John Perry.

Applied mechanics; a treatise for the use of students who have time to work experimental, numerical, and graphical exercises, illustrating the subject: online

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LIBRARY

OF THE

UNIVERSITY OF CALIFORNIA.



Class



IJmVERSITY OF CALIFORNIA






Received



Accessions No...



LIBRARY



DEPARTMENT OF PHYSICS

6 1908 s \ i



Book No..!.






APPLIED MECHANICS

A Treatise for the use of Students who have time to work

Experimental, Numerical, and Graphical Exercises

illustrating the subject



BY



JOHN PERRY, M.E., D.Sc., LL.D., F.R.S.

WHIT. Sen., Assoc. M.lNST.C.E.

PROFESSOR OF MECHANICS AND MATHEMATICS IN THE ROYAL COLLEGE OF

SCIENCE, SOUTH KENSINGTON; PAST PRESIDENT OF THE INSTITUTION

OF ELECTRICAL ENGINEERS ', PRESIDENT OF THE PHYSICAL SOCIETY



WITH 372 ILLUSTRATIONS
NEW EDITION REVISED AND ENLARGED



OF THE

UNIVERSITY



NEW YORK
D. VAN NOSTRAND COMPANY

23 MURRAY, AND 27 WARREN STS

1907



T/
?

n



X^>V

/ . V OF THE A

\ UNIVERSITY I
/

X4L<FOR^X



PREFACE.



Tins book describes what has for many years been the course
of instruction in Applied Mechanics at the Finsbury Technical
College. All mechanical and electrical engineering students
in their first year had two lectures a week ; the substance of
these lectures is here printed in the larger type. Mechanical
engineers had three lectures a week in their second year ; the
substance of these lectures is here printed in small type. As
I found our arrangement of hours per week to work fairly
well, I give it here :





Mechanical Engineers.


Electrical Engineers.




1st year.


2nd year.


1st year.


2nd year.


Mathematics


4


4


4


4


Graphics and Machine Drawing


7


13


4


3


Mechanics: Lectures ...


2


3


2





Mechanical Laboratory


3


2


2





Mechanics: Numerical Exercises


2


1


2





Mechanism


1











Wood and Iron Workshops ...


4


7


3


6



Chemists and building trade students also attended in the
mechanical department, and the mechanical engineering
students had courses of study in the physics and chemistry
departments of the College. The Mechanics course included
work on the steam and gas engine not given in this book.
When, after much experience in teaching at an English public



173200



IV PREFACE.

school, in the Imperial College of Engineering, Japan, and
other places, I ventured sixteen years ago to publish my
method of teaching mechanics, it was met with some ridicule.
Even without encouragement I was prepared to pursue the
course which I had tested and found to be good, but I met
with a great deal of encouragement from thoughtful men of
about my own age. I had myself to scheme out and make
drawings for every piece of apparatus for the laboratory. I
knew of no collection of numerical and graphical exercises
which were suitable for students, but gradually a collection
was made from those given out in the lectures which seemed
to me to be less objectionable, less academic, less misleading,
than those hitherto available. I had difficulty in getting
clever assistants trained in academic ways to sympathise
with me. It was found in time that students took very eagerly
to the quantitative experimental work, and that the whole
system, faithfully followed, produced men whose knowledge was
always ready for use in practical problems, and who knew the
limits of usefulness of their knowledge. I am glad to say that
more than twenty complete sets of the apparatus have been
made and sent to various institutions by my workshop assistant,
Mr. Shepherd.

It would have made this book too large if I had included
in it, as I should have liked to do, copies of the instructions
which each student receives when he begins on a new piece of
apparatus.

Professor (now Sir Robert) Ball, at the Royal College of
Science, Dublin, started quantitative experimental mechanical
work. He used the well-known frame of the late Professor
Willis, which was taken to pieces and built up in new forms
for fresh experiments. What I have done has been to carry
out Professor Ball's idea, using a distinct piece of apparatus
for each fresh kind of experiment. A student measures things
for himself ; illustrates mechanical principles ; finds the limits
to which the notions of the books as to friction and properties
of materials are correct ; learns the use of squared paper, and



PREFACE. V

the accuracy of graphical methods of calculation ; and, above
all, really learns to think for himself. Professor Ewing, at
Cambridge, has developed the ideas of Professor Ball to a far
greater extent than what I have had opportunity for, and I
know of no place in which a better engineering education can
be obtained at the present time than at Cambridge. I am
glad to think that a system begun under the Science and Art
Department by Sir Robert Ball is now likely to be adopted
generally in science classes.

I am under great obligations to my assistant, Mr. G. A.
Baxandall, who has been to great trouble in adding to the
exercises, verifying answers, and correcting proofs. Professor
Willis, D.Sc., has been kind enough to read through the
proofs, and consequently I feel that there can be no im-
portant mistake anywhere.

I should like to think that, before a student begins the part
in small type, he has worked through Thomson and Tait'a
small book on " Natural Philosophy," and that he has read the
early part of my book on " The Calculus for Engineers."

JOHN PERRY.

IMh July, 1897.

Royal College of Science,

London S.W.



CONTENTS.



CHAP. PAGE

I. INTRODUCTORY 1

II. VECTORS : RELATIVE MOTION 29

III. WORK AND ENERGY 38

IV. FRICTION 56

V. EFFICIENCY 88

VI. MACHINES: SPECIAL CASES 98

VII. ELEMENTARY ANALYTICAL AND GRAPHICAL METHODS . . 120

VIII. EXAMPLES IN GRAPHICAL STATICS 151

IX. HYDRAULIC MACHINES 170

X. MACHINERY IN GENERAL 220

XL KINETIC ENERGY 242

XII. MATERIALS USED IN CONSTRUCTION 275

XIII. TENSION AND COMPRESSION .290

XIV. SHEAR AND TWIST 323

XV. MORE DIFFICULT THEORY 361

XVI. BENDING 381

XVII. STRENGTH AT ANY SECTION OF A BEAM . . . .393
XVIII. SOME WELL-KNOWN RULES ABOUT BEAMS .... 410

XIX. DIAGRAMS OF BENDING MOMENT AND SHEARING FORCE . 427

XX. MORE DIFFICULT CASES OF BENDING OF BEAMS . . .441

XXI. BENDING AND CRUSHING ....... 457

XXII. METAL ARCHES 478

XXIII. MEASUREMENT OF A BLOW ..... . 486

XXIV. FLUIDS IN MOTION 505

XXV. PERIODIC MOTION 546

XXVI. MECHANISM 570

XXVII. CENTRIFUGAL FORCE . . 597

XXVIII. SPRINGS 613

XXIX. RESILIENCE OF SPRINGS . . 641

XXX. CARRIAGE SPRINGS , * .... 646

APPENDIX . . . . . . * . , . . 654

.667



LIST OF TABLES.



PAGE

I. Normal Pressure of Wind against Roofs 157

II. Moments of Inertia and the M and k of Rotating Bodies . . 251

III. Young's Modulus (Wertheim) 302

IV. Factors of Safety for Different Materials and Loading . . 305
V. Ultimate Stresses for Loads Repeated a Great Number of Times 310

VI. Moment of Inertia and Strength Modulus of Various Sections .

398-400

VII. Strength and Stiffness of Rectangular Beams Supported at the

Ends and Loaded in the Middle 411

VIII. Diagrams of Bending Moment, with Strength and Stiffness, of
a Uniform Beam when Supported or Fixed and Loaded in

Various Ways 414-16

IX. Breaking Stress of Iron and Timber Struts . . . .468
X. Values of the Constant B for Different Lengths of Struts and

Different Materials .468

XI. Values of the Constant n for Struts of Different Sections . 469

XII. Relative Amplitudes for Different Frequencies in a Case of

Forced Vibration 616

XIII. Spring Materials Subjected to Bending . .. . . . 621

XIV. Materials for Cylindric Spiral Springs . . ... . 622

XV. Values of Torsional and Flexural Rigidity and Axial Load for

Various Sections of Wire used in Springs .... 635

XVI. Relative Elongations of Various Springs for Same Axial Loads. 638
XVII. Proof Load and Resilience of Various Springs .... 640

XVIII. Resilience of Spring Materials for Different Kinds of Loading . 642
XIX. Useful Constants . . . . . . . 654-5

XX. Moduli of Rigidity . . *, : ,. . . . 656

XXI. Moduli of Compressibility ... . . . . . 657

XXII. Melting Points, Specific Gravity, Strength, etc., of Materials 658-61

XXIII. Logarithms. . , V- . - > > - - G62-3

XXIV. Antilogarithms . . . . . *; .3 . . 6G4-5
XXV. Angles in Degrees and Radians, Sines, Tangente, etc. . . 606



APPLIED MECHANICS.



CHAPTER I.

INTRODUCTORY.

1. THE student of Applied Mechanics is supposed to have some
acquaintance already with the principles of mechanics ; to be
able to multiply and divide numbers and to use logarithms ;
to have done a little practical geometry; to know a little
algebra and the definitions of sine, cosine, and tangent of an
angle ; and to have used squared paper. He is supposed to
be working many numerical and graphical exercises ; to be
spending four hours a week at least in a mechanical laboratory ;
to be learning about materials and tools in an iron and wood
workshop ; and to be getting acquainted with gearing and
engineering appliances in a drawing-office and elsewhere.

Unfortunately, many students are deceived as to their fitness
to begin the study of A.pplied Mechanics, and we think it
necessary in this introductory chapter to suggest some pre-
paratory exercise work, and also to state certain definitions
and facts which will be afterwards referred to, perhaps as if
they were still unknown.

When we think of what goes on under the name of teaching
wo can almost forgive a man who uses a method of his own,
however unscientific it may seem to he. Nevertheless, it is not
easy to forgive men who, because they have found a study interest,
ing themselves, make their students waste a term upon it, when
only a few exercises are wanted on what is sometimes called the
scientific study of arithmetic, for example, or of mensuration.

In our own subject of Applied Mechanics there are teachers
who spend most of the time on graphical statics, or the graph-
ing of functions on squared paper, or the cursory examination of
thousands of models of mechanical contrivances. One teacher seems
to think that applied mechanics is simply the study of kinematics
and mechanisms ; another, that it is simply exercise work on pure
mechanics ; another, that It is the hreaking of j specimens on a large
testing machine ; another, that it is the trying to do in a school or
college what can only he done in real engineering works ; another,
that it is mere graphics; another, that it is all calculus and no



APPLIED MECHANICS.

graphics; another, that it is all shading and colouring and the
production of pretty pictures without centre lines or dimensions.
Probably the greatest mistake is that of wasting time in a school
in giving the information that one cannot help picking up in one's
ordinary practical work after leaving school.

We belieA r e that the principles which an engineer really
recollects and keeps ready for mental use are very few. By
means of lectures, models, drawing-office and laboratory, and
numerical exercise work, we show a man how these simple
principles enter, in curiously different-looking shapes, into his
engineering practice. We give him the use of all the necessary
. methods of study, and we send him out into practical life pre-
pared to study things for himself. We ought to recognise the
fact that his real study of his profession is not at school or college.
We ought to teach him how to learn for himself. Any child can
state Newton's second law of motion, and the other half-dozen
all-important principles of mechanics, so as to get full marks in an
examination paper; the engineer knows that the phenomena he
deals with are exceedingly complex, and that only a long ex-
perience will enable him to utilise the so easily stated principles.
Schools and colleges are the places in which men ought to learn
the uses of all mental tools ; they are sure to specialise afterwards,
but in the meantime we ought to give them plenty of tools to
choose from. The average student cannot take in more than the
elementary principles , the best students need not take in more.

2. The most important lesson for a beginner, however he
may have studied mathematics and mechanics, and however able
he may be as a mathematician, is this that he must not go on
merely assuming that he knows how to do things ; he must know
things by actual trial and not mere hearsaj. He must actually
calculate certain numerical results ; he must actually illustrate
principles with laboratory apparatus : and, if there is a school
workshop, he must get to know the properties of materials by
chipping and filing and paring and planing and turning. It
is just the same as in one's after-school work. There is no
great mechanical engineer who has not himself worked like
a workman with other workmen, and got to understand men
and things by actual contact with them. The man who shirks
the following exercises and laboratory work will lose a great
deal more than he is aware of.

Teachers will notice that things requiring even a little pre-
paration more than other things will gradually become neglected.
Therefore, let it be part of the daily work for every student to use
logarithms, drawing instruments in graphical exercise work, and
squared paper. In the drawing-office, blue prints cught to be
made by some student or other every day ; the planimeter ought



APPLIED MECHANICS. 3

to be used every day, and some student ought to be resetting his
drawing-pens and other instruments every day. It ought to be a
rule that all apparatus must always be ready for use, and that it is
always in use. Teachers can arrange their own work in such a
way that they cannot help seeing every day how the practical work
of students is being done. When we find our system to be going
with clockwork regularity and we feel no worry, we ought to
believe that some change is necessary. If we find that the students
are not absorbed in their work, we must understand that we
teachers are in fault. (See Appendix.)

3. Students cannot spend too much time in multiplying,
factoring, and simplifying algebraical and trigonometrical
expressions. These are our tools, and we must get familiar
with them. We may easily spend too much time in studying
roots of equations, permutations and combinations, etc., and
in the solution of triangles ; and therefore, if it is possible, we
try to learn all our mathematics, mechanics, physics, and
chemistry from teachers who are engineers. What acquaint-
ance with these subjects we have, ought to be a real knowledge,
not the glib pretence which suffices for examinations ; it must
not be something apart from our life and work. To effect this
object we must work many numerical and graphical exercises,
and try to conquer our contempt for simple laboratory experi-
ments, and illustrate the forty-seventh proposition of the Fir^t
Book of Euclid by actually drawing some right-angled triangles
and measuring their sides to illustrate rules about triangles
by actual measurement. In this way we learn much more
than the ordinary geometrician knows ; among other things,
we obtain a valuable knowledge of the errors we are likely to
make in graphical calculation.

4. Every student is supposed to be able to calculate the
values of any algebraical or trigonometrical expression when
numerical values are given. He ought to be able, when given
at random such an expression as

w = 2/ 3 fl-Va j. ,/m 2 + n 2 a log e b . cos. 0,

to be able to calculate w when he knows the values of a, b, m,
n, and 0. He ought to be able to use logarithms in multiplica-
tion and division and extraction of roots ; to know all the usual
mathematical symbols, and how it is sometimes convenient to
use ^/a and sometimes ah. It is pedantic to say that a
man must not use a formula unless he is able to prove its
truth. It is usually of great help in learning to prove a



* APPLIED MECHANICS.

formula to have previously used the formula and know the
value and meaning of what we are to prove. A living
Northern professor of great eminence has declared that a boy
ought not to be allowed to use logarithms until he is able
to calculate them ; he has not said that a boy ought not to
use a watch or wear a coat until he is able to make them.

EXERCISES.

1. Find 4-326 x 0*003457 to four significant figures, leaving out all
unnecessary figures in the work. Find p'01584 -j- 2-104 to four signifi-
cant figures. Also do these using logarithms. Find log e 7. Calculate

5 2 ' 43 , 3-' 246 , -042 ' 476 , /\/246-3, 31-01* x 0-02641**

Ans., 0-01495, 0'007529, 1-94591, 49-95, 0-7632, 0-2211, 3-008, 5'872.

2. If m = (a 3 + 2 a?b + * - '345)* -j- (a 8 J 2 ) 5

find m if a = 0'504, 6 = 0'309, * = 1'567.

Ans., 1-453 x 10 8 . '

3. What errors are there in assuming

(1 + a ) = 1 + na

to be true in (1-001) 3 = 1-003, (1-01)* = 1-0033.

(0-99) 2 = (1 - -Ol) 2 = 1 -02 = -98.

= (1 -01) ~ l = 1 + -01 = 1 01.




= (1 + -01) "~ = (1 - -0033) = -9967.
'1-01

- -01) = 10 (1 - -01)* = 10 (1 - -005).

= 9-95 ?

The above answers are very nearly correct ; the student is expected to
find the correct answers.

4. How much, error is there in the assumptions

1+Jj = 1 + a _ IB, (1 + a) (1 + /B) = 1 + a + 0,

when a = '01 j8 = -01, a = - '003 = - '005 ?

Ans., No error : -01 per cent., '004 per cent., -0015 per cent.

5. If d is the diameter of the bore or the " calibre " of a gun, it is
usually assumed that the weight of the gun is proportional to d s , and
that the thickness of armour which its projectile will pierce is propor-
tional to d. If an 8-inch gun weighs 14 tons and can pierce 11 inches of
armour, what thickness will be pierced by a 10-inch gun, and what is the
weight of the gun? Am., 13*75 inches, 27 '34 tons.

5. The linear expansion of bodies by heat is practically
proportional to the rise of temperature. The values of a, the
co-efficient for linear expansion (the fractional increase in
length for a rise in . temperature of 1 Centigrade), are the

* See Appendix.



APPLIED MECHANICS. 5

following numbers divided by 10 5 : Aluminium, 2-34 ; copper,
1-79; gold, 1-45; iron, 1-2; lead, 2-95 platinum, 0-9 ; silver,
1-94; tin, 2-27 ; zinc, 2-9 j brass (71 copper to 29 zinc), 1-87;
bronze (86 copper to 10 tin to 4 zinc), 1-8; German silver,
1-8; steel, 1-11; brick, 0-5; glass, 0-9; granite, 0-9; sand-
stone, 1*2; slate, 1*04; boxwood (across the fibre), 6*1; box-
wood (along the fibre), 0'3 ; oak (across), 5 '4 ; oak (along).,
0*5 ; pine (across), 3'4; pine (along), 0'5.

The co-efficient, k, of cubical expansion is three times the
co-efficient of linear expansion, because (1 + a) 3 = 1 + 3 a
is practically correct for these small values of a. The average
values of k between and 100 0. are the following numbers
divided by 10 3 : Alcohol, 1'26 ; mercury, O18 ; olive oil, 0-8;
petroleum, 1-04 ; pure water, 0*43 ; sea- water, 0*5.

The student is supposed to have worked many exercises
like the following ones :

1. Steel rails of 0. have an aggregate length of 1 mile. What is
the length at 33 0. ? Ans., 1 mile 24-2 inches.

2. A ring of wrought iron has an inside diameter of 5 feet when at a
temperature of 970 C. What is the diameter at C. ? Ans., 4-9 feet.

3. A cylindric plug of copper just fits into a hole 4 inches diameter in
a piece of cast iron. After heating the mass to 1,240 C., by how much
is the diameter of the hole too small for the plug? Ans.^ -0293 inch.

4. A har of iron is 70 centimetres long at C. What is its length in
boiling water (100 C.) ? What is its length at 50 C. ?

Ans., 70'079 centimetres, 70*039 centimetres.

5. Two rods one of copper, the other of iron measure 98 centimetres
each at C. What is the difference in their lengths at 57 0. ?

An,*., '027 centimetre.

6. Bars of wrought iron, each 3 '4 metres long, are laid down at a
temperature of 10 C. What space is left between every two if they are
intended to close up completely at 40 0. ? Ans., 1*26 millimetres.

7. A wrought-iron connecting-rod is 12 feet long at 10 C. What is
the increase of length at 80 C. ? Ans., 0-121 inch.

8. A wrought-iron Cornish toiler is 33 feet long ; the shell is at 0.,
the flue at 100 C. What would the difference of the lengths be if the flue
were not prevented from expansion ? Ans., 0*475 inch.

9. A steel pump rod is 1,000 feet long. What is its change of length
for a change of 10 C. Ans., 1'44 inch.

10. In a thermometer "01 cubic inch of mercury at 10 C. is raised to
15 C., and rises 1 inch in the tube. What is the cross- section of the tube ?

Ans., 9 x 10~ 6 S( l uare inch -

11. The volume of a lump of iron being 5 cubic feet at 10 0., find its
volume at 80 C. Ans., 5-0126 cubic feet.

6. A student's knowledge of mathematics ought to be such
that he can work out for himself all the rules given in such an



6 APPLIED MECHANICS.

excellent book on mensuration as that of Professor A. Lodge.
The thorough study of such a book is one of several ways
which may be recommended of getting familiar with mathe-
matical principles. But nobody's life is long enough to use
all these ways, and, besides, unnecessary study leads to dul-
ness. Hence, if a student has taken some other way, he need
not be alarmed at his ignorance of the more complex rules in
mensuration ; he may feel absolutely certain that he can work
out such rules for himself, given time and necessity. He will
study the more complex rules, such as prismoidal formulae, if
he needs to use them practically, not otherwise. The following
rules are in constant use and must be familiar to the student,
whether or not he knows the reasons for them. If he is
familiar with the rules and does not anxiously search for the
reasons for them, he lacks the necessary spirit of the practical
engineer.

RULES IN MENSURATION.

An area is found in square inches if all the dimensions are
given in inches. It is found in square feet if all the dimen-
sions are given in feet.

Area of a parallelogram. Multiply the length of one side
by the perpendicular distance from the opposite side.

The centre of gravity of a parallelogram is at the point of
intersection of its diagonals.

Draw a right-angled triangle ; measure very accurately the
lengths of the sides. You will find that, no matter what scale
of measurement you use, the square of the length of the
hypothenuse is equal to the sum of the squares of the lengths
of the other two sides.

B Area of a triangle. Any side multiplied
by its perpendicular distance from the oppo-
site corner and divided by two.

The centre of gravity, or, rather, the centre
Jig. i. of area, of a triangle is found by joining

(Fig. 1) any corner, A, with the middle
point, D, of the opposite side, B c, and making D G one-third
of D A. G is the centre of gravity.

Area of an irregular figure. Divide into triangles, and
add the areas of the triangles together.

Circumference of a circle. Multiply the diameter by
3-1416.




APPLIED MECHANICS. 7

Arc of a circle. From eight times the chord of half the
arc subtract the chord of the whole arc ; one-third of the
remainder will give the length of the arc, nearly.

Area of a trapezium. Half the sum of the parallel sides
multiplied by the perpendicular distance between them.

Area of a circle. Square the radius, and multiply by
3 '141 6 ; or square the diameter, and multiply by O7854.

Area of a sector of a circle. Multiply half the length of
the arc by the radius of the circle.

Area of a segment of a circle. Find the area of the sector
having the same arc, and the area of the triangle formed by
the chord of the segment and the two radii of the sector.
Take the sum or difference of these areas as the segment is
greater or less than a semicircle.

Otherwise, for an approximate answer : Divide the cube
of the height of the segment by twice the chord, and add the
quotient to two-thirds of the product of the chord and height
of the segment. When the segment is greater than a semi-
circle, "subtract the area of the remaining segment from the
area of the circle.

The areas of curves may be found by Simpson's rule.
Divide the area into any even number of parts by an odd
number of equidistant parallel lines or ordinates, the first and
last touching the bounding curve. Take the sum of the
extreme ordinates (in many cases each of the extreme
ordinates is of no length), four times the sum of the even
ordinates, and twice the sum of the odd ordinates (omitting
the first and last) ; multiply the total sum by one-third of the
distance between any two successive ordinates.

The ordinary rule for an indicator diagram is : Draw
lines at right angles to the atmospheric line, touching the
extreme ends of the diagram. Divide the distance between
them into ten equal parts (a parallel ruler with ten pieces is



Online LibraryJohn PerryApplied mechanics; a treatise for the use of students who have time to work experimental, numerical, and graphical exercises, illustrating the subject: → online text (page 1 of 61)