, water, mica, moving
balls, pumps, boiling liquids, compressed air, mirrors, steam engines,
magnets, dynamos, violins, flutes, and a host of other things. His
studies should, whenever possible, be made first hand upon the things
themselves. The text is an aid to study, never a substitute for the
thing studied.
It is an excellent plan for each student to select some one thing for
special study, the telephone for example. By observation, experiment,
and reading, he may acquire a large amount of valuable information about
such a subject while pursuing his course in Physics. Every part of the
science will be found to bear some relation to it.
The student who takes up the study of Physics in the way suggested will
find himself at the end of a year of study in possession of much new and
valuable knowledge about the physical world in which he lives. By virtue
of this knowledge he will be better able to enjoy the world, to control
it, and to use it.
THOMAS D. COPE.
PHILADELPHIA.
CONTENTS
CHAPTER I. INTRODUCTION AND MEASUREMENT. PAGE
(1) Introduction 1
(2) States of Matter 4
(3) The Metric System 8
CHAPTER II. MOLECULAR FORCES AND MOTIONS.
(1) Molecular Motions in Gases 13
(2) Molecular Motions in Liquids 18
(3) Molecular Forces in Liquids 21
(4) Molecular Forces in Liquids and Solids 27
(5) Molecular Forces in Solids 31
CHAPTER III. MECHANICS OF LIQUIDS.
(1) Liquid Pressure 36
(2) Transmission of Liquid Pressure 41
(3) Archimedes' Principle 47
(4) Density and Specific Gravity 52
CHAPTER IV. MECHANICS OF GASES.
(1) Weight and Pressure of the Air 55
(2) Compressibility and Expansibility of the Air 62
(3) Pneumatic Appliances 66
CHAPTER V. FORCE AND MOTION.
(1) Force, how Measured and Represented 79
(2) Motion. Newton's Laws 85
(3) Resolution of Forces 96
(4) Moment of Force and Parallel Forces 99
(5) Gravitation and Gravity 103
(6) Falling Bodies 109
(7) The Pendulum 115
CHAPTER VI. WORK AND ENERGY.
(1) Work and Energy 119
(2) Power and Energy 123
(3) The Lever and Simple Machines 129
(4) Wheel and Axle and Pulley 136
(5) Efficiency and the Inclined Plane 142
(6) Friction and its Uses 147
(7) Water Power 152
CHAPTER VII. HEAT, ITS PRODUCTION AND TRANSMISSION.
(1) Sources and Effects of Heat 159
(2) Temperature and Expansion 162
(3) Expansion of Gases, Liquids and Solids 167
(4) Modes of Transmitting Heat 173
(5) Convection, Heating and Ventilation 179
(6) The Moisture in the Air, Hygrometry 191
(7) Evaporation 196
CHAPTER VIII. HEAT AND WORK.
(1) Heat Measurement and Specific Heat 200
(2) Heat and Changes of State 205
(3) Heat and Work 212
(4) Heat Engines 222
CHAPTER IX. MAGNETISM.
(1) General Properties of Magnets 228
(2) Theory of Magnetism, Magnetic Fields 232
(3) The Earth's Magnetism 238
CHAPTER XI. STATIC ELECTRICITY.
(1) Electrification and Electrical Charges 243
(2) Electric Fields and Electrostatic Induction 247
(3) Electric Theories, Distribution and Electric Charges 252
(4) Potential, Capacity, and the Electric Condenser 257
(5) Electrostatic Generators 262
CHAPTER XI. ELECTRIC CURRENTS PRODUCED BY VOLTAIC
CELLS.
(1) Electrical Currents and Circuits 267
(2) The Simple Voltaic Cell and its Action 270
(3) Practical Voltaic Cells 274
CHAPTER XII. MAGNETIC EFFECTS OF ELECTRIC CURRENTS,
AND ELECTRICAL MEASUREMENTS.
(1) The Magnetic Effect of Electric Currents 279
(2) Electrical Measurements 289
(3) Ohm's Law and Electrical Circuits 298
(4) Grouping of Cells and Measuring Resistance 302
CHAPTER XIII. CHEMICAL AND HEAT EFFECTS OF ELECTRIC
CURRENTS.
(1) The Chemical Effect of Electric Currents 307
(2) The Storage Battery and Electric Power 312
(3) The Heat Effect of Electric Currents 318
CHAPTER XIV. INDUCED CURRENTS.
(1) Electromagnetic Induction 326
(2) The Dynamo and the Motor 335
(3) The Induction Coil and the Transformer 343
(4) The Telephone 349
CHAPTER XV. SOUND.
(1) Sound, Source, Speed, Media 354
(2) Waves and Wave Motion 357
(3) Intensity and Pitch of Sound 363
(4) Musical Scales and Resonance 368
(5) Interference, Beats, Vibration of Strings 374
(6) Tone Quality, Vibrating Plates and Air Columns 384
CHAPTER XVI. LIGHT.
(1) Rectilinear Propagation of Light 388
(2) Photometry and Law of Reflection 393
(3) Mirrors and Formation of Images 400
(4) Refraction of Light 410
(5) The Formation of Images by Lenses 416
(6) Optical Instruments 423
(7) Color and Spectra 430
(8) Nature of Light 442
CHAPTER XVII. INVISIBLE RADIATIONS.
(1) Electric waves and Radioactivity 448
CHAPTER XVIII. WIRELESS TELEPHONY AND ALTERNATING
CURRENTS.
(1) Wireless Telephony 460
(2) Alternating Currents 466
INDEX 487
PHYSICS
CHAPTER I
INTRODUCTION AND MEASUREMENT
(1) INTRODUCTION
=1. Physics, an Explanation of Common Things.=--Many students take up
the study of physics expecting to see wonderful experiments with the "X"
rays, wireless telegraphy, dynamos, and other interesting devices.
Others are dreading to begin a study that to them seems strange and
difficult, because they fear it deals with ideas and principles that are
beyond their experience and hard to comprehend.
Each of these classes is surprised to learn that _physics is mainly an
explanation of common things_. It is a study that systematizes our
knowledge of the forces and changes about us; such as the pull of the
earth, the formation of dew, rain and frost, water pressure and pumps,
echoes and music, thermometers and engines, and many other things about
us with which people are more or less familiar. Physics is like other
school subjects, such as mathematics and language, in having its own
peculiar vocabulary and methods of study; these will be acquired as
progress is made in the course.
The most useful habit that the student of physics can form is that of
connecting or relating each _new idea_ or _fact_ that is presented to
him to _some observation_ or _experience_ that will illustrate the new
idea. This relating or connecting of the new ideas to one's own personal
experience is not only one of the best known means of cultivating the
memory and power of association, but it is of especial help in a subject
such as physics, which deals with the systematic study and explanation
of the facts of our every-day experience.
=2. Knowledge--Common and Scientific.=--This leads to the distinction
between _common knowledge and scientific knowledge_. We all possess
common knowledge of the things about us, gained from the impressions
received by our senses, from reading, and from the remarks of others.
_Scientific_ knowledge is attained when the bits of common knowledge are
connected and explained by other information gained through study or
experience. That is, common knowledge becomes scientific, when it is
_organized_. This leads to the definition: _Science is organized
knowledge_.
Common knowledge of the forces and objects about us becomes scientific
only as we are able to make accurate measurements of these. That is,
science is concerned not only in _how_ things work, but even more in
_how much_ is involved or results from a given activity. For example, a
scientific farmer must be able to compute his costs and results in order
to determine accurately his net profits. The business man who is
conducting his business with efficiency knows accurately his costs of
production and distribution.
This book is written in the hope that it will make more scientific the
student's common knowledge of the forces and changes in the world about
him and will give him many ideas and principles that will help him to
acquire the habit of looking from effects to their natural causes and
thus tend to develop what is called the _scientific habit of thought_.
=3. Hypothesis, Theory, and Law.=--Three words that are frequently used
in science may be mentioned here: _hypothesis_, _theory_, and _law_. An
hypothesis is a supposition advanced to explain some effect, change, or
condition that has been observed. For example, the Nebular Hypothesis of
which many high-school students have heard, is an attempt to explain the
origin of the sun, the earth, the planets, and other solar systems.
A theory is an hypothesis which has been tested in a variety of ways and
which seems to fit the conditions and results so that it is generally
accepted as giving a satisfactory explanation of the matter in question.
The Molecular Theory of Matter which states that matter of all kinds is
composed of very small particles called molecules (see Art. 6), is a
familiar example of a theory.
A theory becomes a law when it may be definitely proved. Many laws are
expressed in mathematical language, e.g., the law of gravitation. (See
Art. 88.) Many of the laws of physics are illustrated by laboratory
experiments, which show in a simple way just what the law means.
Exercises
Explain what is meant by the following terms and expressions:
1. Common knowledge.
2. Scientific knowledge.
3. Science.
4. Topics in physics.
5. Scientific habit of thought.
6. Value of relating new ideas to former experiences.
7. Hypothesis.
8. Theory.
9. Law.
(2) THE STATES OF MATTER
=4. Physics Defined.=--In the study of any science or field of
knowledge, it is helpful to have a basis for grouping or classifying the
facts studied. In physics we are to study the objects, forces, and
changes about us, to understand them and their relations to one another.
Accordingly, physics, dealing with the material world about us, is often
defined as _the science of matter and energy_, _matter_ being _anything
that occupies space_ and _energy_ the _capacity for doing work_. This
definition of physics while not strictly accurate is sufficiently
comprehensive for our present purpose.
=5. The Three States of Matter.=--Our bodies are _matter_ since they
occupy space. Further, they possess _energy_ since they are able to do
work. In beginning the study of physics it will simplify our work if we
study one of these topics before the other. We will therefore begin with
matter and consider first its three states.
Some bodies are _solid_; as ice, iron, wax. Others are _liquid_; as
water, mercury, oil. Still others are in the state of _gas_; as steam,
air, and illuminating gas. Further we notice that the same substance may
be found in any one of the three states. For example water may be either
ice, water or steam; that is, either a solid, a liquid, or a gas.
Most persons have heard of _liquid air_ and possibly some know of _ice
air_, _i.e._, air cooled until it not only liquefies, but is solidified.
On the other hand, iron may be melted and, if heated hot enough, may be
turned into iron vapor. In fact most substances by heating or cooling
sufficiently may be changed into any one of the three states.
Before defining the three states, let us consider the _structure_ of
matter. This may help us to answer the question: How is it possible to
change a hard solid, such as ice, into a liquid, water, and then into
an invisible gas like steam? This is explained by the molecular theory
of matter.
=6. The Molecular Theory of Matter.=--It is believed that all bodies are
made up of very small particles called _molecules_, and that these
instead of being packed tightly together like square packages in a box,
are, strange as it may seem, very loosely packed even in solids and do
not _permanently_ touch their neighbors. The size of these molecules is
so minute that it has been estimated that if a drop of water could be
magnified to the size of the earth, the molecules magnified in the same
proportion would be in size between a baseball and a football. The air
and all other gases are believed to be made up of molecules in _rapid
motion_, striking and rebounding continually from one another and from
any objects in contact with the gas.
=7. States of Matter Defined.=--These ideas of the structure of matter
assist us in understanding the following definitions: _A solid is that
state of matter in which the molecules strongly cling together and tend
to keep the same relative positions_. (This of course follows from the
tendency of a solid to retain a definite form.) _A liquid is that state
of matter in which the molecules tend to cling together, yet move about
freely._ Hence a liquid takes the form of any vessel in which it is
placed. _A gas is that state of matter in which the molecules move about
freely and tend to separate indefinitely._ Hence a gas will fill any
space in which it is placed.
=8. Effect of Heat on Matter.=--It is further believed that when a body
is heated, that the action really consists in making its molecules move
or vibrate faster and faster as the heating progresses. This increase of
motion causes the molecules to push apart from one another and this
separation of the molecules causes an expansion of the body whether it
be solid, liquid, or gas. Fig. 1 shows the expansion of air in an air
thermometer. Fig. 2 shows the expansion of a solid on heating.
[Illustration: FIG. 1.--When the bulb is heated, the air within expands
forcing down the water in the tube.]
=9. Physical and Chemical Changes.= A change of state such as the
freezing or boiling of water is called a _physical_ change, for this
change has not affected the identity of the substance. It is water even
though it has become solid or gaseous. Heating a platinum wire red hot
is also a physical change for the wire when on cooling is found to be
the same substance as before. Further if salt or sugar be dissolved in
water the act of _solution_ is also a physical change since the
identical substance (salt or sugar) is in the solution and may be
obtained by evaporating the water.
[Illustration: FIG. 2 (_a_) represents a straight bar made of a strip of
brass and a strip of iron riveted together and attached to a handle.
Upon heating the compound bar in a gas flame, the brass expands faster
than the iron causing the bar to bend toward the latter as in Fig. 2
(_b_).]
If some sugar, however, is heated strongly, say in a test-tube, it is
found to blacken, some water is driven off and on cooling _some black
charcoal is found in the tube instead of the sugar_. This action which
has resulted in a _change in the nature of the substance_ treated is
called a _chemical_ change. To illustrate further, if some magnesium
wire is heated strongly in a flame, it burns, giving off an intense
light and when it cools one finds it changed to a light powdery
substance like ashes. Chemical changes, or those that change the nature
of the substance affected, are studied in _chemistry_. In _physics_ we
have to do only with physical changes, that is, with those changes that
do not affect the nature of the substance.
Important Topics
1. Physics defined.
2. The three states of matter; solid, liquid, gas.
3. Molecular theory of matter.
4. Physical and chemical changes.
Exercises
Write out in your own words your understanding of:
1. The structure of matter.
2. Some of the differences between solids, liquids, and gases.
3. How to change solids to liquids and gases and _vice versa_.
4. The reason for the changes of size of a body on heating.
5. Why cooling a gas tends to change it to a liquid or a solid.
6. The actual size of molecules.
Which of the following changes are chemical and which physical?
Give reasons.
1. Melting of ice.
2. Burning of a candle.
3. Production of steam.
4. Falling of a weight.
5. Drying of clothes.
6. Making an iron casting.
7. Decay of vegetables.
8. Sprouting of seeds.
9. Flying an aeroplane.
10. Growth of a plant.
11. Grinding of grain.
12. Sawing a board.
13. Pulverizing stone.
14. Making toast.
15. Sweetening tea or coffee with sugar.
16. Burning wood or gas.
(3) THE METRIC SYSTEM
=10. The Metric System.=--In order to study the three states of matter
with sufficient exactness it is necessary to employ a system of
measurement. The system universally employed by scientists is called
_The Metric System_. In many respects it is the most convenient for all
purposes. Every student should therefore become familiar with it and
learn to use it. At the present time, not only do scientists everywhere
use it, but many countries have adopted it and use it in common
measurements. It was legalized in the United States in 1866. The metric
system was originated by the French Academy of Sciences during the
latter part of the 18th century. There were so many different systems of
weights and measures in use, each country having a system of its own,
that commerce was much hindered. It was therefore decided to make a
system based upon scientific principles. The length of the earth's
quadrant passing from the equator to the pole was determined by
surveying and computation. One-ten-millionth of this distance was
selected as the unit of length and called a _meter_. Accurate copies of
this meter were made and preserved as standards.
Later surveys have shown that the original determination of the earth's
quadrant was not strictly accurate; so that after all the meter is not
exactly one-ten-millionth of the earth's quadrant.
=11. The Standard Meter.=--The standard unit of _length_ in the metric
system is the _meter_. It is the distance, at the temperature of melting
ice, between two transverse parallel lines ruled on a bar of platinum
(see Fig. 3), which is kept in the Palace of the Archives in Paris.
Accurate copies of this and other metric standards are also kept at the
Bureau of Standards at Washington, D. C. Fig. 4 shows the relation
between the inch and the centimeter (one-hundredth of a meter).
=12. Units and Tables in the Metric System.=--The metric unit of _area_
commonly used in physics is the _square centimeter_.
[Illustration: FIG. 3--The standard meter.]
The standard unit of _volume_ or capacity is the _liter_. It is a cube
one-tenth of a meter on each edge. It is equal to 1.057 quarts. It
corresponds, therefore, to the quart in English measure.
[Illustration: FIG. 4.--Centimeter and inch scales.]
The standard unit of _mass_ is the _kilogram_. It is the mass of 1 liter
of pure water at the temperature of its greatest density, 4°C. or
39.2°F.
The three principal units of the metric system, the _meter_, the
_liter_, and the _kilogram_, are related to one another in a simple
manner, since the liter is a cube one-tenth of a meter in each dimension
and the kilogram is the mass of a liter of water. (See Fig. 5.)
The metric system is a _decimal_ system that is, one unit is related to
another unit in the ratio of _ten_ or of some power of ten. This is
indicated by the following tables:
_Metric Table of Length_
10 millimeters (mm.) equal 1 centimeter.
10 centimeters (cm.) equal 1 decimeter.
10 decimeters (dm.) equal 1 meter.
10 meters (m.) equal 1 dekameter.
10 dekameters (Dm.) equal 1 hectometer.
10 hectometers (hm.) equal 1 kilometer.
10 kilometers (km.) equal 1 myriameter.
The measures commonly used are the _centimeter_, _meter_ and
_kilometer_.
_Metric Table of Mass_ (_or Weight_)
10 milligrams (mg.) equal 1 centigram.
10 centigrams (cg.) equal 1 decigram.
10 decigrams (dg.) equal 1 gram.
10 grams (g.) equal 1 dekagram.
10 dekagrams (Dg.) equal 1 hectogram.
10 hectograms (hg.) equal 1 kilogram.
10 kilograms (kg.) equal 1 myriagram.
The masses commonly used are the _milligram_, _gram_ and _kilogram_.
Notice in these tables the similarity to 10 mills equal 1 cent, 10 cents
equal 1 dime, 10 dimes equal 1 dollar, in the table of United States
money.
Other tables in the metric system are built upon the same plan. Learn
the prefixes in order thus: milli, centi, deci, deka, hecto, kilo,
myria. The first three prefixes are Latin numerals and represent
divisions of the unit. The last four are Greek numerals and represent
multiples. In these tables, milli means 1/1000, centi means 1/100, deci
means 1/10, deka means 10, hecto, 100, kilo, 1000, myria, 10,000. Two
other prefixes are sometimes used, _micro_ which means 1/1,000,000; as
microfarad or microvolt, and _meg_ which means 1,000,000, as megohm
meaning 1,000,000 ohms.
=13. Advantages of the Metric System.=--_First_, it is a decimal system;
_second_, the same form and prefixes are used in every table; _third_,
the standards of length (meter), volume (liter), and mass (kilogram)
bear a simple relation to one another. This simple relation between the
three standard units may be given thus: _first_, the liter is a cubic
decimeter, and _second_, the kilogram is the mass of a liter of water.
(See Fig. 5) Since the liter is a cubic decimeter, the length of one
side is 10 cm. The liter therefore holds 1000 ccm. (10 × 10 × 10).
Therefore, 1 liter = 1 cu. dm. = 1000 ccm. and since 1 liter of water
has a mass of 1 kg. or 1000 g., then 1000 ccm. of water has a mass of
1000 g., or _1 ccm. of water has a mass of 1 g._
[Illustration: FIG. 5.--One liter of the water has a mass of one
kilogram.]
The following table of equivalents gives the relation between the most
common English and metric units. Those marked (*) should be memorized.
(*) 1 meter = 39.37 inches. 1 cu, in. = 16.387 ccm.
(*) 1 inch = 2.54 cm. 1 cu. ft. = 28315 cm.
1 foot = 30.48 cm. 1 cu. m. = 1.308 cu. yd.
1 mile = 1.609 km. (*) 1 liter = 1.057 qt.
1 sq. in. = 6.45 sq. cm. (*) 1 kg. = 2.204 lbs.
1 sq. cm. = 0.155 sq. in. 1 g. = 15.44 grains.
1 sq. m. = 1.196 sq. yd. 1 lb. = 0.4536 kg.
1 acre = 0.405 ha. 1 oz. = 28.35 g.
1 hectare = 2.45 acres. 1 g. = 0.0353 oz.
THE C. G. S. SYSTEM. Scientists have devised a plan for expressing any
measurement in terms of what are called the _three fundamental units of
length_, _mass_, and _time_. The units used are the _centimeter_, the
_gram_ and the _second_. Whenever a measurement has been reduced to its
equivalent in terms of these units, it is said to be expressed in
_C.G.S._ units.
Important Topics
1. The metric system; how originated.
2. Units; meter, liter, kilogram.
3. Metric tables.
4. Advantages of the metric system.
5. Equivalents.
6. The C.G.S. system.
Exercises
1. Which is cheaper, milk at 8 cents a quart or 8 cents a liter? Why?
2. Which is more expensive, cloth at $1.00 a yard or at $1.00 a meter?
Why?
3. Which is a better bargain, sugar at 5 cents a pound or 11 cents a
kilogram? Why?
4. Express in centimeters the height of a boy 5 ft. 6 in. tall.
5. What is the length of this page in centimeters? In inches?
6. What is the mass of a liter of water? Of 500 ccm.? Of 1 ccm.?
7. From Chicago to New York is 940 miles. Express in kilometers.
8. A 10-gallon can of milk contains how many liters?
9. What will 100 meters of cloth cost at 10 cents a yard?
10. What will 4 kg. of beef cost at 15 cents a pound?
11. What will 5-1/2 lbs. of mutton cost at 40 cents a kilogram?
12. How can you change the state of a body? Give three methods.
13. Correct the statement 1 ccm. = 1 g.
14. How many liters in 32 quarts?
CHAPTER II
MOLECULAR FORCES AND MOTIONS
(1) EVIDENCES OF MOLECULAR MOTION IN GASES
=14. Size of Molecules.=--The difference between solids, liquids, and
gases has been explained as due to the different behavior of molecules
in the three states of matter. That is, in solids they cling together,
in liquids they move freely, and in gases they separate. At this time we
are to consider the _evidences_ of molecular motion in gases. It must be
kept in mind that molecules are exceedingly small. It has been said that
if a bottle containing about 1 ccm. of ordinary air has pierced in it a
minute opening so that 100,000,000 molecules (a number nearly equal to
the population of the United States) pass out every second, it would
take, not minutes or hours, but nearly 9000 years for all of the
molecules to escape. The number of molecules in 1 ccm. of air at 0°C.
and 76 cm. pressure has been calculated by Professor Rutherford to be
2.7 × 10¹⁹. It is evident that such minute particles cannot be seen or
handled as _individuals_. We must judge of their size and action by the
results obtained from experiments.
=15. Diffusion of Gases.=--One line of evidence which indicates that a
gas consists of moving particles is the rapidity with which a gas having
a strong odor penetrates to all parts of a room. For example, if
illuminating gas is escaping it soon diffuses and is noticed throughout
the room. In fact, the common experience of the diffusion of gases
having a strong odor is such that we promptly recognize that it is due
to motion of some kind. The gas having the odor consists of little
particles that are continually hitting their neighbors and are being
struck and buffeted in turn until the individual molecules are widely
scattered. When cabbage is boiled in the kitchen soon all in the house
know it. Other illustrations of the _diffusion_ of gases will occur to
anyone from personal experience, such for instance as the pleasing odor
from a field of clover in bloom.
The following experiment illustrates the rapid diffusion of gases.
[Illustration: FIG. 6_a_.--Diffusion of gases.]
[Illustration: FIG. 6_b_.--Effusion of gases.]
Take two tumblers (see Fig. 6_a_), wet the inside of one with a few
drops of strong ammonia water and the other with a little
hydrochloric acid. Cover each with a sheet of clean paper. Nothing
can now be seen in either tumbler. Invert the second one over the
first with the paper between, placing them so that the edges will
match. On removing the paper it is noticed that both tumblers are
quickly filled with a cloud of finely divided particles, the two
substances having united chemically to form a new substance,
ammonium chloride.
On account of their small size, molecules of air readily pass through
porous solids, cloth, unglazed earthenware, etc. The following
experiment shows this fact strikingly. (See Fig. 6_b_.)
A flask containing water is closed by a rubber stopper through
which pass the stem of a glass funnel and a bent glass tube that
has been drawn out to a small opening (_J_). The funnel has
cemented in its top an inverted porous clay jar (_C_), over the top
of the latter is placed a beaker (_B_). A piece of flexible rubber
tubing (_H_) leading from a hydrogen generator is brought up to the
top of the space between the jar and the beaker. When hydrogen gas
is allowed to flow into the space between _C_ and _B_, the level of
the water in _W_ is seen to lower and a stream of water runs out at
_J_ spurting up into the air.
On stopping the flow of hydrogen and removing _B_, the water falls
rapidly in _J_ and bubbles of air are seen to enter the water from
the tube. (The foregoing steps may be repeated as often as
desired).
This experiment illustrates the fact that the molecules of some
gases move faster than those of some other gases. Hydrogen
molecules are found to move about four times as fast as air
molecules. Hence, while both air and hydrogen molecules are at
first going in opposite directions through the walls of _C_, the
hydrogen goes in much faster than the air comes out. In consequence
it accumulates, creates pressure, and drives down the water in _W_
and out at _J_. On removing _B_, the hydrogen within the porous cup
comes out much faster than the air reënters. This lessens the
pressure within, so that air rushes in through _J_. This experiment
demonstrates not only the fact of molecular motion in gases but
also that molecules of hydrogen move much faster than those of air.
(This experiment will work with illuminating gas but not so
strikingly.)
Careful experiments have shown that the speed of ordinary air molecules
is 445 meters or 1460 ft. per second; while hydrogen molecules move at
the rate of 1700 meters or 5575 ft. or more than a mile per second.
=16. Expansion of Gases.=--Gases also possess the property of indefinite
expansion, that is, if a small quantity of gas is placed in a vacuum,
the gas will expand immediately to fill the entire space uniformly. This
is shown by an experiment with the air pump. On raising the piston the
air follows instantly to fill up the space under it. As the air is
removed from the receiver of an air pump the air remaining is uniformly
distributed within.
=17. How Gases Exert Pressure.=--It is further found that air under
ordinary conditions exerts a pressure of about 15 lbs. to the square
inch. In an automobile tire the pressure may be 90 lbs. and in a steam
boiler it may be 200 lbs. or more to the square inch.
How is the pressure produced? The molecules are not packed together
solidly in a gas, for when steam changes to water it shrinks to about
1/1600 of its former volume. Air diminishes to about 1/800 of its volume
on changing to liquid air. The pressure of a gas is not due then to the
gas filling all of the space in which it acts, but is due rather to the
_motion_ of the molecules. The blow of a single molecule is
imperceptible, but when multitudes of molecules strike against a surface
their combined effect is considerable. In fact, this action is known to
produce the pressure that a gas exerts against the walls of a containing
vessel. Naturally if we compress twice as much gas into a given space
there will be twice as many molecules striking in a given time, which
will give twice as much pressure.
If gas is heated, it is found that the heat will cause a swifter motion
of the molecules. This will also make the molecules strike harder and
hence cause the gas to expand or exert more pressure.
=17a. Brownian Movements.=--Direct photographic evidence of the motion
of molecules in gases has been obtained by studying the behavior of
minute drops of oil suspended in stagnant air. Such drops instead of
being at rest are constantly dancing about as if they were continually
receiving blows from many directions. These motions have been called
_Brownian Movements_ (see Fig. 7).
It has been proved that these movements are due to the blows that these
small drops receive from the swiftly moving molecules of the gas about
them. If the drops are made smaller or the gas more dense, the movements
increase in intensity. These effects are especially marked at a pressure
of 0.01 of an atmosphere.
[Illustration: FIG. 7.--Photograph of Brownian movement. This record is
prepared by the aid of Siedentopf's ultra-microscope and a plate moving
uniformly across the field from left to right.]
Important Topics
It is assumed that air and all gases are made up of molecules in rapid
motion; that this motion is dependent upon temperature and pressure.
Evidence of this is shown by (a) diffusion, (b) expansion, (c) pressure.
Brownian Movements.
Questions
1. What is the molecular (kinetic) theory of gases?
2. What three kinds of evidence help to confirm the theory?
3. What have you seen that seems to show that a gas consists of
molecules in motion?
4. How many meters long is a 10-ft. pole?
5. A 50-kg. boy weighs how many pounds?
6. What are three advantages of the metric system?
7. What will 12 qts. of milk cost at 8 cents a liter?
8. A cube 1 meter each way will contain how many cubic centimeters? How
many liters? What will a cubic meter of water weigh?
(2) MOLECULAR MOTION IN LIQUIDS
=18. Diffusion of Liquids.=--From the evidence given in Arts. 14-17, (a)
of diffusion of odors, (b) of the continued _expansion_ of air in the
air pump, and (c) of the pressure exerted by a gas in all directions,
one may realize without difficulty that a _gas consists of small
particles in rapid motion_. Let us now consider some of the evidence of
molecular motion in liquids. If a little vinegar is placed in a pail of
water, all of the water will soon taste sour. A lump of sugar in a cup
of tea will sweeten the entire contents. This action is somewhat similar
to the diffusion of gases but it takes place much more slowly. It is
therefore believed that the motion of liquid molecules is much slower
than that of gas molecules.
Again, if a dish of water is left standing in the open air in fine
weather, within a few days the dish will become dry though no one has
taken anything from it. We say the water has _evaporated_. What was
liquid is now _vapor_. If we were to observe carefully any dish of water
we would find that it continually loses weight on dry days. That is,
there is a constant movement of the molecules of water into the air.
This movement of the molecules is explained as follows. There appear to
be in the dish of water some molecules that by moving back and forth
acquire a greater velocity than their neighbors; when these reach the
surface of the liquid, some vibration or movement sends them flying into
the air above. They are now vapor or gas molecules, flying, striking,
and rebounding like the air molecules. Sometimes on rebounding, the
water molecules get back into the water again. This is especially apt to
happen when the air is damp, _i.e._, when it contains many water
molecules. Sometimes the air over a dish becomes _saturated_, as in the
upper part of a corked bottle containing water. Although molecules are
continually leaving the surface of the water they cannot escape from the
bottle, so in time as many molecules must return to the water from the
space above as leave the water in the same time. When this condition
exists, the air above the water is said to be _saturated_. On very damp
days the air is often saturated. The explanation above shows why wet
clothes dry so slowly on such a day (See Arts. 166-7 on Saturation.)
=19. Cooling Effect of Evaporation.= We have seen that warming a gas
increases its volume. This expansion is due to the increased motion of
the warmed molecules. Now the molecules that escape from a liquid when
it evaporates are naturally the fastest moving ones, _i.e._, the hottest
ones. The molecules remaining are the slower moving ones or colder
molecules. The liquid therefore becomes colder as it evaporates, unless
it is heated. This explains why water evaporating on the surface of our
bodies cools us. In evaporating, the water is continually losing its
warm, fast moving molecules. The _cooling effect_ of evaporation is,
therefore an evidence of molecular motion in liquids.
[Illustration: FIG. 8.--Osmosis Shown by carrot placed in water.]
=20. Osmosis.=--If two liquids are separated by a membrane or porous
partition, they tend to pass through and mix. This action is called
osmose, or _osmosis_.
Such a movement of liquid molecules in osmosis may be illustrated
by filling a beet or carrot that has had its interior cut out to
form a circular opening (see Fig. 8) with a thick syrup. The
opening is then closed at the top with a rubber stopper through
which passes a long glass tube.
If the carrot is immersed in water, as in Fig. 8, a movement of
water through the porous wall to the interior begins at once. Here,
as in the experiment of the hydrogen and air passing through the
porous cup, the lighter fluid moves faster. The water collecting in
the carrot rises in the tube. This action of liquids passing
through porous partitions and mingling is called _osmosis_.
Gases and liquids are alike in that each will _flow_. Each is therefore
called a _fluid_. Sometimes there is much resistance to the flow of a
liquid as in molasses. This resistance is called _viscosity_. Alcohol
and gasoline have little viscosity. They are _limpid_ or _mobile_. Air
also has some viscosity. For instance, a stream of air always drags some
of the surrounding air along with it.
Important Topics
1. Liquids behave as if they were composed of small particles in motion.
2. This is shown by (1) Diffusion, (2) Solution, (3) Evaporation, (4)
Expansion, (5) Osmosis.
Exercises
1. Give an example or illustration of each of the five evidences of
molecular motion in liquids.
2. When is air saturated? What is the explanation?
3. Why does warming a liquid increase its rate of evaporation?
4. Air molecules are in rapid motion in all directions. Do they enter a
liquid with a surface exposed to the air? Give reason.
5. What are some of the inconveniences of living in a saturated
atmosphere?
6. Fish require oxygen. How is it obtained?
(3) MOLECULAR FORCES IN LIQUIDS
=21. Cohesion and Adhesion.=--In liquids "the molecules move about
freely yet tend to cling together." This tendency of molecules to cling
together which is not noticeable in gases is characteristic of =liquids=
and especially of =solids=. It is the cause of the viscosity mentioned
in the previous section and is readily detected in a variety of ways.
For instance, not only do liquid molecules cling together to form drops
and streams, but they cling to the molecules of solids as well, as is
shown by the wet surface of an object that has been dipped in water. The
attraction of like molecules for one another is called _cohesion_, while
the attraction of =unlike molecules= is called _adhesion_, although the
force is the same whether the molecules are alike or unlike. It is the
former that causes drops of water to form and that holds iron, copper,
and other solids so rigidly together. The adhesion of glue to other
objects is well known. Paint also "sticks" well. Sometimes the "joint"
where two boards are glued together is stronger than the board itself.
The force of attraction between molecules has been studied carefully.
The attraction acts only through very short distances. The attraction
even in liquids is considerable and may be measured. The cohesion of
water may be shown by an experiment where the force required to pull a
glass plate from the surface of water is measured.
[Illustration: FIG. 9.--The water is pulled apart.]
Take a beam balance and suspend from one arm a circular glass
plate, Fig. 9. Weigh the plate and its support. Adjust the glass
plate so that it hangs horizontally and just touches the surface of
clean water, the under side being completely wet. Now find what
additional weight is required to raise the glass plate from the
water.
Just as the plate comes from the water its under side is found to be
wet. That is, _the water was pulled apart_, and the plate was not pulled
from the water. The cohesion of the water to itself is not so strong as
its adhesion to the glass.
The cohesion of liquids is further shown by the form a drop of liquid
tends to take when left to itself. This is readily seen in small drops
of liquids. The spherical shape of drops of water or mercury is an
example. A mixture of alcohol and water in proper proportions will just
support olive oil within it. By carefully dropping olive oil from a
pipette into such a mixture, a drop of the oil, an inch or more in
diameter suspended in the liquid, may be formed. It is best to use a
bottle with plane or flat sides, for if a round bottle is used, the
sphere of oil will appear flattened.
[Illustration: FIG. 10 _a_.
FIG. 10 _b_.
FIGS. 10 _a_ AND _b_.--Surface tension of a liquid film.]
=22. Surface Tension.=--The cohesion of liquids is also indicated by the
tendency of films to assume the smallest possible surface. Soap bubble
films show this readily. Fig. 10 _a_ represents a circular wire form
holding a film in which floats a loop of thread. The tension of the
film is shown in Fig. 10 _b_ by the circular form of the loop after the
film within it has been pierced by a hot wire, Fig. 11 shows a
rectangular wire form with a "rider." The tension in the film draws the
rider forward.
[Illustration: FIG. 11.--The rider is drawn forward.]
[Illustration: FIG. 12.--Surface tension causes the pointed shape.]
A soap bubble takes its spherical shape because this form holds the
confined air within the smallest possible surface. A drop of liquid is
spherical for the same reason. Many illustrations of the tension in
films may be given. Users of water colors notice that a dry camel's-hair
brush is bushy. (Fig. 12 _A_). When in water it is still bushy. (Fig. 12
_B_.) But when it is taken from the water and the excess is shaken from
it, it is pointed as in Fig. 12 _C_. It is held to the pointed shape by
the tension of the liquid film about the brush.
[Illustration: FIG. 13.--A needle depresses the surface when floating.]
The surface of water acts as if covered by a film which coheres more
strongly than the water beneath it. This is shown by the fact that a
steel needle or a thin strip of metal may be floated upon the surface of
water. It is supported by the surface film. (See Fig. 13.) If the film
breaks the needle sinks. This film also supports the little water bugs
seen running over the surface of a quiet pond in summer. The surface
film is stronger in some liquids than in others. This may be shown by
taking water, colored so that it can be seen, placing a thin layer of it
on a white surface and dropping alcohol upon it. Wherever the alcohol
drops, the water is seen to pull away from it, leaving a bare space over
which the alcohol has been spread. This indicates that the alcohol has
the weaker film. The _film of greasy benzine is stronger_ than the film
of the pure material. If one wishes to remove a grease spot and places
pure benzine at the center of the spot, the stronger film of the greasy
liquid will pull away from the pure benzine, and spread out, making a
larger spot than before, while if pure benzine is placed _around the
grease spot_, the greasy liquid at the center pulls away from the pure
benzine, drawing more and more to the center, where it may be wiped up
and the grease entirely removed.
[Illustration: FIG. 14.--The molecule at _A_ is held differently from
one within the liquid.]
=23. Explanation of the Surface Film.=--Beneath the surface of a liquid
each molecule is attracted by all the other molecules around it. It is
attracted equally in all directions. Consequently the interior molecules
move very easily over each other in any direction. A molecule at the
surface, as at _A_, Fig. 14, is not attracted _upward_ by other liquid
molecules. Its freedom of motion is thereby hindered with the result
that a molecule at the surface behaves differently from one beneath the
surface. The surface molecules act as if they form an elastic skin or
membrane upon the liquid surface.
[Illustration: FIG. 15.--Capillary attraction in tubes.]
=24. Capillarity.=--A striking action of the surface film of a liquid is
seen in the rise of liquids in tubes of small bore when the liquid
_wets_ them. If the liquid _does not wet_ the tube, as when mercury is
placed in glass, the liquid is depressed. It is found in general that:
_Liquids rise in capillary tubes when they wet them and are depressed in
tubes which they do not wet; the smaller the diameter of the tube the
greater the change of level._ (See Fig. 15.) This action is explained as
follows: The molecules of a liquid have an attraction for each other and
also for the sides of a tube. The former is called "cohesion for
itself," the latter is called "adhesion for the sides of the containing
vessel." If the cohesion for itself is greater than the adhesion for the
side of the containing vessel, the liquid is pulled away from the side
and is depressed. If the adhesion is greater, the liquid is elevated.
This action is called "capillary action" from the Latin word
(_capillus_) signifying hair, since it shows best in fine hairlike
tubes.
There are many common illustrations of capillary action: oil rising in a
wick; water rising in a towel or through clothes; ink in a blotter, etc.
The minute spaces between the fibers composing these objects act as fine
tubes. If cloth is treated with a preparation which prevents water from
adhering to its fibers, the material will not be wet when water is
poured upon it, because the water will not run in between the fibers; a
surface film spreads over the cloth so that no water enters it.
_Cravenette cloth_ has been treated in this way and hence is waterproof.
The action of this film may be shown by the following experiment.
Dip a sieve of fine copper gauze in melted paraffin, thus coating
each wire so that water will not adhere to it. Water may now be
poured into the sieve, if a piece of paper is first laid in it to
break the force of the water. On carefully removing the paper the
surface film of the water will prevent the passage of the water
through the sieve.
=25. Capillary Action in Soils.=--The distribution of moisture in the
soil depends largely upon capillary action. When the soil is compact the
minute spaces between the soil particles act as capillary tubes, thus
aiding the water to rise to the surface. As the water evaporates from
the surface more of it rises by capillary action from the damper soil
below. Keeping the soil loose by cultivation, makes the spaces between
the particles too large for much capillary action, thus the moisture is
largely prevented from rising to the surface.
In the semi-arid regions of the West "_dry farming_" is successfully
practised. This consists in keeping the surface covered with a "dust
mulch" produced by frequent cultivation. In this way the moisture is
kept below the surface, where it can be utilized during the hot dry
summer by the roots of growing plants.
Important Topics
1. Attractive forces between liquid molecules.
2. Cohesion (like molecules); adhesion (unlike molecules).
3. Special effects of this force are classified as (a) capillary action,
and (b) surface tension.
Exercises
1. What evidence of capillary action have you seen outside of the
laboratory?
2. What is the explanation for capillary action?
3. Where are surface films found?
4. What are three common effects of surface films?
5. Explain why cravenette cloth sheds water.
6. If a circular glass disc 10 cm. in diameter requires 50 grams of
force to draw it from the water, what is the cohesion of water per
square centimeter?
7. What is the weight in grams of 1 ccm. of water? of a liter of water?
8. Name five examples of adhesion to be found in your home.
9. Under what conditions will a liquid wet a solid and spread over it?
10. When will it form in drops on the surface?
11. Explain the proper procedure for removing a grease spot with
benzine.
12. What difference is there between a liquid and a fluid?
13. Why cannot a "soap bubble" be blown from pure water?
14. Which are larger, the molecules of steam or those of water? Why?
15. Why is the ground likely to be damp under a stone or board when it
is dry all around?
16. Why does any liquid in falling through the air assume the globule
form?
17. Give three examples of capillary attraction found in the home. Three
out of doors.
18. Why does cultivation of the soil prevent rapid evaporation of water
from the ground?
(4) EVIDENCES OF MOLECULAR FORCES IN LIQUIDS AND SOLIDS
=26. Solutions.=--A crystal of potassium permanganate is placed in a
liter of water. It soon dissolves and on shaking the flask each portion
of the liquid is seen to be colored red. The dissolving of the
permanganate is an illustration of the attraction of the molecules of
water for the molecules of the permanganate. We are familiar with this
action in the seasoning of food with salt and sweetening with sugar.
Water will dissolve many substances, but in varying degrees, _i.e._, of
some it will dissolve much, of others, little, and some not at all.
Further, different liquids have different solvent powers. Alcohol will
dissolve resin and shellac, but it will not dissolve gum arabic, which
is soluble in water. Benzine dissolves grease. Beeswax is not dissolved
by water, alcohol or benzine, but is soluble in turpentine.
It is found that the _temperature_ of the liquid has a marked effect
upon the amount of substance that will dissolve. This is an indication
that the _motions_ of the molecules are effective in solution. It
appears that dissolving a solid is in some respects similar to
evaporation, and just as at higher temperatures more of the liquid
evaporates, because more of the molecules will escape from the liquid
into the air above, so at higher temperatures, more molecules of a solid
will detach themselves through greater vibration and will move into the
liquid.
Further, just as an evaporating liquid may saturate the space above it
so that any escape of molecules is balanced by those returning, so with
a dissolving solid, the liquid may become _saturated_ so that the
solution of more of the solid is balanced by the return of the molecules
from the liquid to the solid condition.
=27. Crystals and Crystallization.=--This return from the liquid to the
solid state, of molecules that are in solution, is especially noticeable
when the solution is cooling or evaporating and hence is losing its
capacity to hold so much of the solid. On returning to the solid, the
molecules attach themselves in a definite manner to the solid portion,
building up regular solid forms. These regular forms are _crystals_. The
action that forms them is called _crystallization_.
Each substance seems to have its own _peculiar form of crystal_ due to
the manner in which the molecules attach themselves to those previously
in place. The largest and most _symmetrical crystals are_ those in which
the molecules are deposited slowly with no disturbance of the liquid.
Beautiful crystals of alum may be obtained by dissolving 25 g. of alum
in 50 ccm. of hot water, hanging two or three threads in the solution
and letting it stand over night. The thread fibers provide a foundation
upon which crystals grow.
When a _solution_ of a solid evaporates, the molecules of the _liquid_
escape as a gas, the molecules of the _solid_ remain accumulating as
crystals. This principle has many uses: (a) sea water is purified by
_evaporating the water and condensing the vapor_, which of course forms
pure water. (b) water is forced down to _salt_ beds where it dissolves
the salt. The brine is then raised and evaporated, leaving the salt in
the evaporating pans.
=28. Absorption of Gases by Solids and Liquids.=--If a piece of
heated charcoal is placed in a test-tube containing ammonia gas,
inverted in mercury, the ammonia is seen to disappear, the mercury
rising to take its place. The ammonia has been absorbed by the
charcoal, the gas molecules clinging closely to the solid. The
charcoal being very porous presents a large surface to the action
of the gas.
This experiment indicates that attraction exists between gas molecules
and other molecules. Many porous substances have this power of absorbing
gases. We have all noticed that butter has its flavor affected by
substances placed near it.
That _liquids absorb gases_ is shown by slowly heating cold water in a
beaker. Small bubbles of air form on the sides and rise before the
boiling point is reached. Ammonia gas is readily absorbed in water, the
bubbles disappearing almost as soon as they escape into the water from
the end of the delivery tube. _Household ammonia_ is simply a solution
of ammonia gas in water. On warming the solution of ammonia the gas
begins to pass off; thus, warming a liquid tends to drive off any gas
dissolved in it.
_Soda water_ is made by forcing carbon dioxide gas into water under
strong pressure. When placed in a vessel open to the air the pressure is
lessened and part of the gas escapes. The dissolved gas gives the
characteristic taste to the beverage.
Important Topics
1. The solution of solids is increased by heating.
2. The solution of gases is decreased by heating.
3. Pressure increases the quantity of gas that can be dissolved in a
liquid.
4. The attraction (cohesion) of molecules of a dissolved solid for each
other is shown by crystallization.
Exercises
1. How do fish obtain oxygen for breathing?
2. Why does warming water enable it to dissolve more of a salt?
3. Why does warming water lessen the amount of a gas that will stay in
solution?
4. Will water absorb gases of strong odor? How do you know?
5. Name three solvents. Give a use for each.
6. What liquids usually contain gases in solution? Name some uses for
these dissolved gases.
7. What is the weight of a cubic meter of water?
8. Name three substances obtained by crystallization.
9. How is maple sugar obtained?
10. Name five crystalline substances.
(5) EVIDENCE OF MOLECULAR FORCES IN SOLIDS
=29. Differences between Solids and Gases.=--In studying gases, it is
seen that they behave as if they were composed of small particles in
rapid motion, continually striking and rebounding, and separating to
fill any space into which they are released. This action indicates that
there is practically no attractive force between such molecules.
Between the molecules of a solid, however, the forces of attraction are
strong, as is shown by the fact that a solid often requires a great
force to pull it apart; some, as steel and iron, show this property in a
superlative degree, a high-grade steel rod 1 cm. in diameter requiring
nearly 9 tons to pull it apart. Tests show that the breaking strengths
of such rods are directly proportional to their areas of cross-section.
That is, twice the area has twice the breaking strength.
[Illustration: FIG. 16.--Elasticity of bending.]
=30. Elasticity.=--Fully as important as a knowledge of the breaking
strengths of solids, is the knowledge of what happens when the forces
used are not great enough to break the rods or wires.
Take a wooden rod (as a meter stick) and clamp one end to the table
top, as in Fig. 16. At the other end hang a weight. Fasten a wire
to this end so that it projects out in front of a scale. Add
successively several equal weights and note the position of the
wire each time. Remove the weights in order, noting the positions
as before. The rod will probably return to the first position.
This simple experiment illustrates a characteristic of solids: that of
changing shape when force is applied and of returning to the original
shape when the force is removed. This property is called _elasticity_.
Tests of elasticity are made by subjecting wire of different materials
but of the same dimensions to the same tension. The one changing least
is said to have the greatest _elastic force_ or elasticity. If greater
forces are applied to the wire and then removed, one will finally be
found that will permanently stretch the wire so that it will not return
exactly to the former length. The wire has now passed its _elastic
limit_ and has been permanently stretched.
Just as there are great differences between the _elastic forces_ of
different substances, so there are great differences in the _limits of
elasticity_. In some substances the limit is reached with slight
distortion, while others are _perfectly elastic_ even when greatly
stretched. India rubber is an example of a body having _perfect_
elasticity through wide limits. Glass has great _elastic force_ but its
_limit_ of _elasticity_ is soon reached. Substances like India rubber
may be said to have great "_stretchability_," but little elastic force.
In physics, elasticity refers to the elastic force rather than to
ability to endure stretching.
=31. Kinds of Elasticity.=--_Elasticity may be shown in four ways_:
_compression_, _bending_ or _flexure_, _extension_ or _stretching_,
_twisting_ or _torsion_. The first is illustrated by squeezing a rubber
eraser, the second by an automobile spring, the third by the stretching
of a rubber band, the fourth by the twisting and untwisting of a string
by which a weight is suspended.
_There are two kinds of elasticity_: (1) elasticity of form or shape;
(2) elasticity of volume. Gases and liquids possess elasticity of
volume, but not of shape, while solids may have both kinds. Gases and
liquids are perfectly elastic because no matter how great pressure may
be applied, as soon as the pressure is removed they regain their former
volume. No solid possesses perfect elasticity, because sooner or later
the limit of elasticity will be reached.
=32. Hooke's Law.=[A]--On examining the successive movements of the end
of the rod in Art. 30, we find that they are approximately equal.
Carefully conducted experiments upon the elasticity of bodies have shown
that the changes in shape are _directly proportional_ to the forces
applied, provided that the limit of elasticity is not reached. This
relation, discovered by Robert Hooke, is sometimes expressed as follows:
"_Within the limits of perfect elasticity, all changes of size or shape
are directly proportional to the forces producing them._"
[A] A law is a statement of a constant mode of behavior. It is
often expressed in mathematical language.
=33. Molecular Forces and Molecular Motions.=--If a solid is compressed,
on releasing the pressure the body regains its former shape if it has
not been compressed too far. This indicates that at a given temperature
the "molecules of a solid tend to remain at a fixed distance from each
other, and resist any attempt to decrease or increase this distance."
This raises the question, Why does not the cohesion pull the molecules
tightly together so that compression would be impossible? The reason is
that heat affects the size of solid bodies. On lowering the temperature,
bodies do contract, for as soon as the temperature is lowered the
vibration of the molecule is lessened. On raising the temperature the
molecules are pushed farther apart.
The size of a body, then, is the result of a balance of opposing forces.
The attractive force between the molecules pulling them together is
_cohesion_, while the force which pushes them apart is due to the
motions of the molecules. Raising the temperature and thus increasing
the motion causes expansion; lowering the temperature decreases the
molecular motion and so causes contraction. If an outside force tries to
pull the body apart or to compress it this change of size is resisted by
either cohesion or molecular motion.
=34. Properties of Matter.=--Many differences in the physical properties
of solids are due to differences between the cohesive force of different
kinds of molecules. In some substances, the attraction is such that they
may be rolled out in very thin sheets. Gold is the best example of this,
sheets being formed 1/300,000 of an inch thick. This property is called
_malleability_. In other substances the cohesion permits it to be drawn
out into fine threads or wire. Glass and quartz are examples of this.
This property is called _ductility_. In some, the cohesion makes the
substance excessively _hard_, so that it is difficult to work or scratch
its surface. The diamond is the hardest substance known. Some substances
are _tough_, others _brittle_. These are tested by the ability to
withstand sudden shocks as the blow of a hammer.
Important Topics
1. Molecular forces in solids; (_a_) adhesion, (_b_) cohesion.
2. Elasticity, Hooke's Law.
3. Contraction on cooling.
4. Malleability, ductility, hardness, brittleness, etc.
Exercises
1. Give an illustration of Hooke's Law from your own experience.
2. What devices make use of it?
3. Do solids evaporate. Give reasons.
4. When iron is welded, is cohesion or adhesion acting?
5. When a tin basin is soldered, is cohesion or adhesion acting?
6. Sometimes a spring is made more elastic by _tempering_ and made soft
by _annealing_. Look up the two terms. How is each accomplished?
7. Review the definitions: solid, liquid, and gas. Why do these
definitions mean more to you now than formerly?
8. If a wire is stretched 0.3 cm. on applying 4 kg. of force, what force
will stretch it 0.75 cm? Explain.
9. How long will it take under ordinary conditions for a gas molecule to
cross a room? Give reasons for your answer.
10. What is meant by the elastic limit of a body?
11. Without reaching the elastic limit, if a beam is depressed 4 mm.
under a load of 60 kg., what will be the depression under a load of 400
kg.? Of 600 kg.?
12. Name three substances that possess elasticity of volume.
13. Give three examples of each; elasticity of (1) compression, (2)
stretching, (3) torsion, (4) flexure.
Review Outline: Introduction and Molecules
Physics; definition, topics considered, physical and chemical changes.
Science; hypothesis, theory, law. Knowledge; common, scientific.
Matter; three states, molecular theory. Mass, weight, volume.
Metric system; units, tables, equivalents, advantages.
Evidences of molecular motions; gases (3), liquids (5), solids (3).
Evidences of molecular forces; liquids (3), solids (many) special
properties such as: elasticity, tenacity, ductility, hardness, etc.
Hooke's law; applications.
CHAPTER III
MECHANICS OF LIQUIDS
(1) THE GRAVITY PRESSURE OF LIQUIDS
=35. Pressure of Liquids against Surfaces.=--The sight of a great ship,
perhaps built of iron and floating on water, causes one to wonder at the
force that supports it. This same force is noticed when one pushes a
light body, as a cork, under water. It is quite evident in such a case
that a force exists sufficient to overcome the weight of the cork so
that it tends to rise to the surface. Even the weight of our bodies is
so far supported by water that many persons can float.
[Illustration: FIG. 17.--Water forces the card against the chimney.]
The following experiment provides a means of testing this force:
If an empty can is pushed down into water, we feel at once the
force of the liquid acting against the object and tending to push
it upward. It may be noticed also that so long as the can is not
completely submerged the deeper the can is pushed into the water
the greater is the upward force exerted by the liquid.
We may test this action in various ways: a simple way is to take a
cylindrical lamp chimney, press a card against its lower end and
place it in the water in a vertical position. The force of the
water will hold the card firmly against the end of the chimney.
(See Fig. 17.) The amount of force may be tested by dropping shot
into the tube until the card drops off. At greater depths more shot
will be required, showing that the force of the water increases
with the depth. Or one may pour water into the chimney. It will
then be found that the card does not drop until the level of the
water inside the chimney is the same as on the outside. That is,
before the card will fall off, the water must stand as high within
the chimney as without no matter to what depth the lower end of the
chimney is thrust below the surface of the water.
=36. Law of Liquid Pressure.=--As there is twice as much water or shot
in the chimney when it is filled to a depth of 10 cm. as there is when
it is filled to a depth of 5 cm. the force of the water upward on the
bottom must be twice as great at a depth of 10 cm. as at a depth of 5
cm. Since this reasoning will hold good for a comparison of forces at
any two depths, we have the law: "_The pressure exerted by a liquid is
directly proportional to the depth_."
The amount of this force may be computed as follows: First, the card
stays on the end of the tube until the _weight_ of water from above
equals the force of the water from below, and second, the card remains
until the water is at the same _height_ inside the tube as it is
outside. Now if we find the weight of water at a given depth in the
tube, we can determine the force of the water from below. If for
instance the chimney has an area of cross-section of 12 sq. cm. and is
filled with water to a depth of 10 cm., the volume of the water
contained will be 120 ccm. This volume of water will weigh 120 g. This
represents then, not only the weight of the water in the tube, but also
the force of the water against the bottom. In a similar way one may
measure the force of water against any horizontal surface.
=37. Force and Pressure.=--We should now distinguish between _force_ and
_pressure_. Pressure refers to the force acting against _unit area_,
while force refers to the action against the whole surface. Thus for
example, the atmospheric _pressure_ is often given as 15 pounds to the
square inch or as one kilogram to the square centimeter. On the other
hand, the air may exert a _force_ of more than 300 pounds upon each side
of the hand of a man; or a large ship may be supported by the _force_ of
thousands of tons exerted by water against the bottom of the ship.
In the illustration, given in Art. 36, the upward _force_ of the water
against the end of the tube at a depth of 10 cm. is computed as 120
grams. The _pressure_ at the _same_ depth will be 10 grams per sq. cm.
What will be the pressure at a depth of 20 cm.? at a depth of 50 cm.? of
100 cm.? Compare these answers with the law of liquid pressure in Art.
36.
=38. Density.=--If other liquids, as alcohol, mercury, etc., were in the
jar, the chimney would need filling to the same level outside, with the
_same_ liquid, before the card would fall off. This brings in a factor
that was not considered before, _that of the mass[B] of a cubic
centimeter of the liquid_. This is called the _density_ of the liquid.
Alcohol has a density of 0.8 g. per cubic centimeter, mercury of 13.6 g.
per cubic centimeter, while water has a density of 1 g. per cubic
centimeter.
[B] The _mass_ of a body is the _amount of matter in it_, the
_weight_ is the _pull of the earth upon it_.
=39. Liquid Force against Any Surface.=--To find the force exerted by a
liquid against a surface we must take into consideration the area of the
surface, and the height and the =density= of the liquid above the
surface. The following law, and the formula representing it, which
concisely expresses the principle by which the force exerted by a liquid
against any surface may be computed, should be memorized:
_The force which a liquid exerts against any surface, equals the
area of the surface, times its average depth below the surface of
the liquid, times the weight of unit volume of the liquid._
Or, expressed by a formula, _F = Ahd_. In this formula, "F" stands for
_the force which a liquid exerts against any surface_, "A" _the area
of the surface_, "h," for _the average depth (or height) of the liquid
pressing on the surface_, and "d", for _the weight of unit volume of the
liquid_. This is the first illustration in this text, of the use of a
formula to represent a law. Observe how accurately and concisely the law
is expressed by the formula. When the formula is employed, however, we
should keep in mind the law expressed by it.
We must remember that a liquid presses not only downward and upward but
sideways as well, as we see when water spurts out of a hole in the side
of a vessel. Experiments have shown that at a point the pressure in a
fluid is the same in all directions, hence the rule given above may be
applied to the pressure of a liquid against the side of a tank, or boat,
or other object, provided we are accurate in determining the _average
depth of the liquid_; The following example illustrates the use of the
law.
_For Example_: If the English system is used, the area of the
surface should be expressed in square feet, the depth in feet and
the weight of the liquid in pounds per cubic foot. One cubic foot
of water weighs 62.4 lbs.
Suppose that a box 3 ft. square and 4 ft. deep is full of water.
What force will be exerted by the water against the bottom and a
side?
From the law given above, the force of a liquid against a surface
equals the product of the _area_ of the surface, the _depth_ of the
liquid and its weight per unit volume, or using the formula, _F =
Ahd_. To compute the downward force against the bottom we have the
area, 9, depth, 4, and the weight 62.4 lbs. per cubic foot. 9 × 4 ×
62.4 lbs. = 2246.4 lbs. To compute the force against a side, the
area is 12, the average depth of water on the side is 2, the weight
62.4, 12 × 2 × 62.4 lbs. = 1497.6 lbs.
Important Topics
1. Liquids exert pressure; the greater the depth the greater the
pressure.
2. Difference between force and pressure.
3. Rules for finding upward and horizontal force exerted by a liquid. _F
= Ahd._
4. Weight, mass, density.
Exercises
1. What is the density of water?
2. What force is pressing upward against the bottom of a flat boat, if
it is 60 ft. long, 15 ft. wide and sinks to a depth of 2 ft. in the
water? What is the weight of the boat?
3. If a loaded ship sinks in the water to an average depth of 20 ft.,
the area of the bottom being 6000 sq. ft., what is the upward force of
the water? What is the weight of the ship?
4. If this ship sinks only 10 ft. when empty, what is the weight of the
ship alone? What was the weight of the cargo in Problem 3?
5. What is the liquid force against one side of an aquarium 10 ft. long,
4 ft. deep and full of water?
6. What is the liquid force on one side of a liter cube full of water?
Full of alcohol? Full of mercury? What force is pressing on the bottom
in each case?
7. What depth of water will produce a pressure of 1 g. per square
centimeter? 10 g. per square centimeter? 1000 g. per square centimeter?
8. What depth of water will produce a pressure of 1 lb. per square inch?
10 lbs. per square inch? 100 lbs. per square inch?
9. What will be the force against a vertical dam-breast 30 meters long,
the depth of the water being 10 meters?
10. A trap door with an area of 100 sq. dcm. is set in the bottom of a
tank containing water 5 meters deep. What force does the water exert
against the trap door?
11. What is the force on the bottom of a conical tank, filled with
water, the bottom of which is 3 meters in diameter, the depth 1.5
meters?
12. If alcohol, density 0.8 were used in problem 11, what would be the
force? What would be the depth of alcohol to have the same force on the
bottom as in problem 11?
13. What is the pressure in pounds per square inch at a depth of 1 mile
in sea water, density 1.026 grams per cc.?
14. Find the force on the sides and bottom of a rectangular cistern
filled with water, 20 ft. long, 10 ft. wide, and 10 ft. deep?
15. Find the force on the bottom of a water tank 14 ft. in diameter when
the water is 15 ft. deep, when full of water.
16. Find the force on one side of a cistern 8 ft. deep and 10 ft.
square, when full of water.
17. Find the force on a vertical dam 300 ft. long and 10 ft. high, when
full of water.
18. Find the pressure at the bottom of the dam in question 17.
19. Why are dams made thicker at the bottom than at the top?
20. A ship draws 26 ft. of water, _i.e._, its keel is 26 ft. under
water. What is the liquid force against a square foot surface of the
keel? Find the pressure on the bottom.
(2) TRANSMISSION OF LIQUID PRESSURE
=40. Pascal's Principle.=--Liquids exert pressure not only due to their
own weight, but when confined, may be made to transmit pressure to
considerable distances. This is a matter of common knowledge wherever a
system of waterworks with connections to houses is found, as in cities.
The transmission of liquid pressure has a number of important
applications. The principle underlying each of these was first
discovered by Pascal, a French scientist of the seventeenth century.
Pascal's Principle, as it is called, may be illustrated as follows:
Suppose a vessel of the shape shown in Fig. 18, the upper part of
which we may assume has an area of 1 sq. cm., is filled with water
up to the level _AB_. A pressure will be exerted upon each square
centimeter of area depending upon the depth. Suppose that the
height of _AB_ above _CD_ is 10 cm., then the force upon 1 sq. cm.
of _CD_ is 10 g., or if the area of _CD_ is 16 sq. cm., it receives
a force of 160 g.
[Illustration: FIG. 18.--The force increases with the depth.]
If now a cubic centimeter of water be poured upon _AB_ it will
raise the level 1 cm., or the head of water exerting pressure upon
_CD_ becomes 11 cm., or the total force in _CD_ is 16×11 g.,
_i.e._, each square centimeter of _CD_ receives an additional force
of 1 g. _Hence the force exerted on a unit area at_ _AB_ _is
transmitted to every unit area within the vessel._
The usual form in which this law is expressed is as follows: _Pressure
applied to any part of a confined liquid is transmitted unchanged, in
all directions, and adds the same force to all equal surfaces in contact
with the liquid_.
[Illustration: FIG. 19.--The force is proportional to the area.]
The importance of this principle, as Pascal himself pointed out, lies in
the fact that by its aid we are able to exert a great force upon a large
area by applying a small force upon a small area of a confined liquid,
both areas being in contact with the same liquid. Thus in Fig. 19 if the
area of the surface _CD_ is 2000 times the area of the surface _AB_,
then 1 lb. applied to the liquid on _AB_ will exert or sustain a force
of 2000 lbs. on _CD_.
=41. Hydraulic Press.=--An important application of Pascal's principle
is the _hydraulic press_. See Fig. 20. It is used for many purposes
where great force is required, as in pressing paper or cloth, extracting
oil from seeds, lifting heavy objects, etc. Many high school pupils have
been seated in a _hydraulic chair_ used by a dentist or barber. This
chair is a modified hydraulic press.
[Illustration: FIG. 20.--Cross-section of a hydraulic press.]
The hydraulic press contains two movable pistons, _P_ and _p_ (see Fig.
20). The larger of these, _P_, has a cross-sectional area that may be
100 or 1000 times that of the smaller. The smaller one is moved up and
down by a lever; on each upstroke, liquid is drawn in from a reservoir,
while each down-stroke forces some of the liquid into the space about
the large piston. Valves at _V_ and _V´_ prevent the return of the
liquid. If the area of _P_ is 1,000 times that of _p_, then the force
exerted by _P_ is 1000 times the force employed in moving _p_. On the
other hand, since the liquid moved by the small piston is distributed
over the area of the large one, the latter will move only 1/1000 as far
as does the small piston. The relation between the motions of the two
pistons and the forces exerted by them may be stated concisely as
follows: The _motions_ of the two _pistons_ of the hydraulic press are
inversely proportional to the forces exerted by them. The
_cross-sectional areas_ of the two pistons are, on the other hand,
directly proportional to the forces exerted by them.
An application of Pascal's principle often employed in cities is the
hydraulic elevator. In this device a long plunger or piston extends
downward from the elevator car into a cylinder sunk into the earth,
sometimes to a depth of 300 ft. Water forced into this cylinder pushes
the piston upward and when the water is released from the cylinder the
piston descends.
Fig. 21 represents another form of hydraulic elevator, where the
cylinder and piston are at one side of the elevator shaft. In this
type, to raise the elevator, water is admitted to the cylinder
pushing the piston downward.
=42. Artesian Wells.=--Sometimes a porous stratum containing water in
the earth's crust is inclined. Then if there are impervious strata (see
Fig. 22), both above and below the water-bearing one, and the latter
comes to the surface so that rain may fill it, a well sunk to the
water-bearing stratum at a point where it is below the surface will
usually give an artesian well, that is, one in which the water rises to
or above the surface. Many are found in the United States.
[Illustration: FIG. 21.--A hydraulic freight elevator.]
[Illustration: FIG. 22.--Conditions producing an artesian well.]
[Illustration: FIG. 23.--A standpipe.]
=43. Standpipes and Air Cushions.=--Many who have lived in cities where
water is pumped into houses under pressure know that the water pressure
is changed when several faucets are opened at the same time. Again, if
several persons are using a hose for sprinkling, the pressure may be
lessened so as to be insufficient to force the water above the first
floor. In order to allow for these changes some flexibility or spring
must be introduced somewhere into the water-pipe system. Water is
nearly incompressible and if no means were employed to take care of the
pressure changes, the sudden stopping and starting of the flow would
cause serious jars and start leaks in the pipes. Two common devices for
controlling sudden changes in the water pressure are the _standpipe_ and
the _air cushion_.
The _standpipe_ is simply a large vertical tube connected to the
water mains from which and into which water readily flows. When
many faucets are opened the water lowers; when most faucets are
closed the water rises, giving a simple automatic control of the
surplus water and a supply of water for a short time during a
shut-down of the pumps. Standpipes are often used in towns and
small cities. Fig. 23 represents the standpipe at Jerome, Idaho.
The _air cushion_ (Fig. 24) is a metal pipe or dome filled with air
attached to a water pipe where sudden changes in pressure are to be
controlled. At many faucets in a city water system such an air
cushion is employed. It contains air; this, unlike water, is easily
compressible and the confined air when the tap is suddenly closed
receives and checks gradually the rush of water in the pipe. Even
with an air cushion, the "pound" of the water in the pipe when a
tap is suddenly closed is often heard. If air cushions were not
provided, the "water hammer" would frequently crack or break the
pipes.
[Illustration: FIG. 24.--The short pipe above the faucet contains air
forming an air cushion.]
Important Topics
1. Pascal's law.
2. Hydraulic press.
3. Artesian wells.
4. Standpipes and air cushions.
Exercises
1. Where have you _seen_ an air cushion? Describe it and its use.
2. Where have you _seen_ an hydraulic press? Why and how used?
3. Where have you _seen_ hydraulic elevators? What moves them?
4. Where do you know of liquids under pressure? Three examples.
5. What is the pressure in water at a depth of 1500 cm. Express in grams
per square centimeter and in kilograms per square centimeter.
6. What head[C] of water is required to give a pressure of 200 g. per
square centimeter? 2 kg. per square centimeter?
[C] "Head" is a term used to express the vertical height of water
in pipes.
7. What _pressure_ will be produced by a "head" of water of 20 meters?
8. If 1728 cu. in. of water are placed in a vertical tube 1 sq. in. in
cross section to what height would the water rise? It would give how
many feet of _head_?
9. What would the water in problem 8 weigh? What pressure would it
produce at the bottom, in pounds per square inch? From this, compute how
many feet of "head" of water will produce a pressure of 1 lb. per square
inch.
10. Using the result in problem 9, what "head" of water will produce a
pressure of 10 lbs. per square inch? 100 lbs. per square inch?
11. From the result in 9, 100 ft. of "head" of water will produce what
pressure? 1000 ft. of "head?"
12. If the diameter of the pump piston in a hydraulic press is 2 cm. and
that of the press piston 50 cm. what will be the force against the
latter if the former is pushed down with a force of 40 kg.?
(3) ARCHIMEDES' PRINCIPLE
=44. A Body Supported by a Liquid.=--Among the applications of the force
exerted by a liquid upon a surface, Archimedes' Principle is one of the
most important.
Most persons have noted that a body placed in water is partly or wholly
supported by the force of the water upon it. A stone held by a cord and
lowered into water is felt to have a part of its weight supported,
while a piece of cork or wood is wholly supported and floats.
The human body is almost entirely supported in water, in fact, many
people can easily float in water. It was the consideration of this fact
that led the Greek philosopher Archimedes to discover and state the
principle that describes the supporting of a body in a liquid.
[Illustration: FIG. 25.--Theoretical proof of Archimedes' principle.]
=45. Archimedes' Principle.=--"_A body immersed in a liquid is pushed up
by a force equal to the weight of the liquid that it displaces._" The
proof for this law is simply demonstrated. Suppose a cube, _abcd_, is
immersed in water (Fig. 25). The upward force on _cd_ is equal to the
weight of a column of water equal to _cdef_. (See Art. 39.) The downward
force upon the top of the cube is equal to the weight of the column of
water _abef_. Then the net upward force upon the cube, that is, the
upward force upon the bottom less the downward force upon the top, or
the buoyant force exerted by the liquid is exactly equal to the weight
of the displaced water _abcd_.
=46. Law of Floating Bodies.=--This same reasoning may be applied to any
liquid and to any body immersed to any depth below the surface of the
liquid. If the body weighs more than the displaced liquid it will sink.
If it weighs less than the displaced liquid it will float or rise in the
water. A block of wood rises out of the water in which it floats until
its own weight just equals the weight of the water it displaces. From
this we have the law of floating bodies.
_A floating body displaces its own weight of the liquid in which it
floats._
[Illustration: FIG. 26.--A floating body displaces its own weight of
water.]
To test the law of floating bodies, take a rod of light wood 1 cm.
square and 30 cm. long (Fig. 26). Bore out one end and fill the opening
with lead and seal with paraffin so that the rod will float vertically
when placed in water. Mark upon one side of the rod a centimeter scale,
and dip the rod in hot paraffin to make it waterproof. Now find the
weight of the stick in grams and note the depth to which it sinks in
water in centimeters. Compute the weight of the displaced water. It will
equal the weight of the rod.
=47. Applications of Archimedes' Principle.= There are numerous
applications of Archimedes' Principle and the law of floating bodies.
=(a) To Find the Weight of a Floating Body: Problem.=--A boat 20
ft. long and with an average width of 6 ft. sinks to an average
depth of 3 ft. in the water. Find the weight of the boat. What
weight of cargo will sink it to an average depth of 5 ft.?
=Solution.=--The volume of the water displaced is 20 × 6 × 3 cu.
ft. = 360 cu. ft. Since 1 cu. ft. of water weighs 62.4 lbs., 360 ×
62.4 lbs. = 22,464 lbs., the weight of water displaced. By the law
of floating bodies this is equal to the weight of the boat. When
loaded the volume of water displaced is 20 ft. × 6 × 5 ft. which
equal 600 cu. ft. 600 × 62.4 lbs. = 37,440 lbs. This is the weight
of the water displaced when loaded. 37,440 lbs. - 22,464 lbs. =
14,976 lbs., the weight of the cargo.
=(b) To Find the Volume of an Immersed Solid: Problem.=--A stone
weighs 187.2 lbs. in air and appears to weigh 124.8 lbs. in water.
What is its volume?
=Solution.=--187.2 lbs. - 124.8 lbs. = 62.4 lbs., the buoyant force
of the water. By Archimedes' Principle, this equals the weight of
the displaced water which has a volume of 1 cu. ft. which is
therefore the volume of the stone.
=(c) To Find the Density of a Body:= The density of a body is
defined as the mass of unit volume.
We can easily find the mass of a body by weighing it, but the
volume is often impossible to obtain by measurements, especially of
irregular solids.
Archimedes' Principle, however, provides a method of finding the volume
of a body accurately by weighing it first in air and then in water (Fig.
27), the apparent loss in weight being equal to the weight of the
displaced water. One needs only to find the volume of water having the
same weight as the loss of weight to find the volume of the body.
If the metric system is used, 1 ccm. of water weighs 1 g., and the
volume is numerically the same as the loss of weight.
[Illustration: FIG. 27.--A method of weighing a body under water.]
Important Topics
1. Archimedes' Principle.
2. Law of floating bodies.
3. The applications of Archimedes' Principle are to determine (a) the
weight of a floating body; (b) the volume of an immersed solid, and (c)
the density of a body.
Exercises
1. Look up the story of Archimedes and the crown. Write a brief account
of it.
2. Why is it easier for a fat man to float in water than for a lean
one?
3. A fish weighing 1 lb. is placed in a pail full of water. Will the
pail and contents weigh more than before adding the fish? Why?
4. Why can a large stone be lifted more easily while under water than
when on the land?
5. Why does the air bubble in a spirit level move as one end of the
instrument is raised or lowered?
6. Why does a dead fish always float?
7. A ship is built for use in fresh water. What will be the effect on
its water line when passing into the ocean?
8. Why can small bugs walk on water while large animals cannot?
9. If an object weighing 62.4 lbs. just floats in water, what weight of
water does it displace? What volume of water is displaced? What is the
volume of the body?
10. What is the volume of a man who just floats in water if he weighs
124.8 lbs.? If he weighs 187.2 lbs.?
11. An object weighing 500 g. just floats in water. What is its volume?
How much water does a floating block of wood displace if it weighs 125
lbs.? 125 g.? 2 kg.? 2000 kg.?
12. A flat boat 10 × 40 ft. in size will sink how much in the water when
10 horses each weighing 1250 lbs. are placed on board?
13. A ship 900 ft. long and 80 ft. average width sinks to an average
depth of 25 ft. when empty and 40 ft. when loaded. What is the weight of
the ship and of its load?
14. Will a 1000 cc. block sink or float in water if it weighs 800 g.? If
it weighs 1200 g.? Explain.
15. If a 1000 cc. block of metal weighing 1200 g. is placed in the water
in mid ocean what will become of it?
16. Prove Archimedes' Principle by use of the principles of liquid
pressure.
17. An irregular stone, density 2.5 g. per ccm. displaces 2 cu. ft. of
water. What is its weight? Its apparent weight in water?
18. Will the depth to which a vessel sinks in water change as she sails
from Lake Ontario into the Atlantic Ocean? Why?
19. If the density of sea water is 1.0269 g. per cubic centimeter and
that of ice 0.918 g. per ccm., what portion of an iceberg is above
water?
20. In drawing water from a well by means of a bucket, why is less force
used when it is under water than when entirely above?
21. A stone which weighs 300 lbs. can be lifted under water with a force
of 150 lbs. What is the volume of the stone?
22. The average density of the human body is 1.07 grams per c.c. How
much water will a man who weighs 150 lbs. displace when diving? How much
when floating?
(4) DENSITY AND SPECIFIC GRAVITY
=48. Density.=--The density of a substance is often used as a test of
its purity. Archimedes in testing King Hiero's crown to find out if it
were made of pure gold determined first its density. It is by such tests
that the purity of milk, of alcohol, of gold, and a great variety of
substances is often determined.
Knowledge of methods of finding density is of value to everyone and
should be included in the education of every student. _The density of a
substance is the mass of unit volume of the substance._ In the metric
system, for example, the density of a substance is the mass in grams per
1 ccm. Taking water, 1 ccm. weighs 1 gr. or its density is therefore 1
g. to the cubic centimeter. A cubic centimeter of aluminium weighs 2.7
g. Its density therefore is 2.7 g. per ccm.
=49. Specific Gravity.=--_Specific gravity is the ratio of the weight of
any volume of a substance to the weight of an equal volume of water._
Its meaning is not quite the same as that of density, since specific
gravity is always a _ratio_, _i.e._, an _abstract_ number, as 2.7.
Density of a substance is a _concrete_ number, as 2.7 grams per ccm. In
the metric system the density of water is one gram per cubic centimeter,
therefore we have:
Density (g. per ccm.) = (numerically) specific gravity.
In the English system, the density of water is 62.4 pounds per cubic
foot, therefore in this system we have:
Density (lbs. per cu. ft.) = (numerically) 62.4 × sp. gr.
=50. Methods for Finding Density and Specific Gravity=
=(a) Regular Solids.=--Solids of regular shapes such as cubes, spheres,
etc., whose volumes may be readily found by measurement, may be weighed.
The mass divided by the volume gives the density, or _D = Mμ/v_.
=(b) Irregular Solids.=--with these the volume cannot be found by
measurement but may be obtained by Archimedes' Principle. Weigh the
solid first in the air and then in water. The apparent loss of weight
equals the weight of the equal volume of water displaced. From this the
volume may be found. And then the
density equals mass/volume; the specific gravity =
wt. in air / wt. of equal volume of water = wt. in air / ((wt. in air) -
(wt. in water))
mass
density equals ------; the specific gravity =
volume
wt. in air wt. in air
---------------------------- = -----------------------------
wt. of equal volume of water ((wt. in air) - (wt. in water))
=(c) Solids Lighter than Water.=--This will require a sinker to hold the
body under water. Weigh the solid in air (_w_). Weigh the sinker in
water (_s_). Attach the sinker to the solid and weigh both in water
(w´). The specific gravity equals
(wt. of solid in air)/(loss in wt. of solid in water) or _w/((w +
s) - w´)_
wt. of solid in air _w_
--------------------------- or ------------
loss of wt. of solid in water (_w + s_) - _w´_
The apparent loss of weight of the solid is equal to the sum of its
weight in air plus the weight of the sinker in water, less the combined
weight of both in water.
=(d) The Density of a Liquid by a Hydrometer.=--One may also easily find
the density of any liquid by Archimedes' Principle. If one takes the rod
described in Art. 46, and places it in water, the number of cubic
centimeters of water it displaces indicates its weight in grams. On
placing the rod in another liquid in which it floats, it will of course
displace its own weight and the height to which the liquid rises on the
scale gives the volume. By dividing the _weight_ of the rod as shown by
its position in _water_ by the _volume_ of the _liquid_ displaced we
obtain the density of the liquid. Commercial hydrometers for testing the
density of milk, alcohol and other liquids are made of glass of the form
shown in Fig. 28. The long narrow stem permits small differences in
volume to be noticed, hence they are more accurate than the rod
described in the preceding paragraph. For convenience this rod contains
a paper scale, so that when the height of the liquid on the stem is
noted, the density is read at once.
[Illustration: FIG. 28.--A hydrometer used to find the density of a
liquid.]
=Density of Liquids by Loss of Weight.= Weigh a piece of glass in air
(_W_{a}_), in water (_W_{w}_), and in the liquid to be tested (_W_{l}_).
Then (_W_{a}_ - _W_{w}_)gives the weight of the water displaced.
And (_W_{a}_ - _W_{l}_) gives the weight of the liquid displaced.
Hence, (_W_{a}_ - _W_{l}_)/(_W_{a}_ - _W_{w}_) equals the specific
gravity of the liquid.
Important Topics
1. Definitions of density and specific gravity.
2. Methods of finding density: (a) regular solids; (b) irregular solids;
(c) solids lighter than water; (d) liquids by hydrometer; (e) liquids by
loss of weight.
Exercises
_Note._--Consider that 1 cu. ft. of water weighs 62.4 lbs. Consider that
1 ccm. of water weighs 1 g.
1. What is meant by the statement that a block of wood has a specific
gravity of 0.6?
2. Considering that the density of the human body is the same as that of
water, what is the volume of a 125-lb. boy? Of a 250-lb. man? Of a
62.4-lb. boy? What is the volume of your body?
3. How is the weight of large ships found? Give an example.
4. Mention three cases where determinations of density are important.
5. A body weighs 40 g. in air, 15 g. in water, 5 g. in an acid. Find (a)
the density of the body; (b) its volume; (c) density of the acid.
6. If the specific gravity of a horse is 1, what is the volume of a
horse weighing 500 kg.? Of one weighing 1248 lbs.?
7. A weighted wooden box sinks to a depth of 20 cm. in water and 24 cm.
in alcohol, and to a depth of 18 cm. in brine. What is the density of
the alcohol and of the brine?
8. A glass stopper weighs in the air 25 g., in water 15 g., in oil 18 g.
Find the density and volume of the stopper. Find the density of the oil.
9. What would a cubic foot of wood weigh if the specific gravity were
0.5.?
10. The specific gravity of aluminum is 2.7. Find the weight of a cubic
foot of it.
11. A block of wood weighs 40 g. A piece of lead appears to weigh 70 g.
in water. Both together appear to weigh 60 g. in water. Find the density
of the wood.
12. A stone weighs 30 g. in air, 22 g. in water, and 20 g. in salt
water. Find the density of the salt water.
13. Will iron sink in mercury? Why?
14. A submarine boat weighing 200 tons must have what volume in order to
float?
15. Find the weight of 2 cu. ft. of copper from its density.
16. What is the weight in water of a mass whose specific gravity is 3.3
and whose weight is 50 kg.?
17. A block of granite weighs 1656 lbs.; its volume is 10 cu. ft., what
is its density?
18. If the specific gravity of hard coal is 1.75 how would you determine
how many tons of coal a bin would hold?
19. A hollow copper ball weighs 2 kg. What must be its volume to enable
it to just float in water?
20. A mass having a volume of 100 ccm. and a specific gravity of 2.67 is
fastened to 200 ccm. of wood, specific gravity 0.55. What will the
combination weigh in water?
21. A block weighing 4 oz. in air is tied to a sinker which appears to
weigh 14 oz. in water. Both together appear to weigh 6 oz. in water.
What is the specific gravity of the block?
CHAPTER IV
MECHANICS OF GASES
(1) WEIGHT AND PRESSURE OF THE AIR
=51. Weight of Air.=--It is said that savages are unaware of the
presence of _air_. They feel the _wind_ and hear and see it moving the
leaves and branches of the trees, but of air itself they have little
conception.
To ordinary observers, it seems to have no weight, and to offer little
resistance to bodies passing through it. That it has weight may be
readily shown as follows: (See Fig. 29.) If a hollow metal sphere, or a
glass flask, provided with tube and stopcock, be weighed when the
stopcock is open, and then after the air has been exhausted from it by
an air pump, a definite loss of weight is noticeable.
[Illustration: FIG. 29.--Proof that air has weight.]
If the volume of the sphere is known and it is well exhausted of air, a
fair approximation of the weight of air may be obtained. Under
"_standard conditions_," which means _at the freezing temperature_ and a
barometric pressure of 76 cm., a liter of air weighs 1.293 g. while 12
cu. ft. of air weigh approximately 1 lb.
=52. Pressure of Air.=--Since air has weight it may be supposed to exert
pressure like a liquid. That it does so may be shown in a variety of
ways.
If a plunger fitting tightly in a glass cylinder be drawn upward, while
the lower end of the tube is under water, the water will rise in the
tube (Fig. 30). The common explanation of this is that the water rises
because of "suction." The philosophers of the ancient Greeks explained
it by saying that "nature abhors a vacuum," and therefore the water
rises. Neither explanation is correct. It was found in 1640 that water
would not rise in a pump more than 32 ft. despite the fact that a vacuum
was maintained above the water. Galileo was applied to for an
explanation. He said, "evidently nature's horror of a vacuum does not
extend above 32 ft." Galileo began tests upon "the power of a vacuum"
but dying left his pupil Torricelli to continue the experiment.
Torricelli reasoned that if water would rise 32 ft., then mercury, which
is 13.6 times as dense as water, would rise about 1/13 as much. To test
this, he performed the following famous experiment.
[Illustration: FIG. 30.--Air pressure forces the liquid up the tube.]
=53. Torricelli's Experiment (1643).=--Take a glass tube about 3 ft.
long, sealed at one end, and fill it with mercury. Close the end with
the finger and invert, placing the end closed by the finger under
mercury in a dish (Fig. 31). Remove the finger and the mercury sinks
until the top of the mercury is about 30 in. above the level of the
mercury in the dish. Torricelli concluded that the rise of liquids in
exhausted tubes is due to the pressure of the atmosphere acting on the
surface of the mercury in the dish.
To test this, place the tube with its mercury upon the plate of an air
pump and place a tubulated bell jar over the apparatus so that the tube
projects through a tightly fitting stopper. (See Fig. 32.) If the air
pressure is the cause of the rise of mercury in the tube, on removing
the air from the bell jar the mercury should fall in the tube. This is
seen to happen as soon as the pump is started. It is difficult to remove
all the air from the receiver so the mercury rarely falls to the same
level in the tube as in the dish. A small tube containing mercury is
often attached to air pumps to indicate the degree of exhaustion. Such
tubes are called _manometers_.
[Illustration: FIG. 31.--Torricelli's experiment.]
[Illustration: FIG. 32.--The mercury drops as the air is removed.]
=54. The Amount of Atmospheric Pressure.=--Torricelli's experiment
enables us to compute readily the pressure of the atmosphere, since it
is the atmospheric pressure that balances the column of mercury in the
tube. By Pascal's Law, the pressure of the atmosphere on the surface of
the mercury in the dish is transmitted as an exactly equal pressure on
the mercury column in the tube at the same level as the mercury
outside.
This pressure, due to the air, must balance the weight of the column of
mercury in the tube. It therefore equals the weight of the column of
mercury of unit cross-section. The average height of the column of
mercury at sea-level is 76 cm. Since the weight of 1 cc. of mercury is
13.6 grams, the pressure inside the tube at the level of the surface of
the mercury in the dish is equal to 1 × 76 × 13.6 or 1033.6 g. per
square centimeter. Therefore the _atmospheric pressure_ on the surface
of the mercury in the dish is 1033.6 g. per square centimeter,
approximately _1 kg. per square centimeter or 15 lbs. per square inch_.
=55. Pascal's Experiment.=--Pascal tested in another way the action of
atmospheric pressure upon the column of mercury by requesting his
brother-in-law, Perrier, who lived near a mountain, to try the
experiment on its top. Perrier found that on ascending 1000 meters the
mercury fell 8 cm. in the tube. Travelers, surveyors, and aviators
frequently determine the altitude above sea-level by reading the
barometer, an ascent of 11 meters giving a fall of about 1 mm. in the
mercury column, or 0.1 in. for every 90 ft. of ascent.
[Illustration: FIG. 33.--A standard barometer.]
_56. The Barometer._--The modern barometer (Fig. 33), consists of a
Torricellian tube properly mounted. Reading a barometer consists in
accurately reading the height of the mercury column. This height varies
from 75 to 76.5 cm. or 29 to 30 in. in localities not far from the
sea-level. The atmospheric pressure varies because of disturbances in
the atmosphere. It is found that these disturbances of the atmosphere
pass across the country from west to east in a somewhat regular manner,
hence a series of readings of the barometer may give reliable
information of the movement of these disturbances and so assist in
forecasting the weather. The weather Bureau has observations taken at
the same moment at various stations over the country. These observations
form the basis for the daily forecast of the weather.
[Illustration: FIG. 34.--An aneroid barometer]
Another form of barometer in common use is the _Aneroid Barometer_
(Fig. 34). Its essential parts are a cylindrical air-tight box with
an elastic corrugated cover. Inside the box is a partial vacuum.
This makes the cover very sensitive to slight changes of pressure.
The motion of the top of the box is conveyed by a series of levers
to an indicating hand which moves over a dial. This barometer can
be made so sensitive as to indicate the change of air pressure from
a table top to the floor. It is much used by travelers, explorers,
surveying parties and aviators, since the mercurial barometer is
inconvenient to carry.
Important Topics
1. _Weight_ and _Pressure_ of air in English and metric units. How
shown. Evidences.
2. Work of Galileo, Torricelli, and Perrier.
3. Barometer: construction, action, mercurial, aneroid.
[Illustration: FIG. 35.--Air pressure keeps the water In the tumbler.]
[Illustration: FIG. 36.--Cross-section of a modern drinking fountain.]
Exercises
1. Do you think Archimedes' Principle applies to the air? Does Pascal's
Law? Why?
2. Find the downward pressure of the mercury in a barometer tube if the
cross-section is 1 sq. cm. and the height 75 cm. at the level of the
mercury surface in contact with the air. (The density of mercury is 13.6
grams per cc.)
3. What is the weight of the air in a room if it is 10 × 8 × 4 meters?
4. What weight of air is in a room 10 × 15 × 10 ft.?
5. When smoke rises in a straight line from chimneys, is it an
indication of a high or low barometric pressure? Why?
6. Why does a tumbler filled with water and inverted in a dish with its
rim under water remain full?
7. If the barometer tube is inclined the mercury remains at the same
horizontal level. How can this be explained?
8. When the mercurial barometer stands at 76 cm., how high would a water
barometer stand? Explain.
9. Explain why it is possible for one to suck soda water through a tube?
10. Fill a tumbler with water. Place a sheet of paper over the top and
invert. The paper clings to the tumbler and prevents the water from
escaping. Explain. (See Fig. 35.)
11. Why must a kerosene oil can have two openings in order to allow the
oil to flow freely?
12. Explain the action of the modern drinking fountain (Fig. 36).
(2) COMPRESSIBILITY AND EXPANSIBILITY OF THE AIR
=57. Effect of Pressure on Liquids and Gases.=--Both classes of fluids,
liquids and gases, have many characteristics in common. Both are
composed of molecules that move freely; hence both _flow_. At any point
within a fluid the _pressure is the same in all directions_. Archimedes'
Principle applies, therefore, to both liquids and gases.
We now come to an important _difference_ between liquids and gases.
_Liquids_ are _practically incompressible_. "So much so, that if water
is subjected to a pressure of 3000 kg. per sq. cm., its volume is
reduced only about one-tenth." Gases show a very different behavior from
liquids on being subjected to pressure. They may readily be compressed
to a small fraction of their volume as is noticed on inflating a
pneumatic tire. A gas has also the _ability to spring back_ to a larger
volume as soon as the pressure is released, as when a cork is driven
from a pop gun. Not only is compressed air able to expand, but air
under ordinary conditions will expand if it is released in a space where
the pressure is less.
Hollow bodies, animals and plants, are not crushed by atmospheric
pressure, because the air and gases contained within exert as much force
outward as the air exerts inward.
=58. Boyle's Law.=--The relation between the volume and pressure of a
gas was first investigated by Robert Boyle in the seventeenth century.
The experiment by which he first discovered the law or the relation
between the volume and the pressure of a gas is briefly described as
follows:
[Illustration: FIG. 37 _a_.
FIG. 37 _b_.
FIGS. 37 _a_ AND 37 _b_.--Boyle's law apparatus.]
A glass tube is bent in the form of the capital letter J, the short
arm being closed. A little mercury is poured in to cover the bend.
(See Fig. 37 _a_.) Since the mercury is at the _same level in both
arms_, the pressure in (_A_) is the same as in (_B_). Mercury is
now poured into (_A_) until it stands in the long tube at a height
above that in (B) which is equal to the height of the mercury
column of the barometer. (See Fig. 37 _b_.) The air in (_BC_) is
now under a pressure of two atmospheres (one atmosphere is due to
the mercury column). On measurement the air in (_BC_) will be found
to have just one-half of its original volume.
Thus doubling the pressure to which a gas is subjected reduces its
volume to one-half. Tripling the pressure, reduces the volume to
one-third and so on.
Careful experiments reveal the following law: _The volume of a given
mass of gas at constant temperature is inversely proportional to the
pressure to which it is subjected_.
This law is often expressed mathematically. _P/P´ = V´/V_, or _PV =
P´V´_. Since doubling the pressure reduces the volume one-half, it
doubles the density. Tripling the pressure triples the density. We
therefore have _P/P´ = D/D´_ or the density of a gas directly
proportional to its pressure.
[Illustration: FIG. 38.--Height and density of the air.]
=59. Height of the Atmosphere.=--From its properties of _compression_
and _expansion_, the air varies in density and pressure as one ascends
in it. At a height of 3 miles the pressure is reduced to about one-half.
This is an indication that one-half of the air is below this level.
Balloonists have gone to a height of 7 miles, Glaser and Coxwell in
England in 1862 and Berson in France in 1901. The atmosphere has been
explored to a height of 30,500 meters (18.95 miles) by sending up
self-registering barometers in small balloons which burst at great
altitudes. A parachute protects the instruments from breakage from too
rapid fall. This height of 30,500 meters was reached by a balloon sent
up by William R. Blair, at Huron, South Dakota, September 1, 1910.
At a height of 35 miles, the density is estimated at 1/30,000 of its
value at sea-level. (See Fig. 38.) It is believed that some rarefied air
exists for a considerable distance above this point, some estimates
placing the extent at 100 miles, and others from 200 to 500 miles.
Evidences of some air at such heights are shown by: (a) the height at
which meteors first appear, (b) the height of the Aurora Borealis, and
(c), the distance that the sun is below the horizon when the last traces
of color disappear from the sky in the evening.
Although the exact limits of the atmosphere are unknown, the weight of a
column of air 1 sq. cm. in cross-section, and extending _upward as high
as the atmosphere_, may be accurately computed. For this column of air
exactly balances the column of mercury in the tube of the barometer.
Below sea-level, the air increases rapidly in density and it is
estimated that at a depth of 35 miles, the density of the air would be a
thousand times that at the earth's surface, or more than that of water.
Important Topics
1. Evidence of compressibility of gases and incompressibility of
liquids.
2. Boyle's Law. Proof, applications.
3. Extent of the atmosphere--three evidences.
Exercises
1. Mention three illustrations of the compressibility and expansibility
of air that you know from your own experience.
2. Increasing the pressure increases the amount of a gas that will be
absorbed by a liquid? Explain this. Have you ever observed this fact?
Where?
3. If a toy balloon containing 2000 ccm. of gas at the earth's surface
where the barometer reading is 76 cm., rises to an elevation where the
barometer reads 54 cm., the balloon will tend to expand to what volume?
Explain. Will it attain this volume?
4. If a gas is compressed, it changes in temperature. How do you explain
this?
5. What change in temperature will occur when compressed air is allowed
to expand? Explain.
6. Air blowing up a mountain side has its pressure lessened as it
approaches the top. How will this affect the temperature? Why? What may
result from this change in temperature? Explain.
7. To what pressure must 500 ccm. of air be subjected to compress it to
300 ccm. the barometer reading at first being 75 cm. Explain.
[Illustration: FIG. 39.--The air pump.]
(3) PNEUMATIC APPLIANCES
=60. The Air Pump.=--The air pump is used to remove air or other gases
from a closed vessel. It was invented about 1650 by Otto Von Guericke,
burgomaster of Magdeburg, Germany. One form of air pump is shown in
Fig. 39. _C_ is a cylinder within which slides a tightly fitting piston.
_R_ is the vessel from which the air is to be exhausted. _r_ and _u_ are
valves opening upward. The action of the pump is as follows:
On pushing the piston down, the air in _C_ is compressed. This opens
valve _r_ allowing the confined air to escape above the piston. The
piston is then raised making the space in _C_ a partial vacuum. The
pressure in _R_ now being greater than in _C_, _u_ is pushed up and the
air from _R_ rushes into _C_, until the pressure is equalized. On
pushing down the piston again, valve _u_ closes and the process is
repeated until the pressure in _R_ is no longer able to raise the valve
_u_. Some air pumps are so constructed that the valves are opened and
closed automatically by the movement of the piston. With these pumps a
higher degree of rarefaction can be obtained.
Air is often partially exhausted from receivers or vessels by the use of
a filter pump or _aspirator_. A stream of water flowing through a
constriction causes a reduced pressure, draws in air and carries it
away, and thus produces a partial vacuum. See Fig. 40 for a section of
the device.
[Illustration: FIG. 40.--An aspirator.]
=61. The Condensing Pump.=--This is like the exhaust pump except that
its valves are reversed. It is used in compressing illuminating gases
into cylinders for use in lighting vehicles, stereopticons, Pintsch
lights, gas light buoys, etc., and also for compressing air to operate
air brakes, pneumatic hammers and drills, and for other uses.
The common condensing pump is the kind used for inflating tires. (See
Fig. 41.) In this, a loosely fitting metal piston is attached to a disc
of leather somewhat larger than the cylinder. This device is called a
_cup valve_. On raising the piston, air rushes in from the top past the
valve, but on pushing the piston down, the valve is pressed tightly
against the sides of the cylinder and prevents the escape of any air.
The compressed air pushes open a valve on the tire and enters it. This
valve closes as soon as the pressure is lessened from outside. It is
well to notice in all of these pumps that _two_ valves are used. One
holds the air already secured while the other opens for a new supply.
Both valves are never open at the same time.
[Illustration: FIG. 41.--Condensing pump used in inflating tires.]
=62. Water Pumps.--The Common Lift Pump.= This, the simplest pump for
raising water, consists of a cylinder _C_ (Fig. 42) connected by a pipe
_R_ to a supply of water as a cistern or well. A valve opening upward is
placed at the bottom of the cylinder over the entrance to the pipe. In
the cylinder is a tightly fitting piston connected by a rod to a lever
for ease in action. The piston contains a valve opening upward. In
operating this pump water is usually first poured into the cylinder to
"prime" it. This helps to close the valves and prevents air leaking past
them. When the piston is lowered the lower valve closes, the air in the
cylinder being compressed pushes the upper valve open and passes above
the piston. On raising the piston the upper valve closes. This forms a
partial vacuum in the cylinder.
The air pressing on the surface of the water below forces the water and
air that may be in the tube upward through the lower valve to fill this
partial vacuum.
When the cylinder becomes filled with water, this is lifted out on the
up-stroke, whence its name, "lift pump." Since the atmospheric pressure
at sea-level can only support a column of water about 34 ft. high, the
lower valve must be within this distance of the water surface. In actual
practice the limit is about 27 ft. In deeper wells, the cylinder and
valves are placed so that they are within 25 or 27 ft. of the surface of
the water in the well, a long piston rod reaching above the surface of
the ground and connected to a pump handle operates the piston. A
discharge pipe extends from the cylinder to the surface of the ground
above.
[Illustration: FIG. 42.--The common lift pump.]
[Illustration: FIG. 43.--A force pump with an air chamber (_A_).]
=63. The Force Pump.=--The force pump is used to deliver water under
pressure either for spraying or to an elevated reservoir. The piston is
solid, the second valve being placed at the entrance of the discharge
pipe. (See Fig. 43.) The action is the same as that of the lift pump,
with this exception; the piston in its down stroke forces the water out
through the discharge pipe, the velocity depending upon the pressure
exerted.
A force pump is usually provided with an air chamber which is connected
with the discharge pipe. On the down stroke of the piston, water is
forced into the air chamber. This compresses the air it contains. The
compressed air reacts and exerts pressure on the water forcing it out in
a steady stream.
Force pumps are used in deep wells, being placed at the bottom.
The pumps used in city water works, fire engines, and all steam pumps,
are force pumps. (See Fig. 44.)
[Illustration: FIG. 44.--A steam pump used on a fire engine.]
=64. The Siphon.=--The siphon is a tube used to convey a liquid from one
level over an elevation to a lower level by atmospheric pressure. It is
used to remove liquids from tanks or vessels that have no opening at the
bottom.
The siphon cannot be completely understood until one has mastered the
laws of the flow of liquids. The following is offered as an incomplete
explanation of its behavior. Consider the siphon to be full of water
and closed at _d_ (Fig. 45). Atmospheric pressure on _a_ will hold the
siphon full if _ab_ does not exceed 34 feet. If _d_ is opened the water
falls out with a speed equal to that acquired in falling from the level
of _a_ to that of _d_. This speed is acquired by all the water in the
siphon and results in a drop in pressure throughout it. The pressure at
_a_ inside the siphon becomes less than the pressure at the same level
outside as soon as the water starts flowing. The water in the vessel
then flows into the siphon and out at _d_. This flow continues as long
as there is a fall from the free surface of the water in the vessel to
the outlet at _d_.
[Illustration: FIG. 45.--Cross-section of a siphon.]
[Illustration: FIG. 46.--The Cartesian diver.]
=65. The Cartesian Diver.=--This is a device which illustrates at the
same time transmission of pressure by liquids, Archimedes' principle,
and compressibility of gases. It was invented by Des Cartes (1596-1650).
As ordinarily made, it is a hollow glass image with a small opening in
the foot. It contains air and water in such amounts that the average
density of image and contents is slightly less than that of water. It is
placed in a tall glass jar filled with water and covered with tightly
stretched rubber tissue. (See Fig. 46.) By pressing on the rubber cover
the diver may be made to sink, since the air and water transmit the
pressure on the cover which compresses the air inside the figure
admitting some water to it, thus making the diver more dense than
water. By varying the pressure it can be made to sink, rise, or remain
stationary at will.[D] A small vial can be used instead of the image.
[D] The position of a submarine in or under water is controlled in
a similar manner.
=66. Hydraulic Ram.=--The hydraulic ram (see Fig. 47) is an automatic
device that is much used for raising water from springs to houses
located on higher ground. Water flows through the pipe _A_ through the
opening at _B_. The pressure closes the valve at _B_. The increased
pressure in the pipe due to the closing of _B_ opens the valve _C_ and
some of the water flows into the air chamber _D_. This reduces the
pressure against the valve _B_ so that it drops and allows a little
water to escape. Just as this happens, valve _C_ closes. The pressure in
the pipe then closes _B_ and forces water past _C_. This action being
continually repeated, the air in _D_ becomes so compressed that it has
elastic force enough to raise the water in a steady stream to a height
of many feet.
[Illustration: FIG. 47.--Cross-section of a hydraulic ram.]
=67. The Balloon.=--Since air is a fluid, Archimedes' principle applies
to it as well as to liquids. Therefore any object in the air is lifted
up by a force equal to the weight of the air it displaces. The object
will rise, if it weighs less than this displaced air and will continue
to rise until both weights are equal.
_The Balloon_ (Fig. 48) rises because it weighs less than the air it
displaces, and therefore it is pushed up by the heavier air, the
"lifting power" being the difference between its weight and that of the
air displaced. The neck at the bottom is left open to allow for
expansion of the gas. When the aeronaut wishes to descend, he opens a
valve at the top allowing some of the gas to escape.
[Illustration: FIG. 48.--Winner of international championship race,
Paris, 1913.]
_Hydrogen_ is the lightest gas, weighing 0.09 kg. per cubic meter, and
so gives the greatest lifting power, but as it is expensive to make,
coal gas, density 0.75 kg. per cubic meter, is ordinarily employed.
Helium has recently been used to fill military balloons because it
cannot be set on fire.
_The Parachute_ (Fig. 49) is an umbrella-shaped device for use in
descending from a balloon. After falling a few seconds it opens, the
large surface exposed to the air causing it to descend slowly. The hole
in the top keeps the parachute upright by allowing the air to escape
through it, thus relieving the pressure.
[Illustration: FIG. 49.--A parachute.]
[Illustration: FIG. 50.--Cross-section of a Westinghouse air brake.]
=68. The Air Brake.=--Compressed air is used to do work in many
machines, such as pneumatic drills, hammers, and air brakes. The
Westinghouse air brake (Fig. 50) uses air at a pressure of about 70 lbs.
to the square inch. The essential parts as shown are a reservoir _R_,
the brake cylinder _C_ and a triple valve _V_, placed under each car
with an air pipe _P_, leading to the engine. This is connected to _R_ by
the triple valve _V_. When the pressure in _P_ is reduced by the
engineer or by accident, the triple valve operates so as to admit air
from _R_ into the cylinder _C_ pushing the piston _H_ to the left. _H_
is connected to the brakes by levers which press the brake shoes
strongly against the wheels. When the air pressure in _P_ is restored
the triple valve acts so as to permit the air in _C_ to escape while _R_
is filled again from _P_. The hissing sound heard when a train stops is
caused by air escaping from cylinder _C_. The spring in _C_ keeps the
brakes from the wheels except when the "air is on."
[Illustration: FIG. 51.--Cross-section of a gas meter showing its
construction and action.]
=69. The Gas Meter.=--The gas meter consists of a box divided into two
parts by a vertical partition (Fig. 51). Two bellows are attached to
this partition, one on each side. The valves that regulate the flow of
gas to and from the bellows and the chambers _A_ and _D_ are opened and
closed by levers connected with the bellows. These levers also operate
the hands upon the dials. When the inlet to the bellows _B_ is opened,
the outlet of _A_ is also opened. Gas entering _B_ opens the bellows and
forces the gas in _A_ out into the house-pipe _E_. When _B_ is full its
inlet valve closes and its outlet valve opens. The inlet of _A_ also
opens and its outlet closes. Gas now flows into _A_, compressing the
bellows and _B_, and forcing the gas from it into the house-pipe. At
each filling of the bellows _B_ there will be displaced from _A_ and
forced into the house-pipe as much gas as enters _B_. It is evident that
at each emptying of _B_ an equal amount of gas enters _A_. Thus we have
_A_ and _B_ alternately filling and emptying as long as the gas burner
is open. To have a continuous flow of gas in the house-pipes two pipes
and two chambers are necessary, one being filled while the other is
being emptied.
Fig. 52 represents the dials upon a gas meter showing a reading of
54,600 cu. ft.
[Illustration: FIG. 52.--Dials of a gas meter.]
=70. Centrifugal Pumps.= Fluids, such as water and air, are often put in
motion by devices called _centrifugal pumps_ (see Art. 78). These pumps
contain a revolving part, like a wheel without a rim, whose spokes are
replaced by thin blades. This revolving part resembles the paddle wheel
of some steam boats and is enclosed in a case or cover having one
opening at the rim and another opening on one side about the axle.
[Illustration: FIG. 53.--A vacuum sweeper. (_Courtesy of the Hoover
Suction Sweeper Co._)]
When the wheel is rapidly revolved, the fluid is driven out with
considerable force through the opening at the rim, while a partial
vacuum is produced at the axle causing a rapid flow into the device at
this point.
This is the principle of the action of the _vacuum cleaner_. Fig. 53 is
a section of a vacuum sweeper showing the revolving wheel and the
current of air passing into the wheel at the lower side and out of the
rim of the case at the rear.
_Centrifugal water pumps_ work on the same principle and furnish a
continuous flow of water, often large in volume and at considerable
pressure.
Important Topics
1. Air pump.
2. Condensing pump.
3. Lift and force pumps.
4. Siphon.
5. Cartesian diver.
6. Hydraulic ram.
7. Balloon.
8. Air brake.
9. Gas meter.
10. Vacuum cleaner.
Exercises
1. Explain why smoke settles to the ground before storms.
2. Why does the water rise in the suction pipe of a pump?
3. Why is it easier to float in water when the lungs are filled with air
than when they are not filled?
4. Why is it easier to swim in salt water than in fresh water?
5. How are submarines made to sink? to rise to the surface?
6. How can a fish rise or sink in water?
7. Explain why a life preserver made of cork will enable a person to
float.
8. Hold the open hand out flat with the fingers together. Place
underneath the fingers a piece of paper. Blow between the first and
second fingers against the paper. As long as you blow hard the paper
will not fall but will stick to the hand. Explain.
9. Why does pressing the bulb of an atomizer force out the liquid in a
fine spray?
10. Why is air that contains a large amount of water vapor lighter than
air that only contains a small amount?
11. How are heights above sea-level ascertained by a barometer?
12. Oil floats on water but sinks in alcohol. Explain.
13. In a balloon the lower end is often open to the air. Why does not
the gas escape and prevent the balloon from rising?
14. How long will a balloon continue to rise?
15. If the pressure against the 8-in. piston of an air brake is 70 lbs.
per square inch, how much force does the piston exert?
16. The capacity of a balloon is 40,000 cu. ft. The weight of the
balloon, car, etc., is 600 lbs.; specific gravity of the gas used is
0.46 that of the air. Find how much weight the balloon can carry.
17. The so-called Magdeburg hemispheres were invented by Otto von
Guericke of Magdeburg, Germany. When the hemispheres (see Fig. 54) are
placed in contact and the air exhausted it is found very difficult to
pull them apart. Explain.
18. Von Guericke's hemispheres had an inside diameter of 22 in. What
force would be required to pull them apart if all the air were exhausted
from them? (Find the atmospheric force on a circle, 22 in. in diameter.)
19. Von Guericke made a water barometer whose top extended through the
roof of his house. On the top of the water in the tube was placed a
wooden image. In fair weather the image appeared above the roof, but it
descended before a storm. Explain.
20. The balloon "Goodyear" (Fig. 48), which won the International
championship race at Paris in 1913, has a capacity of 80,000 cu. ft. The
gas bag weighs 653 lbs., the net 240 lbs. and the basket 92 lbs. How
large a load can it carry when filled with hydrogen specific gravity
0.069 (compared with air).
[Illustration: FIG. 54.--Magdeburg hemispheres.]
Review Outline: Liquids and Gases
Liquids: Force, pressure, and density. Floating and immersed bodies.
Laws: Liquid force, _F = A.h.d_, Pascal's, Archimedes. Illustrations and
Applications:
Specific gravity, _W_{a}/(W_{a} - W_{w})_, _(W_{a} - W_{l})/(W_{a}
- W_{w})_, Boyle's, _PV = P´V´_
Devices: Hydraulic press, air cushion, barometer--mercurial and aneroid.
Pumps, lift, force, vacuum, compression, centrifugal, balloon, siphon,
etc. Construction and action of each.
CHAPTER V
FORCE AND MOTION
(1) FORCE, HOW MEASURED AND REPRESENTED
=71. Force.=--We have been studying various forces, such as air
pressure, pressure in liquids, and the force of elasticity in solids,
and have considered them simply as pushes or pulls. A more formal study
of forces in general and of devices for representing and measuring them
will be helpful at this point of the course.
_A force is that which tends to cause a change in the size or shape of a
body or in its state of motion._ In other words a force is a push or a
pull. That is, force tends to produce distortion or change of motion in
a body. Force itself is invisible. We measure it by the effect it
produces. Forces are usually associated with the objects exerting them.
Thus we speak of _muscular force_, _air pressure_, _liquid pressure_,
the force of a spring, the force of the earth's attraction and so on.
Forces are classified in various ways.
I. With respect to the _duration and steadiness_ of the force.
(a) Constant, as the earth's attraction. (b) Impulsive, as the stroke of
a bat on a ball. (c) Variable, as the force of the wind.
II. With respect to the _direction_ of the force.
(a) Attractive, as the earth's attraction. (b) Repulsive, as air
pressure, liquid pressure, etc.
=72. Methods of Measuring Force.=--Since forces are measured by their
effects which are either distortion or change of motion, either of these
effects may be used to measure them. For example, the force exerted by
a locomotive is sometimes computed by the _speed_ it can develop in a
train of cars in a given time, or the force of the blow of a baseball
bat is estimated by the _distance_ the ball goes before it strikes the
ground.
The more common method of measuring force, however, is by _distortion_,
that is, by measuring the change of shape of a body caused by the force.
In doing this, use is made of Hooke's Law (Art. 32), in which it is
stated that "within the limits of perfect elasticity," changes of size
or shape are directly proportional to the forces employed. That is,
twice as great a force will produce twice as great a change of shape and
so on.
[Illustration: FIG. 55.--A spring balance.]
A common contrivance using this principle is the spring balance (Fig.
55), with which all are familiar, as ice scales, meat scales, postal
scales, etc. The object which changes shape in this device is a coiled
spring contained in the case of the instrument. The balance is so
constructed that when the spring is pulled out as far as possible it has
not reached its limit of elasticity, since, if the spring were stretched
so as to exceed its elastic limit, the index would not return to its
first position on removing the load. (See Arts. 30-32.)
=73. Graphic Representation of Forces.=--A force is said to have three
elements. These are (a) _its point_ of _application_, (b) _its
direction_, and (c) _its magnitude_. For example, if there is hung upon
the hook of a spring balance a weight of 5 lbs., then we have: (a) its
point of application on the hook of the balance, (b) its downward
direction and (c) its magnitude, or 5 lbs. These three elements may be
represented by a line. Thus in Fig. 56_a_, a line _AB_ is drawn as
shown, five units long; _A_ represents the point of application; _B_,
the arrow head, shows the direction; and the length of the line (five
units) shows the magnitude of the force.
This is called a _graphic representation_ since it represents by a line
the quantity in question. If another weight of 5 lbs. were hung from the
first one, the graphic representation of both forces would be as in Fig.
56_b_. Here the first force is represented by _AB_ as before, _BC_
representing the second force applied. The whole line represents the
_resultant_ of the two forces or the result of their combination. If the
two weights were hung one at each end of a short stick _AC_ (Fig.
56_c_), and the latter suspended at its center their combined weight or
_resultant_ would of course be applied at the center. The direction
would be the same as that of the two weights. The resultant therefore is
represented by _ON_. In order to exactly balance this resultant _ON_, a
force of equal magnitude but opposite in direction must be applied at
the point of application of _ON_, or _O_. _OM_ then represents a force
that will just balance or hold in equilibrium the resultant of the two
forces _AB_ and _CD_. This line _OM_ therefore represents the
_equilibrant_ of the weights _AB_ and _CD_. The resultant of two forces
at an angle with each other is formed differently, as in Fig. 57 _a_.
Here two forces _AB_ and _AC_ act at an angle with each other. Lay off
at the designated angle the lines _AB_ and _AC_ of such length as will
accurately represent the forces. Lay off _BD_ equal to _AC_ and _CD_
equal to _AB_. The figure _ABCD_ is then a parallelogram. Its diagonal
_AD_ represents the resultant of the forces _AB_ and _AC_ acting at the
angle _BAC_. If _BAC_ equals 90 degrees or is a right angle, _AD_ may be
_computed_ thus: _AB² + BD² = AD²_. Why?
and _AD_ = √_([line]AB² + [line]BD²)._
[Illustration: FIG. 56.--Graphic representation of forces acting along
the same or parallel lines.]
[Illustration: FIG. 57.--Graphic representation of two forces acting
(_a_) at a right angle, (_b_) at an acute angle.]
This method of determining the resultant by _computation_ may be used
when the two forces are at right angles. (In any case, _AD may be
measured_ using the same scale that is laid off upon _AB_ and _AC_, as
shown in Fig. 57 _b_.) The three cases of combining forces just given
may be classified as follows: The _first_ is that of _two forces acting
along the same line_ in the same or opposite direction, as when two
horses are hitched tandem, or in a tug of war. The _second_ is that of
_two forces acting along parallel lines_, in the same direction, as when
two horses are hitched side by side or abreast. The _third_ is that of
_two forces acting at the same point at an angle_. It may be represented
by the device shown in Fig. 58, consisting of two spring balances
suspended from nails at the top of the blackboard at _A_ and _B_. A cord
is attached to both hooks and is passed through a small ring at _O_ from
which is suspended a known weight, _W_. Lines are drawn on the
blackboard under the stretched cords, from _O_ toward _OA_, _OB_, and
_OW_ and distances measured on each from _O_ to correspond to the three
forces as read on balance _A_ and _B_ and the weight _W_. Let a
parallelogram be constructed on the lines measured off on _OA_ and _OB_.
Its diagonal drawn from _O_ will be found to be vertical and of the same
length as the line measured on _OW_. The diagonal is the _resultant_ of
the two forces and _OW_ is the equilibrant which is equal and opposite
to the resultant.
[Illustration: FIG. 58.--Experimental proof of parallelogram of forces.]
Again, the _first_ case may be represented by a boat moving up or down a
stream; the resultant motion being the combined effect of the boat's
motion and that of the stream. The _second_, may be represented by two
horses attached side by side to the same evener. The resultant force
equals the sum of the two component forces. The _third_, may be
represented by a boat going across a stream, the resultant motion being
represented by the diagonal of the parallelogram formed by using the
lines that represent the motion of the stream and of the boat.
=74. Units for Measuring Force.=--Force is commonly measured in units of
weight: in pounds, kilograms, and grams. For example, we speak of 15
lbs. pressure per square inch and 1033.6 g. pressure per square
centimeter as representing the air pressure. It should be noted here
that the words pound, kilogram, and gram are used not only to represent
_weight_ or _force_ but also the masses of the objects considered. Thus,
one may speak of a pound-mass meaning the amount of material in the
object.
It will help to avoid confusion if we reserve the simple terms "gram"
and "pound" to denote exclusively an amount of matter, that is, a mass,
and to use the full expression "gram of force" or "pound of force"
whenever we have in mind the pull of the earth upon these masses. Or,
one may speak of a _pound-weight_ meaning the amount of attraction
exerted by the earth upon the object. The same is true of _gram-mass_
and _gram-weight_. The mass of a body does not change when the body is
transferred to another place. The weight, however, may vary, for on
moving a body from the equator toward the poles of the earth the weight
is known to increase.
Important Topics
1. Definition of force.
2. Classification of forces. (a) Duration: constant, impulsive,
variable. (b) Direction: attractive, repulsive.
3. Methods of measuring force. (a) By distortion. (b) By change of
motion.
4. Graphic representation of forces: component, resultant, equilibrant.
5. Three cases of combining forces. (1) Two forces acting on the same
line. (2) Two forces acting in parallel lines. (3) Two forces acting at
the same point at an angle.
6. Units for measuring force, pound, gram.
Exercises
1. Name five natural forces. Which produce a tension? Which a pressure?
2. How much can you lift? Express in pounds and kilograms.
3. Show graphically the resultant of two forces at right angles, one of
12 lbs., the other of 16 lbs. What is the magnitude of this resultant?
Then determine the answer, first by measurement and then by computation.
Which answer is more accurate? Why?
4. Represent by a parallelogram the two forces that support a person
sitting in a hammock and draw the line representing the resultant.
5. Find graphically the resultant of the pull of two forces, one of 500
lbs. east and one of 600 lbs. northwest.
6. Determine the equilibrant of two forces, one of 800 lbs. south and
one of 600 lbs. west.
7. Would the fact that weight varies on going from the equator to either
pole be shown by a spring balance or a beam balance? Explain.
(2) MOTION. NEWTON'S LAWS OF MOTION
=75. Motion a Change of Position.=--Motion is defined as a continuous
_change in the position_ of a body. The _position_ of a body is usually
described as its _distance_ and _direction_ from some fixed point. Thus
a man on a boat may be at rest with respect to the boat and moving with
respect to the earth. Or, if he walks toward the stern as fast as the
boat moves forward, he may keep directly over a rock on the bottom of
the lake and hence not be moving with reference to the rock and yet be
in motion with respect to the boat. Motion and rest, therefore, are
_relative_ terms. The earth itself is in motion in turning on its axis,
in moving along its orbit, and in following the sun in its motion
through space. Motions are classified in several ways:
(A) MODES OF MOTION
1. _Translation._--A body is said to have motion of _translation_ when
every line in it keeps the same direction.
2. _Rotation._--A body has motion of _rotation_ when it turns upon a
fixed axis within the body, as a wheel upon its axle or the earth upon
its axis.
3. _Vibration_ or _Oscillation_.--A body is said to have _vibratory_ or
_oscillatory_ motion when it returns to the same point at regular
intervals by reversals of motion along a given path, _e.g._, a pendulum.
(B) DIRECTION OF MOTION
1. _Rectilinear._--A body has rectilinear motion when its path is a
straight line. Absolute rectilinear motion does not exist, although the
motion of a train on a straight stretch of track is nearly rectilinear.
2. _Curvilinear._--A body has _curvilinear_ motion when its path is a
curved line, _e.g._, the path of a thrown ball.
(C) UNIFORMITY OF MOTION
1. _Uniform._--A body has uniform motion when its speed and direction of
motion do not change. Uniform motion for extended periods is rarely
observed. A train may cover, on an average, 40 miles per hour but during
each hour its speed may rise and fall.
2. _Variable._--A body has variable motion when its speed or direction
of motion is continually changing. Most bodies have variable motion.
3. _Accelerated._--A body has accelerated motion when its speed or
direction of motion continually changes. If the speed changes by the
same amount each second, _and the direction of motion does not change_
the motion is said to be _uniformly_ accelerated, _e.g._, a falling
body.
Uniformly accelerated motion will be studied further under the topic of
falling bodies.
_Velocity_ is _the rate of motion_ of a body in a given direction. For
example, a bullet may have a velocity of 1300 ft. a second upwards.
_Acceleration_ is _the rate of change of velocity_ in a given direction,
or the change of velocity in a unit of time. A train starting from a
station gradually increases its speed. The gain in velocity during one
second is its acceleration. When the velocity is decreasing, as when a
train is slowing down, the acceleration is opposite in direction to the
velocity. A falling body falls faster and faster. It has _downward
acceleration_. A ball thrown upward goes more and more slowly. It also
has _downward acceleration_.
=76. Momentum.=--It is a matter of common observation that a heavy body
is set in motion with more difficulty than a light one, or if the same
force is used for the same length of time upon a light and a heavy
body,[E] the light body will be given a greater velocity. This
observation has led to the _calculation_ of what is called the "quantity
of motion" of a body, or its _momentum_. It is computed by multiplying
the mass by the velocity. If the C.G.S. system is used we shall have as
the momentum of a 12 g. body moving 25 cm. a second a momentum of 12 ×
25 or 300 C.G.S. units of momentum. This unit has no name and is
therefore expressed as indicated above. The formula for computing
momentum is: _M = mv_.
[E] By a light body is meant one of small mass, a heavy body
possessing much greater mass.
Newton's Laws of Motion
=77. Inertia, First Law of Motion.=--One often observes when riding in a
train that if the train moves forward suddenly the passengers do not get
into motion as soon as the train, and apparently are jerked backward.
While if the train is stopped suddenly, the passengers tend to keep in
motion. This tendency of matter to keep moving when in motion and to
remain at rest when at rest is often referred to as the property of
_inertia_. _Newton's first law of motion_, often called the _law of
inertia_, describes this property of matter as follows:
_Every body continues in a state of rest or of uniform motion in a
straight line unless it is compelled to change that state by some
external force._ This means that if an object like a book is lying on a
table it will remain there until removed by some outside force. No
inanimate object can move itself or stop itself. If a ball is thrown
into the air it would move on forever if it were not for the _force_ of
attraction of the earth and the resistance of the air.
It takes time to put a mass into motion, a heavy object requiring more
time for a change than a light object. As an example of this, note the
movements of passengers in a street car when it starts or stops
suddenly. Another illustration of the law of inertia is the so-called
"penny and card" experiment. Balance a card on the end of a finger.
Place on it a coin directly over the finger, snap the card quickly so as
to drive the card from beneath the coin. The coin will remain on the
finger. (See Fig. 59.)
[Illustration: FIG. 59.--The ball remains when the card is driven away.]
According to Newton's first law of motion a moving body which could be
entirely freed from the action of all external forces would have uniform
motion, and would describe a perfectly straight course. The curved path
taken by a baseball when thrown shows that it is acted upon by an
outside force. This force, the attraction of the earth, is called
_gravity_.
[Illustration: SIR ISAAC NEWTON "By Permission of the Berlin
Photographic Co., New York."
Sir Isaac Newton (1642-1727) Professor of mathematics at Cambridge
university; discovered gravitation; invented calculus; announced the
laws of motion; wrote the Principia; made many discoveries in light.]
[Illustration: GALILEO GALILEI "By Permission of the Berlin Photographic
Co., New York."
Galileo Galilei (1564-1642). Italian. "Founder of experimental science";
"Originator of modern physics"; made the first thermometer; discovered
the laws of falling bodies and the laws of the pendulum; invented
Galilean telescope.]
[Illustration: FIG. 60.--Cross-section of the DeLaval cream separator.]
=78. Curvilinear Motion.=--Curvilinear motion occurs when a moving body
is pulled or pushed away from a straight path. The pull or push is
called _centripetal_ (center-seeking) force. A moving stone on the end
of a string when pulled toward the hand moves in a curve. If the string
is released the stone moves in a tangent to the curve. The string pulls
the hand. This phase of the pull is called _centrifugal_ force. The
_centripetal_ force is the pull on the stone. Centripetal and
centrifugal force together cause a tension in the string. Examples of
curvilinear motion are very common. The rider and horse in a circus ring
lean inward in order to move in a curve. The curve on a running track in
a gymnasium is "banked" for the same reason. Mud flying from the wheel
of a carriage, the skidding of an automobile when passing rapidly around
a corner, and sparks flying from an emery wheel, are illustrations of
the First Law of Motion.
Cream is separated from milk by placing the whole milk in a rapidly
revolving bowl, the cream being lighter collects in the center and is
thrown off at the top. (See Fig. 60.) Clothes in steam laundries are
dried by a centrifugal drier. In amusement parks many devices use this
principle. (See centrifugal pumps, Art. 70.)
[Illustration: FIG. 61.--The two balls reach the floor at the same
time.]
=79. The Second Law of Motion,= sometimes called the _law of momentum_,
leads to the _measurement of force_, by the momentum or the quantity of
motion, produced by it. The law is stated as follows:
_Change of motion, or momentum, is proportional to the acting force and
takes place in the direction in which the force acts._ In other words,
if two or more forces act at the same instant upon a body each produces
the same effect that it would if acting alone. If a card be supported on
two nails driven horizontally close together into an upright board (see
Fig. 61), and two marbles be so placed on the ends as to balance each
other, when one marble is snapped horizontally by a blow, the other will
fall. Both reach the floor at the same time. The two balls are equally
pulled down by the earth's attraction and strike the ground at the same
time, though one is shot sidewise, and the other is dropped vertically.
As gravity is a constant force, while the blow was only a momentary
force, the actual path or resultant motion will be a curved line.
The constant relation, between the acting force and the change of
momentum it produces in a body, has led to the adoption of a convenient
C.G.S. unit of force called the _dyne_. _The dyne is that force which
can impart to a mass of one gram a change of velocity at the rate of one
centimeter per second every second._ This definition assumes that the
body acted upon is free to move without hindrance of any kind, so that
the acting force has to overcome only the _inertia_ of the body.
_However_, the _law_ applies in every case of application of force, so
that each force produces its full effect independently of other forces
that may be acting at the same time upon the body.
=80. Newton's Third Law.=--This law has been experienced by everyone who
has jumped from a rowboat near the shore. The muscular action that
pushes the body forward from the boat also pushes the boat backward,
often with awkward results. The law is stated: _To every action, there
is always an opposite and equal reaction, or the mutual actions of any
two bodies are always equal and opposite in direction_. Many
illustrations of this law are in every one's mind: a stretched rope
pulls with the same force in one direction as it does in the opposite
direction. If a bat hits a ball, the ball hits the bat with an equal and
opposite force. The third law is therefore sometimes called the law of
_reaction_. When a weight is hung upon a spring balance the action of
the weight pulls down the spring until it has stretched sufficiently
(Hooke's Law) to produce an elastic _reaction_ that equals and hence
supports the weight. When a man stands at the center of a plank
supported at its ends, the action of the man's weight bends the plank
until the elastic force developed in the plank equals the weight
applied. Further, when a train or a wagon is on a bridge the bridge
yields until it has developed an elastic reaction equal to the weight
applied. If a person stands in the center of a room, the floor beams
yield until the third law is satisfied. In fact, whenever a force acts,
a contrary equal force always acts.
=81. Stress and Strain.=--A pair of forces that constitute an action and
a reaction is called a _stress_. The two forces are two parts of one
_stress_. If the two forces act away from each other, as in the breaking
of a string, the stress is called a _tension_, but if they act toward
each other as in crushing anything, the stress is called a _pressure_.
In order for a body to exert force it must meet with resistance. The
force exerted is never greater than the resistance encountered. Thus one
can exert but little force upon a feather floating in the air or upon
other light objects. A fast moving shot exerts no force unless it
encounters some resistance.
Forces, then, are always found in pairs. Thus to break a string, to
stretch an elastic band, to squeeze a lemon, one must exert two equal
and opposite forces. Such a thing as a single force acting alone is
unknown. Usually, however, we give our attention mainly to one of the
forces and ignore the other. When a force acts upon a body the change of
shape or size resulting is called a _strain_. Hooke's law (Art. 32) is
often expressed as follows: "The strain is proportional to the stress,"
_e.g._, the stretch of the spring of a spring balance is proportional to
the load placed upon it.
Important Topics
1. Motion a change of position. Kinds of motion.
2. Newton's laws of motion.
3. Momentum.
4. Inertia. First law of motion. Curvilinear motion.
5. Second law of motion.
6. Third law of motion. Action and reaction, stress and strain.
Exercises
1. Mention three illustrations of the third law, different from those
given.
2. A rifle bullet thrown against a board standing upon edge will knock
it down; the same bullet fired at the board will pass through it without
disturbing its position. Explain.
3. A hammer is often driven on to its handle by striking the end of the
latter. Explain.
4. Consider a train moving 60 miles an hour, with a gun on the rear
platform pointing straight backward. If a ball is fired from the gun
with a speed of 60 miles an hour, what will happen to the ball?
5. Could one play ball on the deck of an ocean steamer going 25 miles an
hour without making allowance for the motion of the ship? Explain.
6. On a railroad curve, one rail is always higher. Which? Why?
7. Why can a small boy when chased by a big boy often escape by dodging?
8. Will a stone dropped from a moving train fall in a straight line?
Explain.
9. A blast of fine sand driven against a sheet of glass soon gives it a
rough surface. Explain.
10. Explain the use of fly-wheels in steadying the motion of machinery
(for example, the sewing machine).
11. Is it easier to walk to the front or rear of a passenger train when
it is stopping? Why?
12. Why does lowering the handles of a wheel-barrow on the instant of
striking make it easier to go over a bump?
13. Why should a strong side wind interfere with a game of tennis? How
can it be allowed for?
14. On which side of a railroad track at a curve is it the safer to walk
while a train is passing? Why?
15. Why does a bullet when fired through a window make a clean round
hole in the glass, while a small stone thrown against the window
shatters the glass?
16. A tallow candle can be fired through a pine board. Why?
17. In cyclones, straws are frequently found driven a little distance
into trees; why are the straws not broken and crushed instead of being
driven into the tree unbroken?
18. A bullet weighing one-half oz. is fired from a gun weighing 8 lb.
The bullet has a velocity of 1800 ft. per second. Find the velocity of
the "kick" or recoil of the gun.
18. When football players run into each other which one is thrown the
harder? Why?
20. A railroad train weighing 400 tons has a velocity of 60 miles per
hour. An ocean steamer weighing 20,000 tons has a velocity of one half
mile per hour. How do their momenta compare?
21. Why is a heavy boy preferable to a lighter weight boy for a football
team?
22. Why does a blacksmith when he desires to strike a heavy blow, select
a heavy sledge hammer and swing it over his head?
23. Why does the catcher on a baseball team wear a padded glove?
(3) RESOLUTION OF FORCES
=82. Resolution of Forces.=--We have been studying the effect of forces
in producing motion and the results of combining forces in _many_ ways;
in the _same line_, in _parallel lines_, and in _diverging lines_.
Another case of much interest and importance is _the determination of
the effectiveness of a force in a direction different from the one in
which it acts_. This case which is called _resolution of forces_ is
frequently used. To illustrate: one needs but to recall that a sailor
uses this principle in a practical way whenever he sails his boat in any
other direction than the one in which the wind is blowing, _e.g._, when
the wind is blowing, say from the north, the boat may be driven east,
west, or to any point south between the east and west and it is even
possible to beat back against the wind toward the northeast or
northwest. Take a sled drawn by a short rope with the force applied
along the line _AB_ (see Fig. 62); part of this force tends to lift the
front of the sled as _AC_ and a part to draw it forward as _AD_. Hence
not all of the force applied along _AB_ is used in drawing the sled
forward. Its effectiveness is indicated by the relative size of the
component _AD_ compared to _AB_.
[Illustration: FIG. 62.--_AD_ is the effective component.]
The force of gravity acting upon a sphere that is resting on an
_inclined plane_ may be readily resolved into two components, one, the
_effective_ component, as _OR_, and the other, the _non-effective_ as
_OS_. (See Fig. 63.) If the angle _ACB_ is 30 degrees, _AB_ equals 1/2
of _AC_ and _OR_ equals 1/2 of _OG_, so that the speed of the sphere
down the plane developed in 1 second is less than (about one-half of)
the speed of a freely falling body developed in the same time. Why is
_OS_ non-effective?
[Illustration: FIG. 63.--The effective component is _OR_.]
[Illustration: FIG. 64.--Resolution of the forces acting on an
aeroplane.]
=83. The Aeroplane.=--The aeroplane consists of one or two frames _ABCD_
(see Fig. 64), over which is stretched cloth or thin sheet metal. It is
driven through the air by a propeller turned by a powerful gasoline
motor. This has the effect of creating a strong breeze coming toward the
front of the aeroplane. As in the case of the sailboat a pressure is
created at right angles to the plane along _GF_ and this may be resolved
into two components as _GC_ and _GE_, _GC_ acting to lift the aeroplane
vertically and _GE_ opposing the action of the propeller. Fig. 65
represents the Curtis Flying Boat passing over the Detroit river.
[Illustration: FIG. 65.--The Curtis hydroplane.]
Exercises.
1. If a wagon weighing 4000 lbs. is upon a hill which rises 1 ft. in 6,
what force parallel to the hill will just support the load? (Find the
effective component of the weight down the hill.)
2. If a barrel is being rolled up a 16-ft. ladder into a wagon box 3 ft.
from the ground, what force will hold the barrel in place on the ladder,
if the barrel weighs 240 lbs. Show by diagram.
3. Show graphically the components into which a man's push upon the
handle of a lawn mower is resolved.
4. Does a man shooting a flying duck aim at the bird? Explain.
5. What are the three forces that act on a kite when it is "standing" in
the air?
6. What relation does the resultant of any two of the forces in problem
five have to the third?
7. Into what two forces is the weight of a wagon descending a hill
resolved? Explain by use of a diagram.
8. A wind strikes the sail of a boat at an angle of 60 degrees to the
perpendicular with a pressure of 3 lbs. per square foot. What is the
effective pressure, perpendicular to the sail? What would be the
effective pressure when it strikes at 30 degrees?
9. How is the vertical component of the force acting on an aeroplane
affected when the front edge of the plane is elevated? Show by diagram.
(4) MOMENT OF FORCE AND PARALLEL FORCES
=84. Moment of Force.=--In the study of motion we found that the
quantity of motion is called _momentum_ and is measured by the product
of the _mass times the velocity_. In the study of _parallel forces_,
especially such as tend to produce _rotation_, we consider a similar
quantity. It is called a _moment of force_, which is the term applied to
the _effectiveness_ of a force in producing change of rotation. It also
measured by the product of two quantities; _One, the magnitude of the
force itself_, and the other, _the perpendicular distance from the axis
about which the rotation takes place to the line representing the
direction of the force_.
[Illustration: FIG. 66.--The moments about _S_ are equal.]
_To illustrate:_ Take a rod, as a meter stick, drill a hole at _S_ and
place through it a screw fastened at the top of the blackboard. Attach
by cords two spring balances and draw to the right and left, _A_ and _B_
as in Fig. 66. Draw out the balance _B_ about half way, hold it
steadily, or fasten the cord at the side of the blackboard, and read
both balances. Note also the distance _AS_ and _BS_. Since the rod is at
rest, the tendency to rotate to the right and left must be equal. That
is, the moments of the forces at _A_ and _B_ about _S_ are equal. Since
these are computed by the product of the _force times the force_ arm,
multiply _B_ by _BS_ and _A_ by _AS_ and see if the computed moments are
equal. _Hence a force that tends to turn or rotate a body to the right
can be balanced by another of equal moment that acts toward the left._
[Illustration: FIG. 67.--Law of parallel forces illustrated.]
=85. Parallel Forces.=--Objects are frequently supported by two or more
upward forces acting at different points and forming in this way a
system of parallel forces; as when two boys carry a string of fish on a
rod between them or when a bridge is supported at its ends. The
principle of moments just described aids in determining the magnitude of
such forces and of their resultant. To illustrate this take a wooden
board 4 in. wide and 4 ft. long of uniform dimensions. (See Fig. 67.)
Place several screw hooks on one edge with one set at _O_ where the
board will hang horizontally when the board is suspended there. Weigh
the board by a spring balance hung at _O_. This will be the resultant in
the following tests. Now hang the board from two spring balances at _M_
and _N_ and read both _balances_. Call readings _f_ and _f´_. To test
the forces consider _M_ as a fixed point (see Fig. 67) and the weight of
the board to act at _O_. Then the moment of the weight of the board
should be equal the moment of the force at _N_ since the board does not
move, or _w_ times _OM_ equals _f´_ times _NM_. If _N_ is considered the
fixed point then the moment of the weight of the board and of _f with
reference to the point N_ should be equal, or _w_ times _ON_ = _f_ times
_NM_. Keeping this illustration in mind, the law of parallel forces may
be stated at follows: 1. _The resultant of two parallel forces acting in
the same direction at different points in a body is equal to their sum
and has the same direction as the components._
_The moment of one of the components about the point of application of
the other is equal and opposite to the moment of the supported weight
about the other._
=Problem.=--If two boys carry a string of fish weighing 40 lbs. on
a rod 8 ft. long between them, what force must each boy exert if
the string is 5 ft. from the rear boy?
=Solution.=--The moment of the force _F_ exerted about the opposite
end by the rear boy is _F_ × 8. The moment of the weight about the
same point is 40 × (8 - 5) = 120. Therefore _F_ × 8 = 120, or _F_ =
15, the force exerted by the rear boy. The front boy exerts a force
of _F_ whose moment about the other end of the rod is _F_ × 8. The
moment of the weight about the same point is 40 × 5 = 200. Since
the moment of _F_ equals this, 200 = _F_ × 8, or _F_ = 25. Hence
the front boy exerts 25 lbs. and the rear boy 15 lbs.
[Illustration: FIG. 68.--A couple.]
=86. The Couple.=--If two equal parallel forces act upon a body along
different lines in opposite directions, as in Fig. 68, they have no
single resultant or there is no one force that will have the same effect
as the two components acting together. A combination of forces of this
kind is called a _couple_. Its tendency is to produce change of rotation
in a body. An example is the action upon a compass needle which is
rotated by a force which urges one end toward the north and by an equal
force which urges the other end toward the south.
Important Topics
1. Moment of force, how measured.
2. Parallel forces.
3. The two laws of parallel forces.
4. The couple.
Exercises
1. Show by diagram how to arrange a three-horse evener so that each
horse must take one-third of the load.
2. Two boys support a 10-ft. pole on their shoulders with a 40-lb.
string of fish supported from it 4 ft. from the front boy. What load
does each boy carry? Work by principle of moments.
3. If two horses draw a load exerting a combined pull of 300 lbs., what
force must each exert if one is 28 in. and the other is 32 in. from the
point of attachment of the evener to the load?
[Illustration: FIG. 69.--Forces acting upon a stretched rope.]
[Illustration: FIG. 70.--A crane with horizontal tie.]
4. A weight of 100 lbs. is suspended at the middle of a rope _ACB_ 20
ft. long. (See Fig. 69.) The ends of the rope are fastened at points _A_
and _B_ at the same height. Consider _D_ as the center of the line _AB_.
What is the tension of the rope when _CD_ is 3 ft.? When _CD_ is 1 ft.?
When _CD_ is 1 in.?
5. A crane is set up with the tie horizontal. (See Fig. 70.) If 1000
lbs. is to be lifted, find the tie stress and the boom stress if the
boom angle is 30 degrees? If 45 degrees? 60 degrees?
6. A ball is placed on a plane inclined at an angle of 30 degrees to the
horizontal. What fraction of its weight tends to cause motion down the
plane? What effect does the other component of the weight have? Why?
7. A person weighing 150 lbs. is lying in a hammock. The distance
between the supports is 15 ft. The hammock sags 4 ft. What is the
tension in the supports at each end? What is the tension when the sag is
only 1 ft.?
8. A ladder 30 ft. long and weighing 80 lbs. leans against the side of a
building so that it makes an angle of 30 degrees with the building. Find
the direction and magnitude of the component forces on the ground and at
the building.
9. A traveling crane 50 ft. long weighing 10 tons moves from one end of
a shop to the other, at the same time a load of 4000 lbs. moves from end
to end of the crane. Find the pressure of the trucks of the crane on the
track when the load is at a distance of 5, 10, 15, and 25 ft. from
either end.
[Illustration: FIG. 71--A truss.]
10. Resolve a force of 500 lbs. into two components at right angles to
each other, one of which shall be four times the other.
11. A truss (see Fig. 71), carries a load of 1000 lbs. at _C_. Find the
forces acting along _AC_, _BC_, and _AB_. If _AC_ and _BC_ are each 12
ft. and _AB_ 20 ft., which of these forces are tensions and which are
pressures?
(5) GRAVITATION AND GRAVITY
=87. Gravitation.=--Gravitation is the force of attraction that exists
between all bodies of matter at all distances. This attraction exists
not only between the heavenly bodies, the stars and planets, etc., but
is also found between bodies on the earth. A book attracts all objects
in a room and outside of a room as well, since its weight shows that it
is attracted by the earth itself. The gravitational attraction between
ordinary bodies is so slight that it requires careful experiments to
detect it. In fact, it is only when one of the attracting bodies is
large, as for example the earth, that the force becomes considerable.
Careful studies of the motions of the heavenly bodies, especially of
that of the moon in its orbit about the earth, led Sir Isaac Newton to
the statement of the _law of gravitation_ which is well expressed in the
following statement:
=88. Law of Gravitation.=--_Every particle of matter in the universe
attracts every other particle with a force that is directly proportional
to the product of their masses and inversely proportional to the square
of the distance between them._
The law may be separated into two parts, one referring to the masses of
the bodies concerned, the other to the effect of the distance between
them. The first part is easily understood since we all know that two
quarts of milk will weigh just twice as much as one quart. To illustrate
the second part of the law, suppose that the moon were removed to
_twice_ its present distance from the earth, then the attraction between
the earth and the moon would be _one-fourth_ its present attraction. If
removed to _three_ times its present distance, the attraction would be
_one-ninth_, etc.
The attraction of the earth for other bodies on or near it is called
_gravity_. The _weight_ of a body is the measure of the earth's
attraction for it; or it is the force of gravity acting upon it.
Newton's third law of motion states that every action is accompanied by
an equal and opposite reaction (Art. 80). Hence, the attraction of the
earth for a book or any other object is accompanied by an equal
attraction of the book for the earth.
=89. Weight.=--In advanced physics it is proved that a sphere attracts
as if it were concentrated at its center. Thus if the earth's radius be
considered as 4000 miles, then a body 4000 miles above the earth's
surface would be 8000 miles above the earth's center, or twice as far
from the center of the earth as is a body upon the earth's surface. A
body then 4000 miles above the earth's surface will weigh then but
one-fourth as much as it will at the surface of the earth.
Since the earth is flattened at the poles, the surface at the equator
is farther from the center of the earth than at points north or south.
Thus a body weighing 1 lb. at the equator weighs 1.002 lb. at Chicago,
or about 1/500 more. The rotation of the earth also affects the weight
of a body upon it so that at the equator the weight of a body is 1/289
less than at the pole. Both effects, that of flattening and of rotation,
tend to diminish the weight of bodies at the equator, so that a body at
the latter place weighs about 1/192 less than at the poles.
In studying the effect of the earth's gravity, the following
illustration will be helpful: Imagine an open shaft a mile square
extending through the earth. What would happen to a stone thrown
into the shaft? At first it would have the attraction of the whole
earth drawing it and continually increasing its speed downward. As
it descends from the surface, the pull toward the center grows less
and less. Halfway to the center the body has lost half its weight.
When the stone reaches the center, the pull in all directions is
the same, or in other words, _it has no weight_. It would, however,
continue moving rapidly on account of its inertia, and as it
continues on from the center, the greater part of the earth being
left behind, the attraction pulling toward the center will
gradually stop it. It will then fall again toward the center and be
stopped again after passing it, and after repeatedly moving up and
down will finally come to rest at the center of the earth. At this
point it will be found to be a body without weight since it is
pulled equally in all directions by the material of the earth. What
force brings the body to rest?
=90. Center of Gravity.=--A body is composed of a great many particles
each of which is pulled toward the center of the earth by the force of
gravity. A single force that would exactly equal the combined effect of
the pull of the earth for all the particles of a body would be their
resultant. The _magnitude_ of this resultant is the weight of the body.
The _direction_ of this resultant is in a line passing toward the
earth's center, while the _point of application_ of this resultant is
called the _center of gravity_ of the body. The center of gravity of a
body may also be briefly defined as _the point about which it may be
balanced_. As the location of this point depends upon the distribution
of matter in the body, the center of gravity is also sometimes called
the _center of mass_ of the body.
The earth's attraction for a body is considered for the sake of
simplicity, not as a multitude of little forces, but as a single force
applied at its center of gravity. To find the center of gravity of a
body find two intersecting lines along which it balances, see Fig. 72,
and the center of gravity will be at the intersection. A vertical line
through this point is sometimes called the _line of direction of the
weight_.
[Illustration: FIG. 72.--The center of gravity is at the intersection of
the lines of direction.]
=91. Equilibrium of Bodies.=--Equilibrium means equally balanced. A body
at rest or in uniform motion is then in equilibrium. An object is in
equilibrium under gravity when a vertical line through its center of
gravity passes through the point of support. A trunk is an example of a
body in equilibrium since a vertical line from its center of gravity
falls within the base formed by the area upon which it rests. Work will
be necessary to tip the trunk from its position. The amount of work
required will depend upon the weight of the body and the location of the
center of gravity.
=92. Kinds of Equilibrium.--(a) Stable.=--A body is in stable
equilibrium under gravity if its center of gravity is raised whenever
the body is displaced. It will return to its first position if allowed
to fall after being slightly displaced. In Fig. 73, _a_ and _b_ if
slightly tipped will return to their first position. They are in stable
equilibrium. Other examples are a rocking chair, and the combination
shown in Fig. 74.
[Illustration: FIG. 73.--Stable equilibrium.]
=(b) Unstable.=--A body is in unstable equilibrium under gravity if its
center of gravity is lowered whenever the body is slightly displaced. It
will fall farther from its first position. A pencil balanced on its
point or a broom balanced on the end of the handle are in unstable
equilibrium. The slightest disturbance will make the line of direction
of the weight fall outside of (away from) the point of support (Fig. 75
_a_).
[Illustration: FIG. 74--An example of stable equilibrium. Why?]
[Illustration: FIG. 75.--Unstable equilibrium _a_, neutral equilibrium
_b_.]
=(c) Neutral.=--A body is in neutral equilibrium if its center of
gravity is neither raised nor lowered whenever the body is moved.
Familiar examples are a ball lying on a table (Fig. 75 _b_) and a wagon
moving on a level street (referring to its forward motion).
[Illustration: FIG. 76.--_B_ is more stable than _A_.]
=93. Stability.=--When a body is in stable equilibrium, effort must be
exerted to overturn it, and the degree of stability is measured by the
effort required to overturn it. To overturn a body, it must be moved so
that the vertical line through its center of gravity will pass outside
of its supporting base. This movement in stable bodies necessitates a
raising of the center of gravity. The higher this center of gravity must
be raised in overturning the body, the more stable it is, _e.g._, see
Fig. 76. Thus a wagon on a hillside will not overturn until its weight
falls outside of its base, as in Fig. 77 _B_. The stability of a body
depends upon the position of its center of gravity and the area of its
base. _The lower the center of gravity and the larger the base_, the
more stable the body. What means are employed to give stability to
bodies, in every-day use (such as clocks, ink-stands, pitchers, vases,
chairs, lamps, etc.)?
[Illustration: FIG. 77.--_B_ will overturn; _A_ will not.]
Important Topics
1. Gravitation; law of gravitation, gravity, weight.
2. Center of gravity.
3. The three states of equilibrium. Stability.
Exercises
1. Why is a plumb-line useful in building houses?
2. What is the center of gravity of a body?
3. Explain the action of a rocking chair that has been tipped forward.
4. Is the stability of a box greater when empty or when filled with
sand? Explain.
5. How can you start yourself swinging, in a swing, without touching the
ground?
6. Is the center of gravity of the beam of a balance above, below, or at
the point of a support? How did you find it out?
7. Why are some ink bottles cone shaped with thick bottoms?
8. Would an electric fan in motion on the rear of a light boat move it?
Would it move the boat if revolving under water? Explain.
9. What turns a rotary lawn sprinkler?
10. Why, when you are standing erect against a wall and a coin is placed
between your feet, can you not stoop and pick it up unless you shift
your feet or fall over?
11. What would become of a ball dropped into a large hole bored through
the center of the earth?
12. When an apple falls to the ground, does the earth rise to meet it?
13. How far from the earth does the force of gravity extend?
14. Why in walking up a flight of stairs does the body bend forward?
15. In walking down a steep hill why do people frequently bend backward?
16. Why is it so difficult for a child to learn to walk, while a kitten
or a puppy has no such difficulty?
17. Explain why the use of a cane by old people makes it easier for them
to walk?
(6) FALLING BODIES
=94. Falling Bodies.=--One of the earliest physical facts learned by a
child is that a body unsupported falls toward the earth. When a child
lets go of a toy, he soon learns to look for it on the floor. It is also
of common observation that light objects, as feathers and paper, fall
much slower than a stone. The information, therefore, that all bodies
actually fall at the same rate in a vacuum or when removed from the
retarding influence of the air is received with surprise.
This fact may be shown by using what is called a coin and feather tube.
On exhausting the air from this tube, the feather and coin within are
seen to fall at the same rate. (See Fig. 78.) when air is again
admitted, the feather flutters along behind.
[Illustration: FIG. 78.--Bodies fall alike in a vacuum.]
=95. Galileo's Experiment.=--The fact that bodies of different weight
tend to fall at the same rate was first experimentally shown by Galileo
by dropping a 1-lb. and a 100-lb. ball from the top of the leaning tower
of Pisa in Italy (represented in Fig. 79). Both starting at the same
time struck the ground together. Galileo inferred from this that
feathers and other light objects would fall at the same rate as iron or
lead were it not for the resistance of the air. After the invention of
the air pump this supposition was verified as just explained.
[Illustration: FIG. 79.--Leaning tower of Pisa.]
=96. Acceleration Due to Gravity.=--If a body falls freely, that is
without meeting a resistance or a retarding influence, its motion will
continually increase. The _increase_ in motion is found to be constant
or uniform during each second. This uniform increase in motion or in
velocity of a falling body gives one of the best illustrations that we
have of uniformly accelerated motion. (Art. 75.) On the other hand, a
body thrown upward has uniformly retarded motion, that is, its
acceleration is downward. The velocity acquired by a falling body in
unit time is called its _acceleration_, or the _acceleration due to
gravity_, and is equal to 32.16 ft. (980 cm.) per second, downward, each
second of time. In one second, therefore, a falling body gains a
velocity of 32.16 ft. (980 cm.) per second, downward. In two seconds it
gains twice this, and so on.
In formulas, the acceleration of gravity is represented by "_g_" and the
number of seconds by _t_, therefore the formula for finding the
velocity, _V_,[F] of a falling body starting from rest is _V_ = _gt_. In
studying gravity (Art. 89) we learned that its force varies as one moves
toward or away from the equator. (How?) In latitude 38° the acceleration
of gravity is 980 cm. per second each second of time.
[F] _V_ represents the velocity of a falling body at the end of _t_
seconds.
=97. Experimental Study of Falling Bodies.=--To study falling bodies
experimentally by observing the fall of unobstructed bodies is a
difficult matter. Many devices have been used to reduce the motion so
that the action of a falling body may be observed within the limits of a
laboratory or lecture room. The simplest of these, and in some respects
the most satisfactory, was used by Galileo. It consists of an inclined
plane which reduces the effective component of the force of gravity so
that the motion of a body rolling down the plane may be observed for
several seconds. For illustrating this principle a steel piano wire has
been selected as being the simplest and the most easily understood. This
wire is stretched taut across a room by a turn-buckle so that its slope
is about one in sixteen. (See Fig. 80.) Down this wire a weighted pulley
is allowed to run and the distance it travels in 1, 2, 3, and 4 seconds
is observed. From these observations we can compute the distance covered
each second and the velocity at the end of each second.
[Illustration: FIG. 80.--Apparatus to illustrate uniformly accelerated
motion.]
In Fig. 63, if _OG_ represents the weight of the body or the pull of
gravity, then the line _OR_ will represent the effective component along
the wire, and _OS_ the non-effective component against the wire. Since
the ratio of the height of the plane to its length is as one to sixteen,
then the motion along the wire in Fig. 80 will be one-sixteenth that of
a falling body.
=98. Summary of Results.=--The following table gives the results that
have been obtained with an apparatus arranged as shown above.
In this table, column 2 is the one which contains the results directly
observed by the use of the apparatus. Columns, 3, 4, and 5 are computed
from preceding columns.
(1) (2) (3) (4) (5)
No. of Total Distance Velocity at Acceleration
seconds distance each second end of second each second
moved
Per second Per second
1 30 cm. 30 cm. 60 cm. 60 cm.
2 120 cm. 90 cm. 120 cm. 60 cm.
3 270 cm. 150 cm. 180 cm. 60 cm.
4 480 cm. 210 cm. 240 cm. 60 cm.
Column 5 shows that the acceleration is uniform, or the same each
second. Column 4 shows that the velocity increases with the number of
seconds or that _V_ = _at_. Column 3 shows that the increase in motion
from 1 second to the next is just equal to the acceleration or 60 cm.
This is represented by the following formula: _s_ = 1/2 _a_(2_t_ - 1).
The results of the second column, it may be seen, increase as 1:4:9:16,
while the number of seconds vary as 1:2:3:4. That is, _the total
distance covered is proportional to the square of the number of
seconds_.
This fact expressed as a formula gives: _S_ = 1/2_at_².
Substituting _g_, the symbol for the acceleration of gravity, for _a_ in
the above formulas, we have: (1) _V_ = _gt_, (2) _S_ = 1/2_gt_², (3) _s_
= 1/2_g_(2_t_ - 1).
=99. Laws of Falling Bodies.=--These formulas may be stated as follows
for a body which falls from rest:
1. The velocity of a freely falling body at the end of any second is
equal to 32.16 ft. per sec. or 980 cm. per second multiplied by the
number of the second.
2. The distance passed through by a freely falling body during any
number of seconds is equal to the square of the number of seconds
multiplied by 16.08 ft. or 490 cm.
3. The distance passed through by a freely falling body during any
second is equal to 16.08 feet or 490 cm. multiplied by one less than
twice the number of the second.
Important Topics
1. Falling bodies.
2. Galileo's experiment.
3. Acceleration due to gravity.
4. Laws of falling bodies.
Exercises
1. How far does a body fall during the first second? Account for the
fact that this distance is numerically equal to half the acceleration.
2. (a) What is the velocity of a falling body at the end of the first
second? (b) How far does it fall during the second second? (c) Account
for the difference between these numbers.
3. What is the velocity of a falling body at the end of the fifth
second?
4. How far does a body fall (a) in 5 seconds (b) in 6 seconds (c) during
the sixth second?
5. (a) What is the difference between the average velocity during the
sixth second and the velocity at the beginning of that second?
(b) Is this difference equal to that found in the second problem? Why?
6. A stone dropped from a cliff strikes the foot of it in 5 seconds.
What is the height of the cliff?
7. Why is it that the increased weight of a body when taken to higher
latitudes causes it to fall faster, while at the same place a heavy body
falls no faster than a light one?
8. When a train is leaving a station its acceleration gradually
decreases to zero, although the engine continues to pull. Explain.
9. Would you expect the motion of equally smooth and perfect spheres of
different weight and material to be equally accelerated on the same
inclined plane? Give reason for your answer. Try the experiment.
10. A body is thrown upward with the velocity of 64.32 ft. per sec. How
many seconds will it rise? How far will it rise? How many seconds will
it stay in the air before striking the ground?
11. 32.16 feet = how many centimeters?
12. The acceleration of a freely falling body is constant at any one
place. What does this show about the pull which the earth exerts on the
body?
(7) THE PENDULUM
=100. The Simple Pendulum.=--Any body suspended so as to swing freely to
and fro is a pendulum, as in Fig. 81. A simple pendulum is defined as a
single particle of matter suspended by a cord without weight. It is of
course impossible to construct such a pendulum. A small metal ball
suspended by a thread is approximately a simple pendulum. When allowed
to swing its vibrations are made in equal times. This feature of the
motion of a pendulum was first noticed by Galileo while watching the
slow oscillations of a bronze chandelier suspended in the Cathedral in
Pisa.
[Illustration: FIG. 81--A simple pendulum.]
=101. Definition of Terms.= _The center of suspension_ is the point
about which the pendulum swings. A _single vibration_ is one swing
across the arc. A _complete_ or _double_ vibration is the swing across
the arc and back again. The time required for a double vibration is
called the _period_. The _length_ of a simple pendulum is approximately
the distance from the point of support to the center of the bob.
A _seconds pendulum_ is one making a single vibration per second. Its
length at sea-level, at New York is 99.31 cm. or 39.1 in., at the
equator 39.01 in., at the poles 39.22 in.
A _compound pendulum_ is one having an appreciable portion of its mass
elsewhere than in the small compact body or sphere called a bob. The
ordinary clock pendulum or a meter stick suspended by one end are
examples of compound pendulums.
The _amplitude_ of a vibration is one-half the arc through which it
swings, for example, the arc _DC_ or the angle _DAC_ in Fig. 81.
=102. Laws of the Pendulum.=--The following laws may be stated:
1. The period of a pendulum is not affected by its mass or the material
of which the pendulum is made.
2. For small amplitudes, the period is not affected by the length of the
arc through which it swings.
3. The period is directly proportional to the square root of the length.
Expressed mathematically, _t_/_t´_ = √_l_/√_l´_.
=103. Uses of the Pendulum.=--The chief use of the pendulum is to
regulate motion in clocks. The wheels are kept in motion by a spring or
a weight and the regulation is effected by an escapement (Fig. 82). At
each vibration of the pendulum one tooth of the wheel _D_ slips past the
prong at one end of the escapement _C_, at the same time giving a slight
push to the escapement. This push transmitted to the pendulum keeps it
in motion. In this way, the motion of the wheel work and the hands is
controlled. Another use of the pendulum is in finding the acceleration
of gravity, by using the formula, _t_ = π√(_l_/_g_), in which _t_ is the
time in seconds of a single vibration and _l_ the length of the
pendulum. If, for example, the length of the seconds pendulum is 99.31
cm., then 1 = π√(99.31/_g_); squaring both sides of the equation, we
have 1² = π²(99.31/_g_), or _g_ = π² × 99.31/1² = 980.1 cm. per sec.,
per sec. From this it follows that, since the force of gravity depends
upon the distance from the center of the earth, the pendulum may be used
to determine the elevation of a place above sea level and also the shape
of the earth.
Important Topics
1. Simple pendulum.
2. Definitions of terms used.
3. Laws of the pendulum.
4. Uses of the pendulum.
Exercises
1. What is the usual shape of the bob of a clock pendulum? Why is this
shape used instead of a sphere?
2. Removing the bob from a clock pendulum has what effect on its motion?
Also on the motion of the hands?
3. How does the expansion of the rod of a pendulum in summer and its
contraction in winter affect the keeping of time by a clock? How can
this be corrected?
4. Master clocks that control the time of a railway system have a cup of
mercury for a bob. This automatically keeps the same rate of vibration
through any changes of temperature. How?
5. How will the length of a seconds pendulum at Denver, 1 mile above
sea-level, compare with one at New York? Why?
[Illustration: FIG. 82--Escapement and pendulum of a clock.]
6. What is the period of a pendulum 9 in. long? _Note._ In problems
involving the use of the third law, use the length of a seconds pendulum
for _l_, and call its period 1.
7. A swing is 20 ft. high, find the time required for one swing across
the arc.
8. A pendulum is 60 cm. long. What is its period?
9. If in a gymnasium a pupil takes 3 sec. to swing once across while
hanging from a ring, how long a pendulum is formed?
10. A clock pendulum makes four vibrations a second, what is its
length?
Review Outline: Force and Motion
Force; definition, elements, how measured, units, dyne.
Graphic Representation; typical examples of finding a component, a
resultant, or an equilibrant.
Motion; Laws of motion (3), inertia, curvilinear motion, centrifugal
force, momentum, (_M = mv_), reaction, stress and strain.
Moment of Force; parallel forces, couple, effective and non-effective
component.
Gravitation; law; gravity, center of; weight. Equilibrium 3 forms;
stability, how increased.
Falling Bodies; velocity, acceleration, "g," Laws; _V_ = _gt_, _S_ =
(1/2)_gt_² - _s_ = (1/2)_g_(2_t_ - 1).
Pendulum; simple, seconds, laws (3), _t_ = π√(_l_/_g_).
CHAPTER VI
WORK AND ENERGY
_104. Work._--"Whenever a force moves a body upon which it acts, it is
said to do work upon that body." For example, if a man pushes a
wheelbarrow along a path, he is doing work on it as long as the
wheelbarrow moves, but if the wheelbarrow strikes a stone and the man
continues to push and no motion results, from a scientific point of view
he is then doing no work on it.
"Work signifies the overcoming of resistance," and unless the resistance
is overcome no work is done. Lifting a weight is doing work on it,
supporting a weight is not, although the latter may be nearly as
tiresome as the former. Work as used in science is a technical term. Do
not attach to it meanings which it has in every-day speech.
=105. Measurement of Work.=--Work is measured by the product of the
force by the displacement caused
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