Graphical Methods in Applied Mathematics

PROPERTY OF THE CA1NEGIE INSTITUTE OF TECHNOLOGY

LIBRARY

MAY

GRAPHICAL METHODS IN APPLIED MATHEMATICS

MACMILLAN AND LONDON

CO., LIMITED

BOMBAY CALCUTTA MELBOURNE

THE MACMILLAN COMPANY NEW YORK !v

BOSTON

CHICAGO

ATLANTA -SAN FRANCISCO

THE MACMILLAN

CO.

OF CANADA,

TORONTO

LTD.

"

First published in 1908, under the title of Graphics.' " Reprinted as Graphical Methods," 1909.

GLASGOW PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. :

PREFACE. THE

importance of Graphics in modern mathematical training, numerous uses in practical work, render unnecessary any excuse for the publication of an elementary account of some of its applications, provided these applications are chosen with discretion and treated with clearness.

and

its

The author is hopeful that competent judges will consider that the present book fulfils these requirements. It has not been written with a view to any particular examination ; but the easier parts will be found to meet the needs of secondary schools of candidates in military and naval examinations ; while

and

students in technical colleges and candidates in the examinations of the University of London will, it is believed, find most of

the chapters of definite use to them. All sections and exercises marked with an asterisk should be

omitted in a

first

reading of the volume

further to curtail the course of

Course mapped out on page Special attention

work

;

who wish an easy First

students

will find

ix.

directed to the large number of concrete out in detail, which are supplied in the is

examples, worked It various chapters.

essential that the student should himself

is

work out the graphical constructions according to the instructions given, and afterwards compare his results with those To obtainable by measurement of the figures in the text. avoid

the

tendency

characterise the

work

to of

produce very small figures, which almost all students, the instructions

supplied will be found to determine large drawings in nearly An endeavour should be made so to construct the all cases.

diagrams that figures in the

;

it is

all

lengths are correct to at least three numerical this degree of accuracy has been attained

hoped that

answers given at the end of the book.

Owing

to slight,

PREFACE.

vi

final result, perhaps very slight, errors in construction the obtained by measurement, will often be slightly incorrect in

the third figure. The student is graphical

strongly

methods

only

in

urged not to statics and

to

himself

confine

The

mensuration.

and graphics may be likened to the use of our two hands; no matter how highly developed one instrument may be, much more can be done with the two Of necessity, in this book conjointly than with one alone.

employment

of

calculation

methods and calculations are only incidentally touched upon, but students with a knowledge of Trigonometry will see that even roughly drawn vector polygons can easily be used for purposes of calculation. This opportunity is gladly taken to acknowledge a debt analytical

of gratitude to Prof. Henrici,

College,

London

to

whom

F.ES.,

of the Central Technical

the author's

first

knowledge

of

the

His teaching showed Statics true value of Graphics is due. and Dynamics not merely as a branch of somewhat unsatisfying Mathematics, but as a real and interesting subject with important Those acquainted with Prof. Henrici's work and

applications.

lectures will appreciate the author's obligation to him. Thanks are also due to Mr. E. F. Witchell of the Central

Technical and Goldsmiths'

Colleges for reading most of

the

proof sheets, suggesting improvements, and correcting some of the answers; to Prof. B. A. Gregory and Mr. A. T. Simmons for their unsparing trouble during the preparation of the

MSS.

and while the book was passing through the press; and finally to the Senate of the University of London and the Controller of H.M. Stationery Office for permission to make use of problems set

in

Board

various University, Civil Service, Naval, Military, and Education examinations. The source of each such

of

problem, and the date when given after the question.

subsequent to

1902,

G. C, SIDCUP, November, 1907-

has been

TfJENEK.

CONTENTS. CHAPTER

I.

PAGE

GRAPHICAL ARITHMETIC,-

1

Miscellaneous Examples

41

/.,

CHAPTER GRAPHICAL MENSURATION,

II.

43

-

CHAPTER

-

Miscellaneous Examples III.,

CONCURRENT FORCES,

69 *

-

-

115

-

-

119

-

-

166

IV.

-

Miscellaneous Examples IV.,

~

V.

173

-

THE LINK POLYGON, Miscellaneous Examples

-

F".,

65

VELOCITIES,

-

CHAPTER

-

III.

VECTORS AND THEIR APPLICATION TO ACCELERATIONS AND MASS-CENTRES,

CHAPTER

-

-

Miscellaneous Examples IL,

-

-

-

208

CONTENTS.

viii

CHAPTER

VI. PAGE

STRESS DIAGRAMS,

-

-

-

-

-'-210

FL,

-

-

-

-

-

Miscellaneous Examples

CHAPTER

252

VII.

256

FRICTION, Miscellaneous Examples

VIi

CHAPTER

MOMENTS,

-

-

281

-

'.,

--------VIII.

Miscellaneous Examples Fill.,

CHAPTER

IX.

BENDING MOMENT AND SHEARING FORCE, Miscellaneous Examples IX.,

CHAPTER STRESS DIAGRAMS

....

307

....

337

-

-

-

335

X.

(continued}^

Miscellaneous Examples

284 305

-

X,

353

CHAPTER XL WORK, Miscellaneous Examples XT.,

APPENDIX.

-

EXPERIMENTS ON MOMENTS,

-

-

-

354

-

367

368

INSTRUCTIONS TO THE STUDENT. IfoR the construction of the figures required in this book a set square The side of the 45 set square should be at 3-ineli side is useless. least 6 or 7 inches.

with a

The standard

scale used should be flat

on one side and bevelled on

the other and the scale divisions should reach to the edge. One edge should be divided into fiftieths of an inch and the other into millimetres or half-millimetres.

&

Messrs. Aston

makers, at

An

angle

this

description can be obtained from

6d. each.

Is.

is

Scales of

Mander, Old Compton Street, London, W., and other

best set off or measured

scale of chords.

If a protractor is

used

by means it

of its tangent or by a should be a large semicircular

one of transparent material.

Hard

chisel-pointed pencils should be used for all the constructions.

A FIKST COURSE OF STUDY. Ch. L, pp. 1-17, 28-30, 33-35, 37-40. Oh. II, pp. 4348, 50, 53-59. Ch. III., pp. 69-78, 84-92, 95-111. Ch. IV., pp. 119-134, 137-162.

Ch.

V,

pp. 172-200.

Ch. VI., pp. 210-222, 236-245, Ch. VII., pp. 256-264.

Ch. VIII., pp. 284-303. Ch. XI., pp. 354-367.

CONTENTS.

viii

CHAPTER

VI. PAGE

'-210

STRESS DIAGRAMS, Miscellaneous Examples

-

PL,

CHAPTER

-

252

-

VII.

256

FRICTION, Miscellaneous Examples

VIL,

CHAPTER

-

-

-

-

-

-

-

-

281

VIII.

284

MOMENTS, Miscellaneous Examples

VIII.,

CHAPTER

305

IX.

BENDING MOMENT AND SHEARING FORCE, -

Miscellaneous Examples IX.,

CHAPTER STRESS DIAGRAMS

(continued),

Miscellaneous Examples

307

335

X.

-----

X,

-

337 353

.

.

.

.

.

367

-

-

-

368

CHAPTER XL 354

WORK, Miscellaneous Examples

APPENDIX.

XL,

EXPERIMENTS ON MOMENTS,

INSTRUCTIONS TO THE STUDENT. i\>K the construction of the figures required iri this book a set square is useless. The side of the 45 set square should be at least 6 or 7 inches. The standard scale used should be flat on one side and bevelled on One edge the other and the scale divisions should reach to the edge. should be divided into fiftieths of an inch and the other into millimetres Scales of this description can be obtained from or half-millimetres. Messrs. Aston & Mander, Old Compton Street, London, "W,, and other makers, at Is. 6d. each. An angle is best set off or measured by means of its tangent or by a If a protractor is used it should be a large semicircular scale of chords. one of transparent material. Hard chisel-pointed pencils should be used for all the constructions.

with a 3-inch side

A FIRST COURSE OF STUDY. Ch. Oh. Ch. Ch. Ch. Ch. Ch. Ch. Ch.

pp. 1-17, 28-30, 33-35, 37-40. pp. 43-48, 50, 53-59. III., pp. 69-78, 84-92, 95-111. IV., pp. 119-134, 137-162. V., pp. 172-200. VI., pp. 210-222, 236-245, VII., pp. 256-264. VIII., pp. 284-303. XI., pp. 354-367. I.,

II.,

CHAPTEE

I.

GRAPHICAL ARITHMETIC. Scalar Quantities. such

as

In Mechanics and Physics, quantities

masses, time, temperature, disSome of these placement, velocity and force are dealt with. quantities are related to direction in space and cannot be defined

numbers,

volumes,

without reference to direction, others have no such relation to space.

Mass, time, temperature, volume and number are examples of quantities which are completely given when we know the kind of quantity and how much there is of it; they are called Scalar Quantities. To specify the amount needs reference to some unit, a gramme, a degree centigrade, a cubic centimetre, the number 1 so that Scalar Quantities are specified by giving . . .

(1)

the unit quantity,

(2) the

number

,

of units.

Those quantities which require for their some reference to direction in space are called

Vector Quantities. specification

Vector Quantities. city, acceleration,

An

Examples

of these are

displacement, velo-

force, ....

from a minute only in amount, but the book differs from the pull of a locomotive amount but also in direction. a scalar quantity and force a vector quantity.

hour

difTers

pull of the earth on a on a train not only in

Time

is

Quantities to Scale.

The word

scalar

is

used because these

quantities can be graphically represented to scale by lengths (Latin scalae a ladder divided into equal parts by the rungs). Thus, if we agree to represent unity by a length of 3 cms. then

the

number

3

would be represented by a line 9 cms. long, and a would represent the number 3*5.

line of length 10*5 cms.

GRAPHICS.

2

cases of the representation of physical quantities by the scale of the representation, i.e. the length reprelengths, the unit quantity, must be given either directly or by senting

In

all

implication.

Masses to

Scale.

EXAMPLE.

To

corresponding

to

construct

any

a

length^

scale

and

the

of masses length

so

that

mass

the to

corresponding

any

mass, can be read off at once.

The given line % (Fig. 1) represents 1 Ib. mass. Transfer this length to your drawing paper, by pricking through with needle points or by the aid of dividers (having fine adjustment), and mark the end points Mark off on this (left) and 1 (right). produced, lengths giving 2, 3, 4, ... 10 Ibs., as follows In the figure 01 is (intentionally) not the same length as

line

:

u.

With

dividers accurately adjusted to the length u, and with (i) the right-hand point as centre (marked 1 in figure) describe a

PIG.

1.

semicircle clockwise, pricking a slight mark at the point (marked 2 in figure) where the semicircle cuts the line. With 2 as

centre describe a semicircle at 3,

contraclockwise, pricking through clock- and contraclockwise half

and so on by alternate

MASSES TO SCALE.

3

With a properly revolutions, pricking through points 4, 5,... 10. adjusted straight edge and set square draw fine sharp, short lines perpendicular to 01 through the points marked.

Draw

(ii)

a

straight

and 60

between 20

line through making some angle inches with 01, and mark off from

10' along it. a Adjust straight edge and set square with one edge of the latter passing through 1 and 1', so that when the set square is moved parallel to itself along the straight edge to 10' it

T,

...

2', 3',

still

intersects

when the

Mark

01 produced.

set square passes

through

the points on 01 produced 2',

3',

...

10'

by

short, sharp,

fine lines.

These points so determined should coincide with the points already marked on 01 produced why ? We have now 01 representing 1 Ib. mass, and 07 a mass of ;

7

Ibs., etc.

To

(iii)

obtain,

by method

(ii),

the division marks perpendicular

to 01.

Draw

a fresh straight line

and mark

off

01

u on

it.

Place

the inch scale and set square so that V I is perpendicular to 01 when the scale edge passes through 0, and mark off the points 2, 3,

...

as before.

EXAMPLE. To find the length which (a) With a scale, adjusted at 3*7"

from

this

point parallel to 3

along the

then the length from (5)

Produce 10 and

mark

represents 3-7

as before,

Ibs.

mark

the point

and with set square adjusted at mark a point on 01 produced 3*7,

scale, 3',

to 3*7 represents 3-7 Ibs.

VO

backwards through

0,

tenths of inches along the latter

and with an

up to 1 inch, and from these points draw parallels to 3 3' cutting 01 produced in points marked O'l, 0-2, 0*3, ... 0*9. Then the distance between 0*7 and 3 represents 3*7 Ibs. The final result in (b) is a scale of masses from which the and 11 Ibs. or the length corresponding to any mass between inch scale

mass corresponding to any length

may

be found.

GRAPHICS.

4 (1)

With squared paper

or straight edge ruled in

number represented by the

line a, if

FIG.

(2)

Make

a scale for numbers from

representing unity being

(3)

On a plan and J

feet.

(ii)

mms.

or 2

mms.

find the

2 cms. represents unity.

2.

to 10 on squared paper, the length

3.

inch represents 3 feet. Draw a scale giving |> length represents 7 ft. 6 in., and what length is and by the line a ?

house

of a

What

represented by 3*2

cm.

u.

FIG.

feet

1

(i)

ins.,

FIG.

4,

(4) The areas of certain fields are represented by lengths to the scaie Draw a scale giving 1 to 5 acres, tenths of an acre of 6 cms. to an acre. and hundredths of an acre. Read off from your scale the area represented by 17*3 ems. and the length which represents 4*25 acres.

Addition.

EXAMPLE. -J-

an inch

The

lines a, b, c, d, represent

Find

to unity.

the

sum

numbers

to the scale

of

of the numbers.

JL FIG.

Take a the

lines,

paper with a straight edge and apply in turn to marking with a fine sharp line the beginnings and ends strip of

of the segments so that the

equal to b

The

and so

5,

OD

segment

OA

is

equal to

on.

edge, then,

In Fig.

5.

is

is

marked

OABCD

as in Fig. 5.

one-half the true length.

a,

AB

is

ADDITION.

OD

Measure

by

2), this

5

in half inches (or in inches

number

and multiply mentally

of half inches is the required

Notice that the order of addition

is

sum.

immaterial.

(5) A scale pan is suspended from the hook of a spring balance, and it is loaded with small shot. The shot is put in by means of a small scoop. The weight of shot added each time is given by the lines a, 5, c, d, e, and the line u represents 1 oz.

FIG.

6.

Find graphically the reading of the spring balance at each addition to the load.

(Add the lengths as above and then draw the u

scale along the straight

edge.) (6)

scale

A weight of shot given by the line in Fig. 7 pan ; what is the reading of the balance ? FIG.

(7)

What

is

the

is

taken out of the

7.

peri-

meter of the room of which the accompanying figure is the plan, drawn -to a scale of 0'6" to lift.

FIG.

8.

Subtraction.

EXAMPLE. scale

The

of 1"5 cms.

number and

the

lines a, b, c, d, (Fig, 9) represent

to unity.

sum

of the

Find

numbers

the difference between

the

to the

second

rest.

the lengths a + c + d as before and obtain OD on the to the left D=*b, then OB is straight edge, cut off from

Add

D

GRAPHICS.

6

the length representing the required number. 9, (In Fig. OA, ... are half their true lengths.) Eead off the length of OB on the 1/5 cm. scale.

ABC

O

FIG.

Notice that since addition

is

D

9.

performed as a continuous process

by adding lengths from left to right, subtraction must be performed by setting off distances from right to left, if we wish to measure our result from 0.

EXAMPLE. third numbers

Mark, the

left,

Required the number equal

minus

the

as before,

sum of

the second

on a straight edge,

OD = d and DB = b.

sum

of the first and

ACc,

OA=a,

The point

O

B

to the

and fourth.

B

then to

comes to the

D~A

left of

C

FIG. 10.

the starting point (the origin), and the length OB, measured to the left instead of the right, corresponds to the fact that the Measure OB on the proper required difference is negative. scale and prefix a negative to the number. sign If distances

to the right of

distances to the left

represent positive numbers,

must represent negative numbers.

Scale of Numbers.

Such a

line as

BODAC

(Fig. 10)

when

produced both ways represents numbers to the scale of 1-5 cms. to

unity.

Every distance

to the right of

represents

some

SUBTRACTION. definite positive

some

7

number, every distance to the

definite negative

number; conversely,

corresponds a definite point in the

left represents

to every

number

a, 6, c, d,

the scale

line.

(8) Find the sum of the numbers represented by being 0*4 inches to unity.

dFIG. 11.

(9)

(10)

(11)

Find the algebraic sum corresponding to a + b - c + d. Find the algebraic sum corresponding to a b-c-d. Shew by actual measurement that a + b-c +

da + d -c +

b,

a-b-c-d= ~b-

and that

The construction on page 3 depended on a property of similar triangles, viz. the ratios taken in order, about the equal angles are equal.

Similar Triangles. for its validity

of the sides,

triangles we can always ensure similarity by making them Generally, one figure is similar to another when equiangular. it is a copy of the second drawn to the same or a different

For

scale (in the first case the figures are congruent,

i.e.

identically

equal).

ABO

and by the aid of the right angle of a set (12) Draw any triangle whose sides square and a straight edge construct another triangle are perpendicular to those of the first. Scale the sides and calculate the

A^C^

AB

BG

CA

(13) By aid of the 30 set squa're construct -4 2 #2