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UNIVER5ITY OF PITTSBURGH
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THE
2' G U N G
MILL-WRIGHT & MILLER'S
u
I
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E-
IN
PART «— ji,iVl5ELLlSHED WITH TWENTY FIVE PLATES.
CONTAINING,
Part I.— Techanics and Hydraulics; fhewin; , sir t'.e old,a:ideilablilhing anew I_;;!c... if theoiie; of water-mills, by 'vhlci: 'Wr: :j,)w"er ot' luili-ieati and Che ede.-^s tncyviH prpduce may be af- ciiitained bv calculation. -Part li..-i-i5.uies tor applying the theo- ries to pra^lici ; tables for proportion- ing m'>ii , !G tae power and faii of the waCc- , iiid luies for finding pitch cir- cie"^, -.nth tables fioni 6 to 136 cogs.
Part Ilx — ^Oi/e.iions for conltrufting and uiing all the authors patented im- provements in mills.
jj Part IV.— rThe art of manufafturing meal
' and flour in all its parts, as pradlifed by
J the mofl Ikilful millers in America.
Part V. — The Praftical Mill-wright ;
containing inllruftions for building
mills, with tables of their proportions
fuitablc for all falls from three to
thirty-fix feet.
APPENDIX.
Containing rules for difcovering new im- provements— exemplified in improving the art of thralhing and cleaning grain, hulling rice, varming rooms, and vent- ing fmoke by chimneys, &:c.
By OLIVER EVANS, of Phi ladelphia.
PHILADELPHIA:
PRINTED FOR, AND SOLD BY THE A UTHOR, No. 215, KORTH SECOND STREET.
1795.
Dijlrid of Pennfylvania — to wit:
BE it remembered, that on the nineteenth day of January, in the nineteenth year of the Independence of the United States of Ainerica, OLIVER EVANS, ofthefaid diftrid, hath depofited in this office the title of a Book, the right whereof he claims, as Author and Pro- prietor, in the following words — to wit :
" The Yeung Mill-wright and Miller's Guide : in five parts, embelliflied with twenty- five plates, &c. By Oliver Evans, of Philadelphia" — in confonnity to the aft of the Con- grefs of the United States, intituled, " An at^for the encouragement of learning, by fecur- ing the copies of maps, charts and books, to the authors and proprietors of fuch copies, dur- ing the times therein mentioned.'''
SAMUEL CALDWELL, Clerk of the Diftrift of Pennfylvania,
t> RE F A G * E.
?T
HE reafon <vhy a book of this kind although fo much want-
i ed did not fooner appear, may be — becaufe they who have been '' Verfed in fcience and literature, have not had pradice and expe- i rience in the arts; and they who have had pradice and experimental >^knowledge, have not had time to acquire fcience and theory, thofe ^^ceffary qualifications for compleating the fyflem, and which are ^;not to be found in any one man. Senfible of my deficiences in J both, I fhould not have uridertaken it, w<as I not interelled in the 1 explanation of my own inventions. I have applied to fuch books and men of fcience as I expeded afliftaince from, ill forming a fyftem of theory ; and to pradical mill-wrights and millers for ^the pradice ; but finding no authors who had joined pradice and Inexperience with theory, (except Smeaton whom I have quoted) 'finding many of their theories to be erroneous, and lofing the af- fiftance of the late ingenious William Waring, the only fcientific s^rharader of my acquaintance, who acknowledged that he had in- ^efligated the principles and powers of water ading on mill-wheels, ~^I did not meet the aid I expeded in that part. ^ Wherefore it is not fafe to conclude that this work is without error — but that it contains many, both theoriticai, pradical, and \ grammatical ; is the moft natural, fafe, and rational fuppofition. i The reader whofe mind is free and unbiafl'ed by the opinion of * others, will be mofi likely to attain the truth. Under a momen-^ |ltary difcouragement, finding I had far exceeded the prefcribed 'limits, and doubtful vv'hat might be its fate, I left out feveral ex- penfive draughts, of mills, &:c. — But fmce it went to prefs the profpeds have become fo encouraging that I may hope it will be kwell received : Therefore I requell the reader, who may prove ;-^ny part to be erroneous, can point out its defeds, propofe amend- >ments, or additions; to inform me thereof by letter ; that I may be enabled to corred, enrich, and enlarge it, in cafe it bears ano- ther edition, and I will gratefully receive their communications r For if what is kncu'n on thefe fubieds bv the different in^jenious-
PREFACE,
pradlitioners in America could be colleded in one work, it would be precious indeed, and a fufficient guide to fave thoufands of pounds from being ufelefsly expended. For a work of this kind wilt^never be perfeded by the abilities and labours of one man.
The pracftical part received from Thomas Ellicott will doubtlefs be ufeful, confidering his long experience and known genius.
Comparing this with other original, difEcuit works, with equally expenfive plates, the price will be found to be low.
I CONTENTS.
PART h V;;
MECHANICS. Articles —
1. AXIOMS, or felf evident truths. - - -< Paget
2. Of the firfl: principles of mechanical motion. - - - 2
3. — elafticity, its power unknown. - - - - 4
4. — motion, abfolute and relative, ,- ' - - 5
5. — do. accelerated and retarded. - ^- 6
6. — the momentum, or quantity of motion. - - ibid.
7. — general laws of motion. - - "7
8. — the momentum of elaftic and non-elaftic bodied in motion. - ^
9. — laws of motion and force, of falling bodies ; table and fcale of their mo- tion. - - - - 14
10. — the laws of motion of bodies defcending inclined plains, and curved far- faces. - _ . . - 20 12. — the motion of projeftiles. - - - 21 13. — circular motion and central forces, - - 22 14. — ; centres of motion, magnitude and gravity. - - . 25 IS- — general laws of mechanical powers, - - 27 16 — 21. Of levers, fimple and compound ; their laws applicable to mill- wheel's ; general rule for calculating their power. - 20 21. Power decreafes as the motion increafes. - "35 Z2 — 23. No power gained by enlarging underfhot -wheels, nor by double gear- ing mills. - - - - 36 24. The pulley, 25 the axle and wheel, 26 the inclined plain, 27the wedge, and 28 the fcrew. - - - . ^g ;50. The fty-whcel, its ufe. - - - 42 31 — 33. Of friftion, its laws, and the inventions to reduce it. 44 34. Of maximums, or the greateft effed: of machines. - - 48 J5 — 37. Old theory of the motion of underfliot-wheels inveftigated ; new the- ory propofed ; fcale of experiments. - - 50 ^3 — 39. William Waring's new theory. ■ - 59
\o. ■ theory doubted. - - 63
ji — 42. Search for a true theory on a new plan, and one eftabliihed agreeing with praftice. - - - - 65
!}3 — 44. The maximum motion of overfhot- wheels, with a fcale thereof. 75: .
HYDRAULICS.
\5 — 4.7- Laws of the motion and effeiSs offpouting fluids ; their application tor
underftiot-mllls. - - - - 80
\% — 50. Hydroftatic paradox ; on which is founded a theorem for finding the
preffiire of water on any furface. - - . 87
rt. Rule for finding the velocity of fpoutiiig water. - - 89
;2. Rule for finding the effeft of any gate of water on underfhot-wheels. 90
<3 — 54- Water applied by gravity; the power thereof on the principles of
overfiiot-mills, equal in theoryto the bed application poffible. - 92
CONTENTS*
Articles—, Page*
55. Fridlion of the aperture on fpouting fluids. - "97
56. Prefliire of the lair the caufe of fluids rifing in pumps and eyphons, &c. 98;
57. Direftions for pump-makers, with a table. - - 100
58. Tubes for conveying water over hills and under valleys. - 102 S^. Paradoxical mill explained, that will not move empty ; the diiference of
force of indefinite and definite quantity of water. - - ibid.
60. The motion of bread and pitch-back wheels. They do not run before the
gravity of the water on account of the impulfe. - - 104
,61. Simple rule for calculating the power of a mill -feat. - - 107
62. Theory compared, with a table of experiments of 18 mills in pta&.ite, and found to agree. - _ . - - no
63. Rules for proportioning thefize of mill-ftones to the power ; with a table of their ai-eas, powers required, and quantity ground, &c. - 118
The furface pafled by mill-ftones of diflferent fize and motion. - i ; 9
64 — 65. Of digging canals ; with their proper fall and fize to fuit theftones. 122'
66. Of air-pipes, to prevent trunks from burfting. - - 226
67. Smeaton's experiments concerning underfhot-mills. - - 128
68. . experime/its concerning overfliot-mills. - 145
6^. ^ experiments concerning wind-mills, - - - 154
PART II. «
70. OF underfhot-mills, with a table containing the motion of th6 water and: wheels, and proportion of the gears, fuitable to any head from i to 25 feet, both double and fingle gear; the quantity of water required to turn them, and the fize of the gate and canal. - "3
71. Of tub-mills, with a table fhewing the diameter of the wheels td luit any' fize ftone, or head of water. - - - ix!
72. Of breaft and pitch-back wheels, with a table complete for them. 1 7'
73. Of overfliotmills, with tables for them. ... 25 Ofmills moved by re-a6lion. - - - - 33
74. Rules for calculatingthe motion of wheels^ and humberofcogs to product the defired motion. - - - - 35i
75. Rules for finding the pitch circles. - - - 4d
76. A true, fimple, and expeditiousmethod for finding the diameter of the pitch circle, with a table fhewing the diameter of pitch circles, &c. 41
77. Rules for meafuring garners, hoppers, &c. - - 46
78. Of the diiferent kinds ofgears and forms of cogs. - - 48 79 — 81. Of fpur, face, and bevel gears. - - 49
82. Of mntching wheels, to make them wear even and well. - 56
83. Theories of rolling-fcreens and fans for cleaning the grain, improved appli- cation of them. - - - 5 7
84. Of gudcreons, the caufe of their heating and getting loofe, with the reme- dies therefor. - - - - 60
85. On building mill-dams. - - - 64 ,86. On laying foundations and building mill- wails. ~ - 67
PART IIL
87. GEN ER AL acco'-mt of the newiiiiprov-riTnts. - - 7;^
88. Particular defcription of the macijine.s. - - 7.?
89. Application of the machines in the pvocefi: of la.inufufturJng ^lour. • 7I
CONTENTS.
Articles— ^ Page*
JO. Of elevating grain from fhips. - - - 82
^i. A mill for grinding parcels. ... 85
)2. A grid-mill improved. _ . 88
>3. Of elevating from fhips and ftore-houfesby a horfc. - 90
)4. Of an elevator wrought by a man. - _ - 9*
>5. Conftruftion of the wheat elevator, particularly direfted.^ - 97 ,6 — I 00. Of the meal elevator, the meal conveyer, the grain conveyer, the
.hopper-boy, and the drill. - - - - i^io
[01. Of the utility of the machines. - - - ^ J2I
02. Bills of materials, both of wood and iron, &c. to be prepared for building the machines. - - - - 127
03. A mill for hulling and cleaning rice. - - 13*
PART IV.
04. THE principles on which grinding is performed, explained. 139
05. Of the draught neceflary to be giveuto the furrows of mill-ftones. 143
06. Diredlions for facing new mill-ltones. - - 149
07. Of hanging mill-ftones. * - - 151
08. Of regulating the feed and water in grinding. - - I54
09. Rules for judging of good grinding. - - 155
10. Ofdreilingandfharpening the (tones when dull, - - I57
1 1 . Of the molt proper degree of fmenefs for flour. - - 1 58
12. Ofgarlic, with diretitions for grinding wheat mixed therewith; and for dreffingthe ftones fuitabl'e therfeto. - - - 160
13. Of grinding over the middlings, fluff and bran, or fhorts, if neceflary, to make the moft of them. - - - ^. - 163
1 4. Of the quality of the mill-ftones, to fuit the quality of the wheat. 1 66
15. Of bolting-reels and cloths, with diredions for bolting and infpeding flour. - - - - V69
16. Direftions for keeping the mill, and the bulinefs of it in good order, r 73
1 7. Peculiar accidents by which mills are fubjeft to catch fire. / 75 rS. Obfervations on improving of mill-feats. - - 175
PART V, See the contents at the beginning of it.
CONTENTS of the APPENDIX.
lules for difcovering new improvements — exemplified — I. In improving the art of thrafliing grain. — II. Cleaning do. by wind. — III. Diftillation of fpi- rits. — IV. In venting fmoke from roomsby chimneys.— V. Warming rooms by fire to fave fuel. — VI. Hulling and cleaning rice. — VII. Saving fliips from finking at fea. — VIII. Preferving fruits and liquors from putrefaftion and fermentation.
EXPLANATION OF THE TECHNICAL TERMS, &c. USED IN THIS
WORK,
Aperfeure — The opening by which water iffiies .
Area— Plain furface, fuperficial contents.
Atraofphere — The furrounding air.
Algebraic figns ufed are -|- for more, or addition. — Lefs, or fubftrafted. X Multiplication. .|. Divifion. 3: Equality. l^' The fquare root of. 86 2 for 86 fqua- red, 883 for 88 cubed.
Byquadrate — A number twice fquared : the byquadrate of 2 is 16,
Corollary, Inference.
Cuboch— A name for the unit or interger of power, being one cubic foot of water multiplied into one foot perpendicular de- fcent.
Cubic foot of water— What a veflel one foot wide and one foot deep will hold.
Cube of a number — The produft of the num- ber multiplied by itlelf twice.
Cube root of a iiumber — Say of 8, is the num- ber, which niultiplied into itfelf twice will produce 8, viz. 2. Or it is that number by which you divide a number twice to quote itfelf.
Decimal point , iet at the left hand of a figure fhews the whole number to be di- vided into tens, as ,5 for 5 tenths ; ,57 for 57 hundredths ; ,577 for 577 thoufandth parts.
Equilibrio, Equilibrium — Equipoife, or ba-
" Jance of weight.
Elaftic, Springing.
Fri(ftion-T-The aft of rubbing together.
Gravity— ^That tendency all matter has to
fall downwards. Hydroftatics— fcience of weighing fluids. Hydraulics— Water-works, the fcience of
motion of fluids. Impulfe — Force communicated by a ftroke. Impetus— Violent eitbrt of a body inclining
to move. Momemtnm— The force of a body in motion. Maximum— ^Greatett poffible. Non-elaftic' — Without ipring. Oftuble- — ^Eight times told. Paradox — Contrary to appearance. Percuffion — Striking a ftroke, impulfe, Problem: — A queftion. Quadruple — Four times, fourfold. Radius— Half the diameter of a circle. Right Angle — A line fquare, or perpendicu- lar to another. Squared-r-Multiplied into itfelf; 2 fquared
is 4. Theory — Speculative plan exifting only in
tfee mind. "^
Tangent — A line perpendicular or fquare
with a radius touching the periphery of a
circle. Theorem — Pofition of an acknowledged
truth. Velocity? — Swiftnefs of motion. Virtual or effeftive defcent of water : See
Art. 61.
•«S>o^<^)".
SCx-^LE from which the FIGJURES are draAvn.
PLATE IT, Fig. it, 13 — S feet to an Incli; fig. 19 — lofcettor.n inch.
Ill, Fig. 19, ::r — z^^ 26—10 feet to ditto.
IV, Fig. 28, 29—33, 31, 32, 33—10 feet ditto
VI, Fig. I — 13 feet to an inch ; lig. 2, 3, 8, 9, 10, 1 1 , two feet ditto.
Vir,Fig.i2, 13, 14, 15 — twoieetto an inch ; fig. [6, ten ditto.
X. Fig. I, 2 — 18 feet ditto; fig. H, I in fig. i— four feet to an inch.
XI. Fig. I, 2, 3 — two feet ditto ; fig. 6, 8, one ibot to ditto.
t^iP-> «<^ (-i?-> t<5>^ t<s>> t<:?i t(S>5 «<?'^ v:?^ v:^ «<:P^ <k:5^
THE
Y 0 U N G
Mill-wrighf s & Miller's
GUIDE.
PART THE FIRST.
CHAPTER I.
OF THE FIRST PRINCIPLES OF MECHANICS. Art. l.
"OTION may be faid to be the Beginning or Foundation of all Mechanics, becaiife no Mechanical Operation can be performed without Motion*
AXIOMS, or Self-evident Truths,
I . A body at reft m^II continue fo for ever, un- lefs it is put in motion by fome force imprefled.* A body in motion will continue fo for ever, with the fame velocity in the fame diredlion, un- lefs refifted by fome forcc*t
* This fluggifli, inaftive principle, or force, by which a body inclines to a ftate of reft, is called Inertia.
t The fame principle of inertia, which inclines a body to remain at reft, alfo inclines it to continue in motion for ever, if once put in motion, and that in a right-lined direiflion, unlefs changed by fome force : therefore no l:'ody, moving in a ftrait line, can be turned into a curve line, but by fome force ; the confideration of which may lead us to the knowledge of the true principles of fome milk. See the latter part of art. 73.
I '- B
2 MECHANICS. Chap, I.
Art. I. 2. The impulfe that gives motion, and the re-
fiftance that deftroys it, are equal.
4. Caufes and eiTetts are equal, or diredly pro- portional.
POSTULATUMS, or Pofitions without Proof.
A quadruple impulfe, or moving power, is re- quilite to communicate double velocity to a bo- dy*; therefore a quadruple refiftance is requiiite to deflroy double velocity in a body, by axiom
The impulfe we may call power, and the re- fiftance that it overcomes, the effect produced by that powet-.
COROLLAHY.
Confequently, the powers of bodies in motion, to produce effefts, are as the fquares of their velo- , cities ; that is, a double velocity, in a moving bo- dy^ produces 4 times the effedl.
Art. 2. Of th£ Principles of Mechanics.
THERE are two principles, which are the foundation of all mechanical motion and mechani- cal powers, viz. Gravity andElafticity ; or Weiglit and Spring.
By one or the other of thefe principles or pow- ers every mechanical operation is performed.
* In the courfe of this work, T fliallfliew, that a quadruple impulfe pro- duces Only double velocity. See art. 7 and 46. We ftiould follow philofo- phers only in the paths of truth ; becaufe, if all men are fubjeft to err, even the moft eminent philofophers may have erred.
If a theory will not agree with praftice, we may fufpeft it is not true ; and the theory of the momentum or force of bsdies in motion, being as their velocities fimply, does not agree with praftice, with refped: to the efTecls they produce, either in circular motion, art. 30, falling bodies, art. 9, fpoutin? fluids, art. 45, wind on mill-fails, art. 69, therefore we have reafoii to fufped that this theory may not be true, in every refped.
Chap, /. MECHANICS.
Gravity, in the extent of the word, means ^^^- =• every fpecies of attraction ; but more efpecially that fpecies v/hich is common to, and mutual be- tween, all bodies ; and is evident between the fun and its planetary attendants, as alfo the earth and moon.* But we will only confider it, as it relates to that tendency which all bodies on this earth has to fall towards its centre ; thus far it concerns the mechanical arts, and its lav/s are as .follows, viz.
Laws of Gravity.
1 . Gravity is common to all bodies, and mutual between them.
2. It is in proportion to the quantity of matter jn bodies.
3. It is exerted every way from the centre of jattradiing bodies, in right-lined directions ; there- fore all bodies on the earth tend to the centre of gravity of the earth.t
4. It decreafes as the fquares of the diflance in- crcafe; that is, if a body, on the earth, was to be removed to double the diflance from the centre of gravity of the earth, about 4000 miles high, it would there have but 1-4 of the gravity or > weight it had when on the ground : but a fmall height from the furface of the earth (/^o, 100, or 500 feet) will make no fenhble difference in gra- vity.J
* It is this attraftion of gravity between the heavenly bodies, that, keeps up the order of their motion, in their revolutions round each other. See Fergufon's Leftures, page 23.
t The centre of gravity of a body, is that point on which, if the body be fufpended, it will remain at reft in any pofition ; or it is the centre of the whole weight or matter of the body. Art. 14.
\ The diameter «f the earth is allowed to be about 8000 miles; there- fore we may fuppofe the centre of gravity of the earth to be about 4000 miles from its furface; "and any fmall diltance from its furface, fuch as i mile high, will make no fenfible difference in gravity. But \Then the dif- tance is fo great as to bear a confiderable proportion to the diilance of the centre of gravity of the earth, then the power of gravity will decreafe fenfibly. Thus, at the diftance of the moon, which, at a mean, is about bo femi-diameters of the earth, the power of gravity is to that on the fur- face of the earth, as i to 3600. See Martin\' Philofophy.
MECHANICS. Chap, I,
By the 3d law, it follows, that all bodies de- fcending freely by their gravity, tend towards the earth, in right lines, perpendicular to its furface, and with equal velocities (abating for the refift- ance of the air) as is evident by the 2d law,*
■«@>o(^<S»" —
Elasticity.
Elaflicity is that flrength or power, which any body or quantity of matter, being confined or compreiTed, has to expa^'id itfelf ; fuch as a fpring that is bent or wound up, heated air or fleam con- fined in a vefTel, Sec. and by it many mechanical operations are performed.
Elafticity, in the full fenfe of the word, here means every fpecies of repulfion.
The limits of the prodigious power of repulfion, which takes place between the particles of heated air and fleam, are not yet known. Their eifefts are feen in the explofion of gunpowder, the burfl- ing and cracking of wood in the fire, Sec. In ihort, in every inflance, where fleam could not find room to expand itfelf, it has burfl the vefTel that confined it, endangering the lives of thofe who were near it.f
* This refiftance will be as the furfaces of the bodies ; therefore the fmaller the body of equal matter, the greater will be the velocity of its fall. But it has been proved, by experiment, that a feather will fall with the fame velocity as a guiuea, in vacuo. See Fergufon's Ledlures, page 183.
t A worthy and ingeuius youngman, having prepared a yeffel of wrought iron, about 3 inches diameter, an J 9 inches long, partly filled with water, had put it into a finith's fire, and was trying fomc experiments, when the aperture, by which the fteam was meant to iflue, got flopped by fome means (as is fuppofed) and the veflel burft with noife like a cannon, carried off his right arm, and left it laying acrofs one of the upper beams of the fliop, and otherwife defperately wounded liim. This prodigious power is applieu to raife water out of coal-mines, gcc. from great depths, in fur- prijirg quantities, and to turn mills; it may (in my opinion) be applied to many other ufeful purpoles, which it is not yet applied to.
On tliis .ubjeft much m.ight be faid; but as it does not irmnediately con- cern this work, pei'haps I have faid enough to excite the reader to perufe the fevcral late authors on philofophy, who have treated largely and well on it, and to them I rnuft refer.
Chap. I. MECHANICS.
Having premifed what was neceiFary to the ^^"^^ 3- right undcrfcanding of the fcience of mechanics, which nioftly depends upon tlic principles of gra- vitation,
We come to confider the Objecls tliereof, viz. the Nature, Kinds, and various Eftedls of Motion and moving Bodies, and the Struaure and Mc- chanifm of all Kinds of Machines, called Mecha- nical Powers, whether Simple or Compound.
— '"'Tf ■' -^^^^^gaHtag3f*»
CHAPTER H.
Of Motion and its General Laws.
Art.
OTION is the continual and fucceflive ^^^-^^^^ change of fpace or place, and is cither ab- folute or relative.
Abfolute motion is the change of fpace or place ^bfoiute. of bodies, fuch as the flight of a bird, or the mo- tion of a ball projcded in the air.
Relative motion is the motion one body has with p^eiative. refpett to another, iuch as the difference of mo- tion of the flight of two birds, or of two Ihips failing.*
*■ If two fhips, A and B, move with the fame velocity, iii the fame di- reftion, then their abrolute motion is the fjime, and they have no relative motion, and neither of them will appear, to a peribn on board of the other, to move at all. Hence it is, that although the earth is continually revolving about its axis, with a velocity, at the equator, of about 1042 miles in an hour, and round the fun, in continual abfolute motion, v/ith a velocity of about 58000 miles an hour — yet, as all oljefts on its furface have the fame abfolute motion, they appear to be at reft, and not to move at all : therefore all motion of bodies on the earth, appears to us to be abfolute motion, v/hen compared with objefts fixed on the earth ; yet, if we take into confideration the abfolute motion of the earth, all motion on it will appear to be merely re- lative.
MECHANICS.
Chap, II,
Art. 5- Motion,
Equable.
Accelerated.
Retarded.
MOTION is either Equable, Accelerated, or Retarded.
Equable motion is when a body pafles over equal difiances in equal times.
Accelerated motion, is that which is continually increafed ; fuch is the motion of falling bodies.*
Retarded motion, is that which continually de- creafes ; fuch is the motion of a cannon-ball thrown perpendicularly upwards. T
••*^^oQfi^S»"
Art. 6. THE Momentum or quantity of motion, is all
the ppw^er or force which a moving body has to ftrike an obftacle to produce effefts, and is equal to that imprelfed force by which a body is com- pelled to change its place, by axiom 3, art. i ;
If twd fiiips, A and B, moving with equal velocities, pafs each other, then they will appear, to a fpedlator on board of either, to move with double their refpeftive real velocities.
Hence the reafon, why a perfon, riding againft the wind, finds its force greater, and with it, its force lefs, than it really is.
* A falling body is conftantly afted upon by all the power of its own gra- vity ; therefore its motion is continually increafed.
t A cannon-ball, projedled perpendicular upwards, is conftantly refille^l by the whole power of its o^v•n gravity ; therefore its motion will be conti- nually decreafed, and tctally flopped as foon as the fum of this refiftance amounts to the firft impulfe, by axiom 3d, art. I, when it will begin to de- fcend, and its motion will be continually increafed by the fame po^ver of its own gravity : its motion downwards v.all be equal to its motion upwards, in every part of its path, and it Avill return to the mouth of the cannon with the velocity and force that it left it ; ana the time of its afcent and defcent will be equal, fuppofmg there was no refiftance from the air — but this refift^ ance will make a confiderable difference.
From this principle of accelerated motion in falliEg bodies, may appear the reafon, why water poured from the fpout of a tea-kettle, will not conr tinue in a ftream farther than about two feet, and this ftream becomes fmaller as it approaches the place where it breaks into drops ; becaule the attraftionofcohefion keeps the water together, until the accelerated mo- tion of its fall, which ftretches the ftream fmaller andfmaller, overcomes the cohefion, and then it breaks into drops, and tliefe drops become further afur.der while they continue to fall : therefore, if the clouds were to empty themfelves in torrents, the water would fall on the earth in drops. This may ferve to lliew the diiadvantage of drawing the gate of a water-mill at a great diltance from the float-board : but more of this hereafter. See art. 59.
Ghap. II, MECHANICS. 7
which, I think, ouglit to be diftinguifhed by two Art. 6. lames, viz. Inftant and Effective Momcntums.
1 . The Inftant Momentum, or force of moving ^JgrS^d! Dodies, is in the compound ratio of their quantities tion.
3f matter and fimple velocities conjointly ; that is, as the weight of the body A, multiplied into its velocity, is to the weight of the body B, mul- tiplied into its velocity, fo is the inftant force of A to the inftant force of B. If A has 4lbs. of matter, and i degree of velocity, and B has 2lbs. of matter, and 4 degrees of velocity ; then the momentum of their ftrokes will be as 4 is to 8 ; that is, fuppofing them to be inftantaneoufly flop- ped by an obftacle.
2 . The EfFeftive Moihentuiii, or force of moving bodies, is all the efFeft they will produce by im- piiiging on any yielding obftacle, and is in the compound duplicate ratio of their quantities (or weights) multiplied into the fquares of their ve- locities ; that is, as the weight of the body A, multiplied into the fquare of its velocity, is to the weight of the body B, multiplied into the fquare of its velocity, fo is the efFeftivc momen- tum of A to that of B. If A has 2lbs of matter and 2 degrees of velocity, and B 2lbs of matter and 4 degrees of velocity, then their effe^liv^e mo- tnentums are as 8 to 32 ; that is, a double velocity produces a quadruple effeft.
/
THE general Laws of Motion are the three Art. 7- following, viz.
Law I. Every body will continue in its prefent ^^'**'^°^'"®" ftate, whether it be at reft or moving uniformly in a right line, except it be compelled to change that ftate by fome force imprefted.*'
* By the firft law, a body at reft, inclines to continue fo for ever, by its yifmertia or inaftive power, and a body in motion inclines to continue fo for ever, palFmg over equal diftances in equal times, if it meets with no le-
I 8 MECHANICS. Chap. Ih |
Art. 7- Law 2. The change of motion or velocity is i
Kev/ pofitioii. a}^^ays proportional to the fquare root of the ] moving force impreifed, and in a right line with that force, and not as the force direftly.'*
Law 3. Aftion and re-a6tion are always equal, and in contrary diredtions to each other. t
fiftance, and will move on in aright line-. Foi; v/ant of refiftance the planets and conets continue their motions undiminiftied, while moving bov/les or wheels are reduced to a ftate of reft by the refiftance of the air, and the friclion of the parts on which they move. See Fergufon's Leftures on Me- chanics.
It is this friftion of the parts, and refiftance of the air, which renders it impoflible for us to make a perpetual motion ; becaufe this friftion and re- fiftance are to be overcome, and although it may be reduced to be very fmall, yet man cannot, with all his art, by mechanical combinations, gain as much power as will overcome it. Philofophers have demonftrated the impoffibility of making it ; but I think none ought to affert that it will never be found ; for there are many perpetual motions in the heavens. If any man would ipend his time in this "^vay, it fliould be to feek for a created power that he might apply to this purpofe, and not to create one.
• This is, evident, when we confider that a body muft fall a quadruple diftance to obtain double velocity, by art. 9 ; and a quadruple head or pref- fure of fluid produces a double velocity to the fpout, by art. 46. The velo- city, in both thefe cafes, is as the fquare root of the impulfe, and the im-, pulfe as thefquaresof the velocity, .therefore the change of effeftive mo- tion or velocity will always be as the fquare root of the impulfe or force imprelfed, and the force imprelTedas the fquares of the velocity or effeftive motion.
f Adlioh and re-a-Aion are equal ; that is, if a hammer ftrikes an anvil, the anvil will re-aft againft the hammer with an equal force to the aftion of the hammer.
The ailioh of our feet againft the ground, and the re-aftion of the ground againft our feet, are equal.
The aftion of the hand to project a ftone, and the re-aftion of the ftone againft the hand, are equal.
If a cannon, welgliing 64oolbs. gives a 24lb. ball a velocity of 640 feet par fecnnci, the aftion of the powder on the ball, and its re-adtion againft the cannon, ai-e equal ; and if the cannon has liberty to move^ it will have a velocity, which multiplied into its weight, v/illbe equal to the velocity of the ball multiplied by its weight : their inftant momentums are alway.-; equal- .See Martin-'s Philofophy.
J
•<=-i -iPo ^5>^ i^ ^:?^ <:?-> -i>^-;;>> <S^^^<s^ ^i5^-<:5>-> .<^ '-i>2 -<:^ t<^ .
CHAPTER III.
— ■<^>^ —
Of the Momentum or Forge of Bodies in Art. 8. Motion.
IF two non-elauiic bodies, A and B, iig. i, each pig. i. having the fame e][uantity of matter, move with equal velocities againil each other, they Vvall de- ftroy each other's motion, and remain at reft after ^.. „ the flrpke : becaufe their momentums will be ous momen-" equal: that is, if each has 2lb3 of matter and lo ^^^ff^^odies
1 r i •. 1 • ■ '1. ^ " in motion.
degrees or celerity, their initantaneous momen- tums will each be 20.
But if the bodies be perfcftly elaflic, they will recede from each other with the lame velocity with which they meet ; becaufe action and re- action are equal, by the 3d general law of motion, art. 7.* '
If two non-elaflic bodies, A and B, fig, 2, moving in the fame diredion with different veloci- ties, impinge on each other, they will (after the ftroke) move on. together v/ith fuch velocity, as, being multiplied into the fum of their weights, will produce the fum of their inflant momentums which they iKid before the ftroke 5 that is, if each weigh lib. and A has 8 and B 4 degrees of cele- rity, the fum of their inflant momentums will be 12, then, after the ftroke, their velocity will be 6 ; which, multiplied into their quantity of mat-
** Tlus fliews that non-elaftic bodies communicate only half their original force ; becaufe the force required to caufe the bodies to recede from each other, is equal to tiie force that gave them velocity to meet ; and the force that caufe.d the body to recede with velocity lo, is equal to the force tfeat flopped velocity i o.
c
Fig. 2.
M E C H A N I C S. Chap, III.
ter 2, produces 12, the flmi of their inflant mor mentums. But if they had been elaftic, then A would have moved with 4 and B with 8 degi-ees of velocity after the flroke, and the funi of their inftant momentums would be 12, as before.*
3. If a non-elailic body A, with quantity of matter 1, and 10 degrees of velocity, flrike B at refl-, of quantity of matter i, they will both move on together vv^ith velocity 5; but if they be elaflic, B flies off with velocity 10, and A re- mains at refl, by 3d general lav/ of motion, art. y:\ It is univcrfally true, that v*4iatcver inflant mo- mentum is communicated to a body, is loft by the] body that communicates it.
4. If the body A, fig. 4, receive two ftrokes' or impulfes at the fame time, in different direc- tions, the one fufiicient to propel it from A to B, ' and the other to propel it from A to D, in equal time, then this compound force v/ill propel it in tiie diagonal line A G, and it will arrive at C in the fame time that it would have arrived at B or D, hj one impulfe only ; and the projeftile force i of thefe ftrokes are as the fquares of the fides of the parallelogram, bylaw 2, ait. 7.I
* Becaufe elaftic bodies impinging, recede, after the flroke, with the fam' velocity with v/hich they meet : therelbre, a heavy body in motion, Impinging on a lighter body at reft, will give it a greater velocity than that v/ith which it v.-as ftruck ; for if the heavy body be not ilopped, butmove forward after the ftroke, v/ith a certain velocity, that velocity, added to the velocity before the ftroke, will be the velocity of the lighter body.
t This alfo ihews evidently, that non-elaflic bodies commmiicate only half their force. A knowledge of this is of great ufe in eftabiifhing a true theory of water-mills.
+ This do'ilrine of the momentum of bodies in mction, and communi- cation of motion, being as their velocities fmiply, was taught by Sir Ifaac Newton, and has been received by his followers to this day; which appear: to be true, v/here the whole force is inltantaneoufly fpent or communicated : therefore I have changed the term to inftant momentum. I have tried the experiment, by caufing diiterent weights to flrike each other with different velocities, both on the principle of pendulums, and by caufing them to move in horizontal circles; and, in both cafes, 4lbs. with velocity i, ba- lanced 2lbs. v.'ith velocity 2 ; their momentums each were 4 : fo that the tl-.eory appears to be proved to be true. Yet I think we have reafon to doubt its beirg true in any other feme ; becaufe it does not agree with prac- tice. All the bodies we put in motion, to produce effefts, produce them -.i! proportion to the fc]uarc; of their velocities, or nearly, as will appear in
Chap, III, MECHANICS. 1 1
5, If a perfect elaftic body be let fall 4 feet, to Art. 8. ftrike a per fed: elailic plain, by the laws of falling Adoublevdo- bodies, art. o, it will ftrike the plain with a vc- ^^ty produces lociry of 16,2 feet per fecond, and rife, by its re- cff^a. ' aftioD, to the fame heigiit from whence it fell, in lialf a fecond: If it falls 16 feet, it v/ili ftrike with a velocity of 32,4 feet, and rife 16 feet in one fecond. Now, if we call the rifmg of the body the effe6l, we fhall nnd that a double velo- city, in tliis cafe, produces a quadruple efFetl in double time. Hence it appears, that a body moving through a refifting medium, with a dou- ble velocity, will continue in motion a double time, and go 4 times the diftance ; which will be a quadruple cited:,*
the courfe of this work : But I fear I fhall draw on me the ridicule of fome, if 1 fhould doubt a theory long eftablifhed ; but I think we ihould follow Others only in the paths of truth. Doubtiels Sir Ifaac meant the force to be inftantly fpent : and I have underftood that the Dutch and Italian philofo- phers have held and taught, thele lOO years paft, that the mcnientum of bo- dies in motion, is as the iquares of their velocities ; and I muft confefs it appears to be really the cafe, with refpeit to the etfedlc they produce ; which lis generally as their quantity or Vt'eight multiplied into the fquares of their velocities. I found it impofiibie to reconcile the theory of the force ofbodies in motion, being as their limple velocities, to the laws of circular motion, art. 13, where a double velocity produces a quadruple central ibrcc \ of falling bodies, art. 9, where the velocity is as the fquare root of the impulie or dif- tance fallen, and the eliefts as the fquares of the velocities; of projectiles, where a double velocity produces a quadruple random, art. i2; ofbodies defcending on inclined plains, art. 10, where the velocities are as the fquare roots of the perpendicular delcents, and the effects as the fquares of their velocities ; of fpouting fluids, art- 45? where their velocities are as the i Iquare roots of their perpendicular heights or ptsiiures, and •. heir effedls as I the fquares of their velocities, with equal quantities; of v/iiid on mill- falls, art. 69, where the effefts are as the cube oi the velocity of the wind ; becaufe here the quantity is as the velocity, and the etTe<t of equal quantities being as the fquares of the velocity, amounts the eiie>ts to be as the cubes.
But when I difccvered that a quadruple inipizlfe was i-eqni(ite to give double velocity, both in falling bodies and fpouting fluids, and, by axiom 3, the power that produced amotion in a body, and the power tliat deitroyed faid motion, were equal, I concluded that the euei3:s produced by bodies in motion, were as the fquares of their velocities ; and then I found the whole theory to agree M-ith practice. Hereafter I Ihall iay, that the efTective mo- mentum, or force of bodies in niOtion, is as the fquares of their velo- cities.
* We fhould pay no regard to time, in calculating the efTevlive force of ibodies in motion. Becaufe, if lib cf matter move with i degree of velo- city, it will pi-oducc a certain effect (beiore it ceafes moving) in an Unknown time. Every other pound of matter, mo\ ing v.ith equal velocity, will produce an equal eiTe<5l in equal time. But if each pofi'nd rf rliatter mo\'e v.'ivh d«uble velQcity, it will produce 4 times Che eitjt, bnt requires a dou-
12 M E C H A N I C S. Chap. II L
Gf Non-elasticity in impinging Bodies.
1 . IF A and B, fi^. 3, be two columns of matter in motion, meeting each other, and equal in non- eiaPticity, quantity, and velocity, they will meet at the dotted line e e, deftroy each other's motion, and remain at reft, provided aone of their parts feparate.
2. But if A is elaftic, and B non-elaftic, they will meet at ee, butB will give way by battering up, and both will move a little further 5 that is, half the diftance that B fliortens.
3. But if B is a column of fluid, and, v/hen -it ftrikes A, flies off in a lateral perpendicular direc- tion, then whatever is the fum total of the mo- mentums of thefe particles laterally, has not been communicated to A ; therefore A will continue to move, a:fter the llroke, with that faid momen- tum.
4. But v/ith what proportion of the ftriking ve- locity the fluid, after the ftroke, will move in the lateral direftion, I do not find determined ; but, fromfmall experiments I have made (not fully to- be relied on) I fuppofe it to be more than one half ; becaufe water falling 4 feet, and ftriking a horizontal plain, v/ith 16,2 feet velocity, v/ill cafl fome few drops to the diftance of ^ feet (fa 3; 10 feet, allowing one foot to be loft by fi'iftion, <^c.) which we mufl fuppole'take their dire£tion £it an angle of 45 degrees; becaufe it is fliewn, in Martin's philofophy, page 135, Vol. I, that a body projedled at an angle of 45 degrees, will de- fcribe the greateft poilible horizontal randr.m; alfo, that a body falling 4 feet, and refieded with its acquired velocity 16,2 feet, at 45 degrees,
ble time ; wliich difference in time no way afiefts t-s fum total of the ef- fefts oi" the matter put in motion to move any pracftical machine. There- fore we fhould totally leave time out ci this calculation, Teeing it tends tu lead us into errors.
Chap. in. ivr E C H A N I C S. t%
will reach i6 t(^ct horizontal randimi, or 4 times Art 8. the diflance of the fall. Therefore, by this, 1-4 of 10 feet, equal to 2,5 feet, is the fall that will produce the velocity that pi'odiiced it, viz. Velo- city T 2564 feet per fecond, about 3-4 of the ftrik- iiig velocity.
5. And if the force of flriking fluid;* be as the 6-iothsof fquares of their velocities, as proved in art. 67 power loft by
T • iin^ii ^ / non-elafticity.
by experiment, and demonltratea by art. 46 ; then the ratio of the force of this lide velocity, 12,64 feet per fecond, is to the force of forward velocity, as 160 to 256, more than half (about ,6) of the whole force is here loil by non-elafti- city.
6. This fide force cannot be applied to produce any further forward force., after it has flruck the iirft obftacle ; becaufe its action and re-adtion ba- lance each other afterwards : which I demonftrate by fig. 27.
Let A' be an obftacle, againft which the column of water G A, of quantity t6 and velocity per fecond 16, ftrikes ; as it ftrikes A, fuppofe it to change its direction, at right angles, with 3-4 ve- locity, and ftrike B B ; then change again, and ftrike forward againft CC, and backwards againft D D ; then again in the fide diredlion E E ; and again in the forward and backward diredlions all of which counteract each other, and balance exaftly.
Therefore, if we fuppofe the obftacle A to be the float of an underffiot water-^vheel, the water can be of no further fervice, in propelling it, after the flrft impulfe, but rather a diiadvantage ; be- caufe the elafticity of the float will caufe it to re- bound in a certain degree, and not keep iwWy up with the float it ftruck, but re-aft back againft the float following ; therefore it will be better to let it efcape freely as foon as it has fiiliy made the ftrdke, but not fooner, as it v/ill require a certain
14 MECHANICS. Chap, III,
Art. 8. fpace to ad: in, which will be in dire£l proportion
to the diflance between the floats. Greateft effeft 7. From thefe confiderations, we may conclude, ^uid?^ot° that the greateft efFed to be obtained from ftriking more than fluids, will not amount to more than half the m^^gpower P^wer that gives them motion ; but much lefs, if they be not applied to the befh advantage : And that the force of non-elaftic bodies, ftriking to pro- duce effeds, will be in proportion to their non- elafticity.
Art. 9,
CHAPTER IV.
— ■<-^> —
Of fallikg Bodies.
BODIES defcending freely by their gravity,- in vacuo, or in an unreiifting medium, ard lubject to the following laws:
I ft. They are equably accelerated.* 2d. Their velocity is always in proportion to the time of their fall, and the time is as the fquare root of the diftance fallen. t
3d, The fpaces through which they pafs, arc as the fquare of the times or velocities.^ There- fore,
*Itisevident,that iii every equel part of tiine,the body receives an impiilfc from gravity, that will propel it an equal diftance, and give it an equal additional velocity ; therefore it will produce equal effects in equal times, and their velocity will be proportional to the time.
t If the velocity, at the end of one ifecond, be 32,4 feet, at the end of two feconds it will be 64,8, at the end of tlu'ee feconds 97,2 feet per fecond, and fo on.
t That is, as the fquare of r fecond is to the fpace palTed through 16,2, fo is the fquare of 2 feconds, which is 4, to 64,8 feet, palfed through at the end of 2 feconds, and fo on, for any number of feconds. Therefore the l]5aces palTed through at the end of every fecond, will be as the fquare nuni-
Chap, IV, MECHANICS.
4th. Their velocities are as the fquare root of ^'^*- % the ipace defcended through ;* and their force, to produce effeds, as their diftances fallen direft-
5th. The fpace pafTed through the firft fecond, is very nearly 16,2 feet, and the velocity ac- quired, at the lov/eft point, is 32,4 feet per fe- cond.
6th. A body will pafs through twice the fpace, in a horizontal direftion, with the laft acquired velocity of the defcending body, in the fame time of its fall.t
7th. The total fum of the efFeftive impulfe act- ing on them to give them velocity, is in diredc proportion to the fpace defcended throngh,4 and their velocity being as the fquare root of the fpace defcended through ; or, which is the fame, as the fquare root of the total impulfe. There- fore,
8th. Their momentums, or force to produce effects, are as the fquares of their velocities, || or directly as their diftances fell through ; and the times expended in producing the effects, are as jtlie iquare root of the diftance fallen through.^
bfrs r, 4, 9, 16, 2j, 36, %ic. and the fpaces pafled through, in each fecond feparatoly, will be as the odd numbers 1,3,5,7,9,11,13,15, &c.
* That is, as the fquare root of 4, which is 2, is to 16, 2, the velocity ac- quired in falling 4 feet : fo is the fquare root of any other diftance, to the velocity acquired, in falling that diitance.
t That is, fuppofe the body as it arrives at the loweft point of its fall, and has acquired its greatcft velocity, was to be turned in a horizontal di- reftion, and the velocity to continue uniform, it would pafs over double the diftance, in that direftion that it had defcended through in the fame time.
\ This is evident from the confideration, that in every equal part of dif- tance it defcends through, it receives an equal eiFe^live impulfe from gra- vity. Therefore 4 times the diftance, gives 4 times the efteiTtive (but not inftant) impulfe.
II This is evident, when we confider, that a quadruple diftance or impulle, produces only double velocity, and by axiom 3 a quadruple refiftance will be required, to ftop double velocity; confequently their force is as the fquares of their velocities, which brings them to be direftly as their dif- tances defcended through : and this agrees with the fecond law of fpouting fluids. Art. 45.
§ That is, if a body fall 1 6 feet, and ftrike a non-elaftic body, fuch as h#t iron, Ibft lead, clay, &c. it will ftrike with velocity 32, and produce -i
i6 M P: C H A N I C S. C/iap. IF.
•*'■'* 9- pth. The refinance they meet with in any
given time, in paifing through a refilling medium, is as their furfaces, ar^d as the cubes of their ve- locities.*
certain efFeft in a certain time. Again if it fall 64 feet, it will ftrike with velocity 64, and produce a quadruple effeiR:, in a double time ; becaufe, if aperfeftly elaftic body fall 16 feet in one fecond of time, and llrike a per- fectly elaftic plain, with velocity 32 feet, it will rife 16 feet in one fecond of time. Again, if the body fall two feconds of time, it will fall 64 feet, and ftrike with velocity 64, and rife 64 feet in two feconds of time. Now, if we call the rifing of the body the effect of the ftriking velocity (which it really is) then all will appear clearly. But any thing here advanc- ed, if contrary to the opinion of many learned and ingenious author.', ought to be doubted, unlefs known to agree with practice, * This is evident when we confider,
1. That it is a proportion of the furfaces, that meets the refiftance ; *nd,
2. That a double velocity ftrike s a double quantity of refifting particles in the fame time.
3. That a double velocity ftrikes each particle with double the inftant, and four times the effeiftive force, by art. 6.
Therefore, the inftant refiftance is as the fquares of their velocities, and willfoon amount to the whole force of gravity, and reduce tne motion to be imiform. This is the reafon why hail and rain fells with fuch mode- rate force ; whereas if it was non for the refiftance of the air, they would prove fatal to thofe they fall upon. Compare this with the efFeft of wind on mill-fails, proved by experiment, to be as the cubes of the velocity, Art. 69, and with the efferts of fpouting fluids, proved to be as the cubes of their velocities, with equal apertures. Art. 67, and 7th law of fpout- ing fluids.
Again, conflder that the folid content 'of bodies decreafes, as the cubes of their diameters, while their furfaces decreafe only as the fquares of their diameters ; Consequently the fmaller the body, the greater the refift- ance, in proportion to its weight : and this is the reafon why heavy bodies, rediiced to duft, will float in the air ; as, likewiic, feathers, and many other bodies of great furface and little matter. This feems to Ihow, that air . is, perhaps, as heavy as any other matter whatever, of an equal degree of finenefj or fmallnefs of paiticles.
Thefe are the laws of falling bodies fuppofing them to fall in vacuo, or in an unreflfting me Jium ; and without confidering that gravity increales, as the fquare of the diftance from the center of gravity of the attracling power decreafes (4. law of gravity, art. 2 ;) becaufe any fmall diftance, fuch as comes in our pratliice, will make no fenfibJe dilference. But as they fall in the air, -which is a medium of great refiftance, the inftant refifraHce is as the oppofmg furfaces of the falling body, and as the fquares of their ve- locities, their motion will greatly difler from thefe laws, in falling great diftances, or with light bodies ; but in fmall diftances, fuch as 30 feet or lefs, and heavy bodies, the dilference will be iniperceptable in common prafiice.
A TABLE OP THE MOTION OF FALLING BODIES,
SUPPOSED IN. VACUO.
|
Diftai in fa and |
|||||
|
nee pafled feet. |
1^1 yp- ►a. 0- V^ B e ro |
||||
|
rr |
S 3 5 |
• -a rt |
3 rr |
' D- '° 0 i |
|
|
rj^ 1 |
'-' "-n r; |
0 1^ |
-h l^f' |
"• fB 0 |
|
|
■"1 1 |
ST ni ti- |
;ri ^ |
n> -J |
^.3 3 |
|
|
9. ! |
ro n> "1 |
S-S |
n 0 |
3 C- Cl. |
|
|
I |
8.1 |
.125 |
.25 |
4- |
|
|
2 3 |
11.4 14. |
-25 '5 |
l.oi 4-^5 |
8.1 16.2 |
|
|
4 |
16.2 |
•75 |
9. II |
24-3 |
|
|
5 |
18. |
I |
16.2 |
32.4 |
|
|
6 |
19.84 |
2 |
64.8 |
64.8 |
|
|
7 |
21.43 |
3 |
145.8 |
97.2 129.6 |
|
|
8 |
22.8 |
4 |
259.2 |
||
|
9 |
^4-3 |
5 |
305- |
162. |
|
|
lO |
25-54 |
6 |
583.2 |
194.4 |
|
|
II |
26.73 |
7 |
793.8 |
226.8 |
|
|
12 |
28. |
8 |
1036.8 |
259-2 |
|
|
13 |
29.16 |
9 |
1312.2 |
291 .6 |
|
|
H |
30.2 |
10 |
1620. |
324- |
|
|
^5 |
3^-34 |
50 |
14580. |
972. |
|
|
i6 |
32.4 |
60 |
58320. |
1944. |
|
|
17 |
33-32 |
||||
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18 |
34-34 |
||||
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19 |
35-18 |
||||
|
20 |
36.2 |
||||
|
21 |
37-11 |
||||
|
36 |
48.6 |
> |
|||
|
49 |
56.7 |
||||
|
64 |
64.8 |
||||
|
100 |
81. |
1 |
|||
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iiH4 |
97.2 |
||||
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L- D if»w ..i ■ .1 ^■.■i—i~-^ |
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m |
A SCALE OF THE MOTION of FALLING BODIES.^
16.2 feet is the fpace fallen through the I ft fecond, by law 5, which let be equal .to - - - - Which is alfo the whole fpace fallen through at the end of the ift fecond, which let be equal to - ^32.4 feet per fecond is the velocity ac- lo quired by the fall, ditto - - - . . a
48.6 feet is the fpace fallen through the ov 2d fecond, ditto
i^ 64.8 feet do. at the end of 2 feconds, do. ^4.8 feet is the velocity per fecond, ac- quired at the end of the 2d fecond, do.
81. feet is the fpace fallen through the
3d fecond of time, ditto 1 45.8 feet ditto in 3 feconds of time, do.
97.2 feet is the velocity acquired by the fall at the end of 3 feconds, ditto
1 13.4 feet is the fpace fallen through
in the 4th fecond of time, ditto - 259.2 feet ditto in 4 feconds, ditto -
129.6 feet per fecond, is the velocity acquired at the end of 4 feconds, do.
16
3a"
rf rt it
^•5
n ^
Chap, IV' MECHANICS. 19
This fcalc fiiews, at one view, all the laws to A'^t, 8. be performed by the falling body o, which falls from o to J, 16,2 feet, the firft fecond, and ac- quires a velocity that would carry it 32,4 feet, from I to a, the next fecond, by laws 5 and 6 ; this velocity would alfo carry it down to b in the fame time, but its gravity, producing equal effcd:s in equal times, will accelerate it fo much as to take it to 3 in the fame time, by law i. It will BOW have a velocity of 64,8 feet per fecond, that will take it to c horizontally, or dov/n to d, but gravity will help it on to 5 in the fame time. Its velocity will now be 97,2 feet^ whicll will take it horizontally to c, or down to f, but gravity will help it on to 7 ; and its laft acquired veloci- ty will be 129,6 feet per fecond from 7 to g.
If either of thefe horizontal velocities be con- tinued, the body will pafs over double the diflancc it fell, in the fame time, by law 6.
Again, if o be perfeftly elaftic, and, falling, llrikes a perfect elallic plain, either at 1,3, 5 or 7, the effective force of its ftroke will catiie it to rife again to o in the fame fpace of time it took to fall.
Which fhews, that in every equal part of dif- tance, it received an equal cfFeftive impulfe from gravity, and that the total fum of their efFeftivc impulfe is as the diffcance fallen diredlly — and the effedlive force of their ftrokcs will be as the fquares of their velocities, by laws 7 and 8.
CHAPTER V. — <<^>> —
Art. 10. Of Bodies descending, inclined Plains and
CURVED Surfaces.
ODIES defcending inclined plains and curved _ fiirfaces, are fubjeft to the following laws :
1. They are equably accelerated, becaufe their motion is the efFeft. of gravity.
2. The force of gravity propelling the body A, Fig. J. fig. 5, to defcend an inclined plain A D, is to the
abfoiute gravity of the body, as the height of the plain A C is to its length A D.
3. The fpaces defcended thro' are as the fquares of the times.
4. The times, in which the different plains AD, AH, and A I, or the altitude AC, are palfed over, are as their lengths refpeftively.
5. The velocities acquired in defcending fuch plains, in the loweft points D, H, I or C, are all equal.
6. The times and velocities of bodies defcend- ing through plains alike inclined to the horizon, are as the fquare roots^ of their lengths.
7. Their velocities, in all cafes, are as the fquare roots of their perpendicular defcent.
From thefe laws or properties of bodies defcend- ing inclined plains, are deduced the following co- rollaries, viz.
I. That the time, in which a body defcends through the diameter AC, or any cord A a, A e, or A i, are equal. Hence,
Chap. VI, MECHANICS. Ci
2. All the cords of a circle are dcfcribed in equal Art. lo. times.
3. The velocity acquired in defcending thro' any arch, or cord of an arch, of a circle, as a C, in the lowell point C, is equal to the velocity that v/ould be acquired in falling through the per- pendicular height FC.
The motion of pendulums have the fame pro- perties, the rod or firing ading as the fmooth curved furface.
For demonfiration of thefe properties, fee Mar- tin's Philofophy, vol. I, page iii — 117.
■— »iT|f H'l'jJPjl'-
CHAPTER. VI.
Of the Motion of Projectiles. Art. 12.
APPcOjECTILE is a body thrown or pro- jected in any direftion ; fuch as a ilone from the hand, water Ipouting from any vcilel, a ball from a cannon, Sec. hg. 6. Fig. 6.
Every projectile is acled on by two forces at the lame time, viz. the Impulfe and the Gra- vity.
By the impulfe, or projectile force, the body ofprojeftiies. ■will pafs over equal diilances, A B, B C, Sec. in equal times, by ill general law of motion, art. 7, and, by gravity, it defcends through the fpaces A G, G H, Sec. which are as the fquares of the times, by 3d law of falling bodies, art. 9. There- fore, by thefe forces compounded, the body will dcicribe th^ curve A O^, called a parabola ; and
22 MECHANICS, Chap. VIl,
Art. i^ this will be the cafe in all dircftions, except per- ] pendicular ; but the curve will vary with the] elevation, yet it will ftill be what is called a pa^ | rabola. i
If the body is projedted at an angle of 45 de- grees elevation, it will be thrown to the greateft horizontal diftance poffible; and, if projected with double velocity, it will defcribe a quadruple ran-, doia, '■'■m
For a full account and demonflration, fee Mar- tin's Phil. vol. I, p. 128 — 135.
CHAPTER VII.
Art. 13, Of Circular Motion Sc Central Forces.
Fig, 7. T ^ ^ body A, fig, 7, be fufpcnded by a firing
Central for- I A C, and caufcd to move round the centre C, that tendency which it has to fly off froni the centre, is- called the centrifugal force ; and the adlion of the firing upon the body, which con- flantly folicits it towards the centre, and keeps it in the circle A M, is called the centripital force. Speaking of thefe tv/o forces indefinitely, they are called central forces.*
The particular laws of this fpecies of motion, are,
* It may be well to obferve here, that this central force is no real powfef, but only an cffeft of the power that gives the bodythe motion. Its inertia caufes it to recede framthe centre, and fly ofFin a direft tangent line, with the circle it moves in. Therefore this central force can neither add to,nor diminilh from the power of any mechanical or hydratilic engine, unlefs it be by frldtioa.
Chap, VIL MECHANICS. 23
1. Equal bodies defcribing equal circles in equal Art. 13.. times, have equal central forces. ^ ^ trarforce"""
2. Unequal bodies defcribing equal circles in unequal times, their central forces are as their tjuantities of matter multiplied into their velo- cities.
3. Equal bodies defcribing unequal circles in qual times, their velocities and central forces
ire as their diftances from their centres of motion, f)r as the radius of their circles.*
4. Unequal bodies defcribing unequal circles in squal times, their central forces are as their quan- Liries of matter multiplied into their diftance from che centre or radius of their circles.
5. Equal bodies defcribing equal circles in un- equal times, their central forces are as the fquares >f their velocities ; or, in other words, a double /■elocity generates a quadruple central force.t trherefore,
* This ihews, that when mill-ftones are of unequal diameters, and re- La^vs of cen- plve in. equal times, the largeft Jhould have the draught of their furrows ^j.^j forces to :fs, in praportion as their central force is more, which is inverfe propor- j^g confidered iqn; alfo that the draught of a ftone Ihould vary, and be in inverfe pro- j^^ drauchtin'- ortipn to the diftance from the centre- That is, the greater the diilance jaiH-ftones- he lefs the draught.
Hence we conclude, that if ftones revolve in equal times, their draught ^
laft be equal next the centre : that is, fo much of the large ftones, as is qual to the fize of the fmall ones, muft be of equal draught. But that artwhleh is greater, muft have lei's draught in inverfe proportion, as the iftanpe. from the centre is greater, the furrows muft crofs at fo much lefs jngle ; which will be nearly the cafe (if their furrows lead to an equal dif- ince from their centres) at any confiderable diftance trom the centre of lie ftone ; but near the centre the angles become greater than the propor- on, if the furrows be ftraight, a-s appears by the lines g i,h i,g 2, h 2,g 3, 3,in fig. r, pi. XI. the angles near the centre are too great,which ieems to idicate, that the furrows of mill-ftones ihould not be ftraight, but a little urved ; but what this curve fhould be is very difficult to determine exaol- ' by theory. By theory it fliould be fuch as to caufe the angle of fiurowf; rolling, to change in inverfe proportion with the diftance from the centre,
hichwill require, the furrows to curve more,, as they approach the cen-
:e.
t This ftiows that mill-ftones of equal diameters, having their veloci- es unequal, Ihould have the draught of their furrows, as the fquare roots f their number of revolutions per minute. Thus, fuppofe the revolu- ons ol'one ftone to be 8i per minute, and the mean draught of the fur- )ws 5 inches, and found to be right; the revolutions of the otlierto be >o; theri to find the draught, fay, As the fquare root of 81, v/hich is 9, is > the 5 inches (Jr.iught : fo is the fquave raot of loa, v» hich is i'-, tc 4,5
MECHANICS. Chap. Vlt,
6. Unequal bodies defcribing equal circles in unequal times, their central forces are as their quantities multiplied into the fquares of their Velocities.
y. Equal bodies defcribing unequal circles with equal celerities, their central forces are inverfely as their diftances from the centre of motion or radius of the circles.*
8. Equal bodies defcribing unequal circles, having their central forces equal, their periodical ' times are as the fquare roots of their diftances.
9. Therefore the fquares of the periodical times I are proportional to the cubes of their diftances, when neither the periodical times nor the celeri- ties are given. In that cafe,
inches, the draught required (by inverle proportion) becaufe the draught muft decreafe as the central force increafes.
* That is the greaterthe diftancs the lefs the central force. This fho-.vs that mill-ftones of diS^rent diameters, having their peripheries revolving .' ■with equal velocities, fliould have the angle of draught, with which their furrows crofs each other, in inverfe proportion to tlieir diameters, becaufe their central forces are as their diameters, by inverfe proportion, direftly ; and the angle of draught fiiould increafe, as the central force decreafes ,- '. and decreafe, as it increafes.
But here we muft confider, that, to give ftones of different diameters equal draughts, the diflance of their furrows from the centre, muft be in direft proportion to their diameters. Thus, as 4 feet diameter, is to 4 inches draught : fo is five feet diameter to 5 inches draught. To make the furrows of each pair of ftones crofs each otlier at equal angles, in all proportional diftances from the centre, fee fig. i, plate XI. vi'here gb, gd, gf, ha, he, and he, Ihow the direftion of the furrows of the 4, 5 and 6 feet ftones, with their proportional draughts; now it is obvious that they crofs each other at equal angles, becaufe the refpeftive lines are pa- rallel, and crofs in each ftdne, near the middle of the radius, which fhows that in all pioportional diftances, they crofs at equal angles, confequent- ly their draughts are equal.
But the draught muft be further increafed, with the diameter of the ftone, in order to increafe the angle of draught in the inverfe ratio, as the central force decreafes.
To do which fay : If the 4 feet ftone has central force equal i, what cen- tral force will the 5 feet ftone have ? Anfwer : ,8 by the 7th law.
Then fay) If central foi'ce i requires 5 inches draught, for a 5 feet- ftone, what will central force ,8 require I Anfwer : 6,25 iuches draught. This is, fuppofing the verge of each ftone, to move with equal velocity. This rule may bring out the draught nearly true, provided there bs not much difference between the diameter of the ftones. Eut it appears to me, that neither the angles with which the furrows crofs, nor the dii- tance orthe point from the centre, to which they direft, is a true meafur* «f th? draught.
<Dhap, FIIL ME t li A N I C S. 25
' 10. The central forces arc as the fquares of the Art. 13. diftances inverfely.*
CHAPTER VIII.
<<^>'—
Of the Centres of I^jIagnitude, Motion, and Art. i4« Gravity.
HE centre^pf magnitude is that point which is equally diilant frorh all the external parts of a body.
* Thefe are the laws of circular motion and central forces. For expe- rimental demonftrations of them, fee Fergufon's Le(S:ures on Mechanics,
page 27 to 47-
I may hfere obferve that the whole planetary fyftem is governed by thefe laws of circular motion and central forces. Gravity afting as the ftring, and is the centripetal force ; and as the power of gravity decreafes, as the fquare of the diftance increafes, by the 4th law of gravity, art 2 ; and as the centripetal and centrifugal forces muft always be equal, in order to keep the body in a circle. Hence apears the reafon why the planets moll remote from the fun have their motion fo flow , while thofe near him have their motions fwift; becaufe their celerities muft be fuch as to create a centrifugal force equal to the attraction of gravity.
T may here obferve, that modern philofophers begin to doubt the exi- ftence of inertia, as defined by Newton, to be different and independent from gravity, but feem to conclude that they are both one thhig ; but when we confider that the whole force of gravity is exerted as centripetal force, to keep the heaveiily bodies in a circle. It cannot be tliat fame power,caufe or principle, that caufeS the bodies to continue their motion, unlefs one caufe can produce two effects each equal to itlblfjcontrary to axiom 4- Again we may confider, that gravity decreafes, as the fquares of the diftance of the body from the attraftirig power increafes, but inertia is the fame every where ; and if we fuppofe the body to be removed out of the fphere of at- traction of gravity, there will be no gravity at all, yet inertia v/illaCt in its Attempt to full power, to continue the motion or reft of a body, by axiom i and 2. pj-gy^ j-jjg g Hence in this light gravity and inertia appear to be two very diifercLt jfj-gn^e of i principles and ought to be diltlnguiihed by different names: but here we ej-^ja. may difpute about words, for in other lights they appear to bs the very fame tiling.
E
ex-
MECHANICS. Chap, VIII.
2. The centre of motion is that point which remains at reft, while all other parts of the body move round it.
3. The centre of gravity of bodies, is of great
confeqiience to be well underftood, it being the
'principle of much mechanical motion, and pof-
feffes the following particular properties :
Ni
1 . If a body is fufpended on this point, as its centre of motion, it will remain at reft in any po- fition.
2. If a body is fufpended on any other point than its centre of gravity, it can reft only in fuch, pofition, that a right line drawn from the centre of the earth, through the centre of gravity, will interfeft the point of fufpenfion.
3. When this point is fupported, the whole bo- dy is kept from falling, r
4. When this point is at liberty to defcend, the whole body will fall.
5. The centre of gravity of all homogeneal bodies, as fquares, circles, fpheres. Sec. is the middle point in a line conneding any two oppofite points or angles.
6. In a triangle, it is in a right line drawn from any angle to bifeft the oppofite fide, at the diftance of one third of its length from the fide bifedied.
7. In a hollow cone, it is in a right line pafTing from the apex to the centre of the bafe, and at the diftance of one third of the fide from the bafe.
8. In a folid cone, it is one fourth the fide from the bafe, in a line drawn from the apex to thc^ centre of the bafe.
Hence the folotion of many curious phsenomena, as, why many bodies ftand more firmly on their bafes than others ; and all bodies will fall, when their centre of gravity falls v/ithout their bafe.
Chap, IX. MECHANICS. 27
Hence appears thereafon, why wheel-carriages, Art 14. loaded with Hones, iron, or any heavy matter, !!^Sv'^' ,
• 11 r- r 1 lit., Reafonswhy
Will not overturn lo eaiy, as when loaded with wheeicam- wood, hay, or any light matter; for when the ^s^sover- load is not higher than a b, the centre of gravity will fall within the centre of the bafe at c ; but if the load is as high as d, it will then fell outfide the bale of the wheels at e, confequently it will overturn. From this appears the error of thofe, who halHIy rile in a coach or boat, when likely to overfet, thereby throwing the centre of gravi- ty more out of the bafe, and increafing the dan- ger.
C H A P T E R IX.
Of THE Mechanical Powers. Art. 15.
A V I N G now premifed and confidered all that is neceffary for the better underftand- ing thofe machines called Mechanical PiDwers, we come to treat of them, and they are fix in number, viz.
The Lever, the Pulley, the Wheel and Axle, the Inclined Plain, the Wedge, and the Scre>»''-
They are called Mechanical Powers, bccaufe they increafe our poM^cr of railing or moving hea- But one prin- vy bodies ; and, although they are fix in number, chanicai"^^ow- they feem to be reducible to one, viz. the Lever, ers. and appear to be governed by one fimple principle, v/hich I Hiall call the Firft General Law of Me- Gf"^^^? ^^^^^ ,
ot mechanical
chanical Powers ; which is this, viz. the Momen- powers.
28 MECHANICS, Ghap, IX,
Art. 15. turns of the power and weight are always equal, when the engine is in equilibrio.
Momentum, here means the produft of the weight of the body multiplied into the diftance id moves ; that is, the power multiplied into its diffl tance moved, or into its diftance from the centrfl of motion, or into its velocity, is equal to thf| weight multiplied into its diftance moved, or int^f its diftance from the centre of motion, or into its|^> velocity ; or, the power multiplied into its perr; pendicular defcent, is equal to the weight muitit plied into its perpendicular afcent.
The Second General Law of Mechanical Pov^-j ers, is, ' ■
The power of the engine, and velocity of the^ weight moved, are always in the inverfe propor- tion to each other ; that is, the greater the velo- city of the weight moved, the lefs it muft be ; and the lefs the velocity, the greater the weight may be ; and that univerfally in all cafes. There- fore, .
The Third General Law is,
Part of the original power is always loft in over- coming fridion, inertia. Sec. but no power can be gained by engines, when time is confidered in the calculation.
-<^tO<^>i..
I N the tiieory of this fcience, Vv^e fuppofe all plains to be perfeftly fmooth and even, levers to have no weight, cords to be perfedly pliable, and machines to have no friftion ; in fnort, aliimper- feaions are to be laid afide, until the theory is cftabliflied, and then proper allowances arc to be made.
Chap, /X. MECHANICS. 29
Art. 16. Of the Lever.
A Lever is a bar of iron, wood, &c. one part of which is fupported by a prop, and all other parts turn or move on that prop, as their centre of motion ; and its length, on each iide of the prop, is called its arms : the velocity or motion *j
of every part of thefe arms, is diredly as its dif- tance from its centre of motion, by 3d law of cir- cular motion.
The Lever — Obferve the following laws: Lawsofta^e
1 . The pov/er and weight are to each other, as i^ver. their didances from the centre of motion, or from the prop, refpeftively.*
2. The power is to the weight, as the diftance the weight moves is to the dillance the power pioves, refpedivcly.t
3. The power is to the weight, as^he perpen- dicular afcent of the weight is to the perpendicu- lar defcent of the power. |
4. Their velocities are as their diflances from their centre of motion, by 3d law of circular mo- tion.
Thefe Umple laws hold univerfally true in all Laws of the mechanical powers or engines; tlierefore it is ^everhoiduni- safy (from theie fimple principles) to compute inlitmecha- :he power of any engine, either fimple or com- "^"^^^ P°^^^^^ Dound; for it is only to find hew much fwifter *"^^"^"' ^' ;he power moves than the weight, or how much 'arther it moves in the fame time ; and fo much s the power, (and time of producing it) increaied y the help of the engine.
* That is, the pov»'er P, fig. 8, v/hich is i multiplied into its difiance Y\g. S.
C, from the centre I2, is equal to the weight I2 multiplied into its dil- mce AB r, each produft being i2.
t That is, the power multiplied into its diftance moved, is equal to the weight multiplied into its diftance m.oved.
t That is, the power multiplied into its perpendicular defcent, is equal J the weight multiplied into its perpendicular afcent.
30 MECHANICS. Chap, IX,
Art. 17. General Rules for computing the Power
OF ANY Engine.
1. DIVIDE either the diftance of the power from its centre of motion, by the diftance of the weight from its centre of motion. Or,
2. Divide the fpace paffed through by the pow- er, by the fpace paffed through by the weight. This fpace may be counted either on the arch de- fcribed, or perperdiculars. And the quotient will fliew how much the power is increafed by the help of the engine.
Then multiply the power applied to the en- gine, by that quotient, and the produft will be the power of the engine, whether fimple or com- pound.
EXAMPLES.
Fig. 8. Let ABC, fig. 8, reprefent a lever; then, to
compute its power, divide the diftance of the power Pfrom its centre of motion BC 12, by the diftance of the weight W, A B 1 ; and the quoti- ent is I 2 : the power is increafed i 2 times by the engine ; v/hich, multiply by the power applied I, produces 12, the power of the engine at A, or the weight W^, that will balance P, and hold the engine in equilibrio. But fuppoie the arm A B to be continued to E, then, to find the power of the | engine, divide the diftance BC 12, by BE6; and the quotient is two; Which multiplied by 1, the power applied, produces 2, the power of the en- gine, or weight w to balance P.
Or divide the perpendicular defcent of the pow- er C D equal 6, by the perpendicular afcent E F equal 3 ; and the quotient 2, multiplied by the j power P equal i, produces 2, the power of the en- ,o:ine at E.
Zhap. IX, M E C H A N I t: S. 31
Or divide the velocity of the powei^ P equal 6; Art. 17, 3y the velocity of the weight w equal 3 ; and the :][uotient 2, multiplied by the power i, produces 2, the power of the engine at E. If the power P had been applied at 8, then it would have re- :juired to have been i 1-2 to balance W, or w; becaufe i 1-2 times 8 is 12, which is the momen- tum of both weights W and w. If it had been applied at 6, it muft have been 2 ; if at 4, it muft have been 3 ; and fo on for any other diftance from the prop or centre of motion.
— — •«'^5>o©o<S»" —
There are Four kinds of Levers, Art. 18.
1. THE common kind, where the prop is Different placed between the weight and power, but gene- kinds of 1©- rally neareft the weight.
2. When the prop is at one ead, the power at the other, and the weight between them.
3. When the prop is atone end, the weight at the other, and the power applied between them.
4. The bended lever, which differs only in form, but not in properties, from the others.
Thofe of the firil and fecond kind have the fame properties anti powers, and are real mechanical powers, becaufe they increafe the power ; but the third kind is a decreafe of power, and only ufed to increafe velocity, as in clocks, watches, and mills, where the firfl mover is too flow, and the velocity increafed by the gearing of the wheels.
The machinery of the human frame is compofed Great power bf the lall kind of lever ; for when we lift a weight ^^^^^^ ^^JS^ by the hand, refting the elbow on any thing, the human frame. mufcle that exerts the force to raife the weight, is faftened at about one tenth of the diftance from
32
Art. iS.
MECHANICS.
Chap. IX't
the elbow to the hand, and inufl exert a force ten times as great as the weight raifed ; there- fore, he that can lift 561bs with his arm at a right angle at the elbow^ exerts a force equal to 56olbs. by the mufcles of his arm. 'Wonderful is the power of the mufcles in thefe cafes. Here appears the reafon, why men of low ftature are ftronger than thofe of high, in proportion to their thicknefs, as is generally the cafe.
'•■<<^S>o©o<®»"
A.rt. ig.
Fig. 9-
Compound
Lever.
Compound Lever„
I F feveral levers are applied to a£l one upon another, as 2 i 3, in fig. p, where No. i is of the firfl kind,. No. 2 of the fecond, and No. 3 of the third. The power of thefe levers, united to ad on the weight Wj is thus found by the follow- ing rule, which will hold univerfally true in any number of levers united, or wheels (which is ii- miiar thereto) afting upon one another;
11 U L E,
General rule. ^^' Multiply the power P, into'the length of ^il the driving levers fucceilivelyj and note the prc.'iuft.
2d, Tiieii multiply all the leading fevers intd one another fuccellively, and note the product.
3d. Divide the firfl produdl by the laft, and the quotient will be the weight w, that will hold the machine in equilibrio.
This rule is founded on the firft law of the lever, art. 16, and on this principle, viz. Fundamental If the Weight w, aud powcr P, are fuch, that SeTfbrci?-^^ when fufpended on any compound machine, whe-
Chap, IX' MECHANICS.' 3^
tlier of levers united, or of wheels and axles, Art. 19. tlicv hold the machine in eqiiilibrio. Then, if cuiating the
1 Tj • 1^* T 1 • ^ ^1 !• r* 11 power or ve-
the power P, is iTiiiitipiicd into the radius of all locity of any tile driving wheels, or lengths of the drivino- le- combination
1 Ti ' r^L ^ J J ^1 • ^ of wheels or
vers, and the product noted; and the weight w levers. miiitiplied into the radius of all the leading vv^heels, or length of the leading levers, and the produft noted ; thefe products will be equal. If we had taken the velocities or circumferences of the wheels, inftead of their radius, they would have been equal alfo.
On this principle is founded ail rules for calcu- lating the power and motion of wheels in mills, «^c. See art. 20 &: 74.
E X A M P L E S.
Given, the power P equal to 4, on lever 2, at Fig. 9. 8 diftance from the centre of motion. Required, with what force lever i, faftened at 2 from the centre of motion of lever 2, mufc a<St, to hold the. lever 2 in equilibrio.*
By the rule, 4x8 the length of the long arm, is 32, and divided by 2, the length of the Hiort arm, quotes 16, the force required.
Then 16 on the long arm, lever i, at 6 from the centre of motion. Required, the weight on the fnort arm, at 2, to balance it.
* In order to abreviate die work, I fliall hereafter ufe the following Algebraic figns, viz* The lign -\- more, for addition. — lefs, for fubtraclion. X multiplied, for raultipli cation. .|. divided, for divifion. n equal, for equality- Then, inftead of 8 more 4 equal 12, I Ihall write 84-/.rri2. Inftead of 12 lefs 4 equal 8, 12 — 4 = 8. Inftead of 6 rnulci- Iplied by 4 equal 24, 6x4 = 24- And infteaJ of 24 divided by 3 equal 8, 24 { ^ =:8.
F
34 , MECHANICS. Chap,, IX,
^rt. 19. By the rule, 16x6 — 96^ which divided by 2,
the fhort arm, quotes 48, for the weight re- quired.
Then 48 is on the lever 3, at 2 from the cen- tre. Required, the weight at 8 to balance it. ,"
Then 48X2 = 96, which, divided by 8, th length of the long arm, quotes 12, the Vk^elght required.
Given, the power P=:4, on one end of the com- bination of levers. Required, the weight w, on the other end, to hold the whole in equilibrio.
Then by the rule, 4X8X6X2 = 384 the produft of the pov/er multiplied into the length of all the driving levers, and 2X2X8 = 32 the producfl of all the leading levers, and 384 | 32 = 12 the weight w required.
Art. 20. THE fame rule holds good in calculating the Foundation of powcrs of machines, confining of wheels whether findb^^the'"' ^^i"^ple or compound, by counting the radius of motion of the whccls as the levers ; and becaufe the diame- numbeJ- of tersand circumferences of circles are proportional ; cogs to pro- we may tak^ the circumference inllead of the ra- duce motions. ^\^^^_^ ^ud it wlll be the fame. Then again, be- caufe the number of cogs in the wheels, coaftitute the circle, we may take the number of cogs and rounds inllead of the circle or radius, and the j e- fult will be the fame. Fig. II. Let Fig. 1 1 reprefent a water-mill (for grind-
ing grain) double geared :
Number 8 The water-wheel, J
4 The great cog-whcci, 1:
2 The w allow cr, '1^
3 The counter cog-wheel,
1 The trundle.
2 The mili-iiones.
€hap. XL MECHANICS. 3^
And let the above numbers alfo reprefent the Ait. 20, radius of the wheels in feet.
Now fuppofe there be a power of 5001b. on the water-wheel, required what will be the force ex- erted on the mill-Hone, 2 feet from the centre.
Then, by the rule, 500X8X2X1=8000, and Thepoweroa 14X3X2 = 24, by which divide 8000, and it quotes ^^'heTrgTven, 1333,33 lb. the power or force required, exerted tofiudthe on the mill-lbone two feet from its centre, which Ef,'tJe' ^11?^"^ is tlie mean circle of a 6 feet ftonc. — And as the ftone. velocities are as thedillance from the centre of mo- tion, by 3d law of circular motion, art. 13, there- fore, to lind the velocity of the mean circle of the ftone 2, deduce the following rule, viz,
ill. Multiply the velocity of the watcr-v/heel Ruietofind into the radius or circumference of all the driving; th^veiocity oi
, o tne mean cir-
wheels, iucceifively, and note tiie product. cieofamiil-
2. Multiply the radius or circumference of all *^°'^^"
the leading v/iieels, fuccefiively, and note the pro-
du<0; ; divide the firft by the lait produil, and the
quotient wiJl be the anfwer.
But obferve here, that the driving wheels in
this rule, are the leading levers in the lafl rule.
.EXAMPLES.
Suppofe the velocity of the vi^^ater- wheel to be 12 feet per fecond ; then by the rule 12X4X3X2 = - 288 and 8^2X I =16 by which divide the firft pro- duct 288, and it quotes 1 8 feet per ft-cond, the ve- locity of the flone, 2 feet from its centre.
Povjer decreafes as ^notion increofes. Art. 21.
IT may be proper to obferv here, that as the velocity of the ftone is incrcaferl, the powf r to move it isjdecreafed, and as its velocity is dec reaf-
J
6 MECHANICS. C/iap, IX.
Art. 21. cd, the power on it to move it is increafed, by
2d general law of meciianical powers. This holds univerfally true in all engines that can poffibly be contrived ; which is evident froni tiie ift law of the lever, viz. the power irmltiplied into its ve- locity or diftance moved, is equal to the weight multiplied into its velocity or diftance moved. Rule tajfind Hcncc the genera,] rule to compute the power erted*tr4ove 9^^"y engine, fimple or compound, art. 17. If imiu-ftone. you have the moving power, and its velocity or di.rtance moved, given, and the velocity or dis- tance of the weight, then, to find the weight (which, in mills, is the force to move the flone, <^c.) divide that produft by the velocity of t^e weight or mill-fLon^e, &cq. and it quotes the weight or force exerted on the ftone to move it : But a certain quantity or proportion of this force is lojjl:, in order to obtain a velocity to tli^^ done ; v/hlch is fliewn in art, 29.*
ArL. 22, j^Q Power gained by enlarging Underplot Water-
IV he els.
No power THIS fccms a proper time to fnev/ the abfiu'di-
freafmg Ae" ty of the idea of inerealing the power of the miiJ, diameter of by eularging the diameter of the water- v/ heel, on- tir-vJh2eis7o^i the principle of lengthening the kver, or by dou- the principle blc gearing mills where fingle gears will do \ b^-
of lengtheniiis c ^\ -^i i • <- i t • "
the lever. caule the pov/cr can neither be increaiea nor di- miniihed by the help of engines, v/hile the velo- city of the body moved is to remain the fame,
EXAMPLE.
Fig. ir.
Suppofe wc enlarge the diameter of the water- wheel from 8 to 16 feet radius, fig, 11, and leave
* Philofophers have hitherto attributed this lofs of power to friftion, which is owing; to the viiinertia of matter.
Chap, IX, M E C H A N I C S. 37
the other vv? heels the laine ; then, to. find the vc- Art. a^. locity of the fconc, allowing the velocity of the periphery of tlie water-v/heel to be the Ihrne (12 feet per fecoud) ; by the rule, i2X.iX3X2==288, and 16X2X1—32, by which divide 288, it quotes p feet in a fecond, for the velocity of the ftone.
''I'hen, to find the power by the rule for that purpofe, art. 20, 500 X 16 X 2 X i :::t^QQO, and 4X3x2^:24, by which divi'de 16000, it quotes 666^^6l\y. the power. But as velocity as well as power, is aeceiiary in BiiHs, we fxialil- be ofeiiged, iii order to reftorc the vdocity, to enlarg-e the ^reat cog-wheel froin 4 to .8 radius.
Then, to find the velocity, 12x8X3X2=576^ md 16X2X1=32, by which divide 576, it quotes t&, the velocity xTS before.
:. Th^eoiy tro-fiiid tlie power by the ruie^ ^rt. 20, t will be 333,33, as befafe.
\- Therefore no pov/er can be g.aincd, upon the brinciple of lengthening the lever, by enlarging die v/ater-wk^el.
I- The true- advantages tha^t large whqels lii;ave .The true ad- iver fmali oaes, arifes- from the \vidtko4"the buc- 7^"'^''-? ^^^
, . 1 /- I, • 'J- large wheels
:ets bearing bu.t a imail piroportmn up: tiic ra.uiiis have over >f th,^ wheel ; beca.ufe if thjs. radius af tlije wh^elf'"^^^ "''*'' )e 8 feet, ai'id the v/idth.of the bulcfcet or il.Qat-. • board but i foat, the lioat takes up; but i-S of he arm, and tjlie v-^atcr may he faciei to^(S^ faiiflyr ;ipon the end of the arm and to advantage. But jf the r:;dlus of the v/heel be but 2 feet, and the v^idth of the float r fo£>€, part of the v/ater i^ill act on the middle of the arm, and act to dif- d vantage, as the float takes up half the arm. ^'he large wheel alio ferves the purpofe of a lly- Ihttl ; (art. 30) it likewife keeps a more regular lotion, aaci. cafls off back water better. See art, o.
Bus thfi expencc of tlicfe large wheels is tabe d^cD into confidera:t.ioi3, and then the buildei'
38 MECHANICS. Ghap, IX,
Art. 22. will find that there is a maximum fize, (fee art.
44) or a fize that will yield him thegrealeit pro- fit.
Art. 23. ^0 Power gained by double gearing Mills ^ but feme.
loft,
Nopo-iv-er I might alfo go on to fliew that no power or
biT^elrJ'^b"" ^^"^^.ntage is to be gained by double gearing Mills, feme loft.' upon any other principles than the following, viz.
1. The motion neceffary for the flone, can fometimes be obtained w^ithout having the trun- dle too fmall, becaufe we are obliged to have the pitch of the cogs and rounds, and the fize of the fpindle large enough, to bear the ftrefs of the power. This pitch of gear and fize of fpindle may bear too great a proportion to the radius of the trundle (as does the fize of the float to the ra- dius of the water-wheel, art 22) and may work hard. Therefore there may be a lofs of power on that account ; as there can be a lofs but no gain, by 3. general lav/ of mechanical powers, art. 15.
2. The mill may be made more convenient ior two pair of flones to one water-vvhecli*
•((S'O*^ <^> —
Art, 24. Of the Pulley.
Fig. 10- 2. The pulley is a mechanical power well
known. One pulley, if it be moveable by the
Lofles fuftain- * Many and great have been the lofTes fuftained by mill;builders, on ac- ed by errors- count of their not properly underftanding thefe principles. I have otteu met with great high wheels built, M'here thofe of half the fize and expence would <Jo better ; and double gears, whrt-e llngle would do better, Sec. i;c.
Chap. IX, MECHANICS. 39
weight doubles the power, becaufe each rope fuf- Art. 24. tains half the weight.
But if two or more pulleys be joined together of the pulley, in the common way, then tlie eaiiefl: way of com- aneafyway to
•' ' . •' , f, compute Its
puting their power is, to count the number ot power, ropes that join to the lower or moveable block, and fo many times is the power increafed ; becaufe all thefe ropes have to be fhortened, and all run into one rope (called the fall) to which the mov- ing power is applied. If there be 4 ropes the power js increafed fourfold.*. See plate I, fig. 10.
— ••■«^c<^^g»..~—
Of the TVheel and Axle, Art. 2 5.
3. THE wheel and axle, fig. 17, is a mechanical power, the fame as the lever of the firllkind ; there fore the power is to the weight, as the diameter of the axle is to the diameter of the wheel; or the pow- er multiplied into the radius of the wheel, is Equal to the weight multiplied into the radius of the axlet, in an equilibrium of this engine.
Of the inclined Plain. Art. 26.
4. Tlic inclined plain is the fourth meclianicai inclined plain bower : and in this the power is to the weight, "^ "'^"
as the height of the plain is to its length. This
■ * In this engine there is great lofs of original poM'er, by the gVeat friction ^ .
)f the pullies and ropes in bendina;, &c. But there is a very great improve- """
tient lately difcovercd, on the pulley, which is as follows : Make a fyftem r,n.
if pullies of Inch conftruftion, that when thofe of the upper block all J!|„j^,;,,^,"^ °
iiKed together on one pin will revolve in equal time, and the fame in the
Dv.'sr blofk, which effeetually evades all the friction of the fides of the pidlies
nd ropes pailing through the blocks. But as it is almofi; impollibie to pro-
ortion the diameters of the pullies to the motion of the ropes fo exadtly,
L will be beft to let them have liberty to turn on the pin, fo as to ftretch all
he ropes equally.
t There is but little lofs of original power in this engine, becaufe it lias St little f^-lclian.
pullic
40 M E C H A N I iC S. Ch&p, I]t>
Art. a6. is- of ufe in rolling licavy bodies, fuch as barrels,
iioglheads, cxc. into wheel-carriages, Sec. and for letting th€m down again. S«€ plate V, fig. i. If the height of the plain be half its length, then half the force will roll the body u^ the plain, that would lift it perpendiculaiiy.
——«^i^>:<S> )•■'—*
Art. 27. ^-f ^^'^ fVedge,
^, , c. The v/cdjre' is only ail inclined plain.
The wedge •■j q j ■•
equal to an in- Whence, in the common form of it, the power
ciined plain, applied will be to the rehftance to be overcome,
as the thicknefs of the v/edge is to the length
thereof. This is a very great mechanical power,
and may be faid to excel all the reft ; becaiife with
it we can effeft, v/hat vre cannot v/ith any other
in the fame time, and I think may be computed
in the following manner.
iluietocom^ If the wedge be 12 inches long and 2 inches
putethepow- thick, ■ then the power to hold it in ecuilibrio is
erofthe ' , , ,- n i • ''en
■wedge. as I to oalancc 12 reintance ; that is, 12 rehiLance
preiung on each iide of the vv-edge, ^ and when ftruck with a mallet, the whole force of the gra- vity of the mallet, added to the vt'hoie force of the agent exerted in the fcroke, is communicated to thfe wedge in the time it continues to move : and this force to produce cflbft, is as the fquarc of tlie velocity, with wliich the raalict ftrikes,
* NiTX'.- if -^'C coniiderl'iiat t'le one I2 aftlag on nrre fide of the %ved<:;e re- prefent3the re-aftionof the ground on tlie imdci'fide of tlic inclined plain, we will then plainly fee That the v/edgc and inclined plain lire both one thing; for if this vredge be applied to raife a weiglit of 12, it will require 2 inviead of I to drive it under the M-eight. Eiit if the groimd -sv-ould give way u'.^der the wedge a', eafi'.y, and ir.ove the fame dill^ance that the weight raifes, then tlie weiglvt-.yould be raifcdonly Iialfthe height ; confcquently, i v,-onld drive the wedge under t.iie weiglit, and this yielding of the ground equal to the vaifing of the -^veight, will truly reprefent the yielding of the cleft on each fide of the v/edge. And this is the true principle of the wedge notwithflaud' ing fo rauch has been faid to prove it to fce etjiial to 2 inciiued plane?. See Fergufon's Jet^ures-
Chap. IX. M E C H A N I C S. 4I
mnltipliedinto its weight ; therefore the mallet Art. 27. fhould not be too iarge, (fee art. 44) becaufe it may be too iieavy for the workmaa*s ilrength, and will meet too much reiiftance from the air, fo that it will loofe more by lefTening the veloci- ty, than it vvall gain by its weight. Suppoie a mallet of loib. Itrike \vith 5 velocity, its efi'e6l- ive momentum 250; but if it Ifrike with 10 ve- locity, then its cffeftive momentum is 1000. The effeits produced by the fhrokes will be as 250 to 1000 ; and all the force of each ftroke, ex- cept what may be deftroyed by the fri6lion of the wedge, is added in the wedge, until the fum of thefe forces amount to more than the reiillance of the body to be fplit, therefore it muft give v/ay ; but when the wedge doss not move the whole force is deflroyed by the fri<ftion. Therefore tlie lefs the inclination of the (ides of the wedge, the greater refiftance \ve can overcome by it, becaufe it will be eaiier moved by the fcroke.
■<^^^^>" —
Gf the Screw. Art. 28.
6. THE Screw is the laft mentioned mechani- principles and cal power, and is a circular inclined plain (v/hich P°^^'"'*^» °^ *'^"
•11 1 • -A ^ fcrew.
wiU appear by v/rapping a paper, cut m form of an inclined plain round a cylinder) and the lever of the firft kind combined (the lever being applied to force the weight up in the inclined plain) and is a great mechanical power; its ufe is botii for pref- fure and railing great weights. The power ap- plied is to the v/cight it will raife, as the diftance through which the weight moves, is to the dif- tance through which the power moves ; that is, as thediftait^ceof the threads of the fcrew, is to the cir- cle the power defcribes : fo is the power to the weight it will raife. If the diftance of the thread
G
42 MECHANICS. Chap, /X.
Art. 2g. be half an inch, and the lever be 15 inches radius
and the power applied be lolb. tlien the power will deicribe a circle of 94 inches, while the weight raifes half an inch; then, as half an inch is to 94 inches, fo is jolb to i888lb the weight the engine would raife with lolb power. But this is fuppofing the fcrew to have no fri(n:ion, of which it has a great deal.
Perhaps an improvement might be made on the fcrew, for fome particular ufes, by introducing rollers to take olf the friction. See art. 33.
Art. 2g. WE have hitherto confidered the action and effed: of thefe engines, as they would aniwer to the ftriftnefs of mathematical theory, were there no fuch thing as fridion or rubbing of parts upon each other ; by which means, philofophers have allowed, that one third of the effetl of the ma- chine is, at a medium, deftroyed : w^hicli brings "US to treat of it next in courfe.^'
Art. 30. Of the Fly-'ivhcd, and its Ufe. ^ .1
BEFORE I difmifs the fubjeft of mechanical powers, I fhall take notice of the fly-wheel, the
One third of * ^^^ I think it is evident, that this !ofs of i -3 of the original power in
the original producing efFefts by machines, arifes from the vifinertia of the matterthatis power loft, to to be moved. For fuppofe the machine be an elevator, applied to elevate overcome in- wheat, fig. 17, art. 34, it is evident, that if we apply only as much power as ertia, in many '^vill hold the weight of the wheat in the buckets in equilibrio, we will have machines. ^*^ motion .- then in order to obtain a lively motion, we will be obliged to ap-
ply a farther power, which I expeft we will find will be nearly 1-3 of the i whole, art. 41 > and this 1-3 part of the power will be continually employed in changing the ftate of the wheat from reft to a lively motion. Befides, it is {hewn m art. 3 1 , that the friftion of moft machines is not more than r-20 part of the weight upon a plain; and by the difference betM'een the diame- ters of the wheeh and gudgeons, is reduced to i-iooo part of the weight, or the moving power.
Chap, IX, MECHANICS. 43
life of which is to regulate the motion of engines, Art. 30. and mould be made of cafl metal, of a circular F^y^^'ie'^^ "«» form, that it may not meet with mucii reiiitance power, from the air.
Many have taken this wheel for an increafer of power, wliereas it is, in reality, a confiderable de- flroyer of it ; which appears evident, when we coniider that it has no motion of its own, but re- ceives all its motion from the firll mover, and, as the fridion of the gudgeons and refiilance of the air are to be overcome, it cannot be done v/ithout fonie power ; yet this wheel is of great ufc in <many cafes, viz.
I ft. For regulating tlie power, where it is ir- its ufe. regularly appHed, fach as the treadle or crank moved by foot or hand, asfpinning-wheels, turn- ing lathes, iiax-mills, or where fteam is applied, by a crank, to produce a circular motion.
2d, Where the refiftance is irregular, by jerks, Sec. fiich as faw-mills, forges, fiitting-milis, pov/- der-mills, &c.
The fly-wheel, by its inertia, regulates the motion ; becaufe, if it be very heavy, it will re- quire a great miany little Hiocks or impulfes of pov/er to give it a confiderable velocity, and it will require as many equal Ihocks of refiftance to dellroy faid velocity, by axiom 3, art. i.
While a rolling or flitting mill is running emp- t}/, the force of the v/ater is employed in generating velocity to tiie fly-wheel [a heavy water-wheel will have the Ibme efleft] which force, fummed up in the fly, will be fuiiicicnt to continue the motion, v/ithout much abatement, vhile the flieet is running between the rollers; whereas, had the force of the water been lofl I while the mill was empty, fjje would have flack- ened in motion too much before the fliect got through. This may be the cafe w'hcre water is fcarce.
«.tf?^ V;>1 V?>^ '-ei'^ t<^ '-i?^ '-<i^ Vi>^ «<5>^ t<i^ s^^-j ^i?^ <^;?^ <-<^
CHAPTER
Art. 31
J'riftion.
0F Friction.
ROM what I can gather from different ._ authors,* and by my own experiments, I conclude that the doftrine of fridion is as fellows, and we may fay it is fubjed to the following laws, viz.
Its laws.
Laws of F rid ion.
1 . It is neither increafed nor decreafed by in- crealing or decreaiing the furfaces of contaft of the moving body.t
2. It is in proportion to the weight and veloci- ty, conjointly, of the moving body. J
Philofophers opinions about it.
Fig. 13-
It is equal to near 1-3 of the weight oil a plain.
Ijicreafed by velocity.
* Philofophers, treating of friftion, feera to agree in telling us, that if a perfeffly hard body of any weight could be made perfeiliy fmooth andeVen, and laid on a horizontal plain perfeftly hard, fmooth and even, that then tlie leaft force would move the faid weight in any horizontal direftion ; and that it is the roughnefs of the beft poliftied and fmoothed bodies, that is the whole caufe offriftion; becaufe the body in being moved, has firft to be raifed over the prominent parts, which is of the nature of an inclined plain. They alfo fay, in treating of the attraftion of cehefion, that if two bodies of the fame kind of matter could be made perfe(!:tly fmooth and even, lo that the parts would meet exactly, they would ftrongiy cohere or ftick together by attraftion ; by which it appears that the docT:rine of friftion is not yet well explained.
t They alfo fay, that it is proved by experiment, that if a fquare piece of wood or brafs,as F,fig. 1 3, four inches wide, and i inch thick,be made fmooth, and laid on a fmooth plain, AB C D, and the Vi^eight P hung over a pullie, that it will require the weight P to be nearly 1-3 part of the weight of the body F, to draw it along ; and that the fame, whether it be on its flat fide or edge. This proves law i£, that friction is not increafed by increafng the furface of contaft,
\ It has alfo been proved by experiment, that if we fix the lever L, to draw the weight F, making o its centre of motion, and by a covd make F faft to the lever at the point i.and hang the weight Q^at the end of the lever over a pullie, and make O^juft fufficient to move F ; O v/iU then be found to be 1-7
Chap. Z. M E C H A N I C S. 45
3. This proportion clecreafes as tl.e weight and ^j.^ ^r. velocity increafes, bat by what ratio, is not de- termined,*
of P, becauie it will have to move F but r-y of the diftance. Then move the
cord fiOiiii to 2, and we find the ^T eight Q_niuft now be doubled equal to 2-7 Isdireftlyas
of Pto move F ; (the reafon is evident from the laws of the lever) becaufe the diftance
F is double the diltance from the centre of motion that it v/as at i, and it °^ t^e lub-
will h^ve to move double the diflance if the lever, or power O^uiove the fame bing lurfa-
diftance. This iliews that friction is as the diftance from the centre of mo- ces irom the
tion; that is, it is as the dianieter of the gudgeons, double diameter, double centre of
friclion ; tlrerefore gudgeons ought to be as fmallas poliible, lo as to be fuf- motion.
ficicHtly ftrong to endure the llrefs of the weight.
* They have alfo proved by experiment, that if F be a brafs plate of 6 j . .
ounces, and A F> C D a brafs piate, both well polifhedand oiled, then it will _j.g„rgj :
require the weight P to be nearly 2 ounces to mo\ e F. But if F be loaded jj^-gXi. nro-
with 6, 8 or lolb. then'a fixth part of that weight will be fuitlcient to draw it „ ^^- " -^i ' ,^, ■ , , ■ ,. . ^ ■ ? . • 1 ^ J r- portion with
along. This proves that the ratio ot the triCtion to the weight decieales, as '., ...gj,,]^^
the -weight increafes : the reafon of which decixafe ot pioportion I take to be ''^
'as follows, viz. Great part of the friction ariies iroin the cobefion of the farts, even thegreafe put on to deftroy the coheiion, has a coheiion of its The ratio iown ; and this coheiion of parts «r of the greafe, wiil i:otincreaie with the decreafes as kveightor velocity. — Again, if we allow the friclion to be occafifcned by the the weight kvjeightof the body having to be railed over the prominent parts of the rub- and velocity jbing furface, it is evident, that when it is raifed by being flarted, that it has increales. not to be railed again ; therefore the greater the velocity, the lei's proporti- on will this refiitance (occalioued by the railing of the body) bear to the ve- locity.
I have made an experiment fiird.'ar to that of fig. 13, with a fiat lided „ jglafs bottle, on a fmooth poplar plank, oiled ; alio on a well polilhed Heel ' 5 plate oiled, and when loaded with lolb. it was drawn by lib. and when load- ed with 221b. it was drawn by 2lb. and when loaded with 6olb. it was drawn by 4 1-2 lbs. which is about 1-13 part : and the motion was greatly accele- rated, which gives realon to conclude, that lefs weight would have continu- icd the motion, after once begun.
I. V/e may leafoiiably lupp'Oj'e, that the gudgeons ofmills, &c. well polilhed, Ratio of running on good Itones or brafs boxes, Sec. and well oiled, have as littie fric- fj-j^ion (-„ tion as the bottle and plank ; and as we find that the proportion of fri^-ion ^jjg weiffht decreafes as the weight increafes, we may fuppofe that in great vveights it 'will not amount to more than 1-20 part of the weight, fuppohng the gudge- jsfot more ons to be the full l;ze or diameter of the wheels, for fb they muft be in order ^Uay, j.oq to be on the fame principles of plains rubbing together. Upon theJe princi- ^^.^ ^ pies I compute the fri&ion of the gudgeons of a well hung water-wlieel, as pj^jj^ jj^" ioUov/s : viz. As the diameter of the wheel is to the diameter of the gud- oreat geons, lb is 1-20 part of the %veight of the wheel, to the weight that will ba- "yejo-i^j- laiiG£ the friction. ■
EXAMPLE.
Suppofe a wheel 15 feet diameter, with gudgeons 3 inches diam.eter, and v/e!gl;ing403oro. by fuppoi;tion ; then, lay as 15 feet is to 3 iache:, ibis Not more 4000 I 2 J to 3,3!b. the weight on the periphery of the wheel that v/iU ba- than i-rooo iance the friction of 400olb. : which is lefs than i-iooo part ef the weight, part in great But note that for the fame reaforis, tiiat friftion does ndt increase with the machines velocity in direOl proportion, neithei; v/ill it decreafe in direct: proportion and great iVitn tne velocity of the rubbing furface of the gudgeon : hence v/e ir.iift con- weights.
46
MECHANICS.
C/iap. X,
Art. 31. 4. It is greatly varied by the fmoothnefs or
roughnefs, hardnefs or foftnefs, of the furfaces of contadl of the moving bodies,
5. A body without motion has no friftion • therefore, the lefs the motion, the lefs tlie fric- tion.
Art. 32
To reduce friftion.
By friction wheels. Fig. 14-
Of reducing Fri^ion,
TO reduce friftion, we mufl, by mechanical contrivances, reduce the motion of the rubbing parts as much as poflible ; which is done, either by making the gudgeons imall and the diameter of wheels large, or by fixing the gudgeons to run on fridion-whecls. Thus, let A, fig. 14, reprefent the gudgeon of a wheel fet to run on the verge of two wheels of caft metal paffuig each other a little, and the gudgeon laying between them. It is evident, that as A turns, it will turn both friftion-vv heels ; and, if the diameter of gudgeon A is 2 inches, and that of the wheels 12, then the wheels will turn once while A turns 6 times, fo that the velocity of the gudgeons C C of the v^heels, is to the velocity of the gudgeon A, as i is to 6, fiippollng them to be equal in fize ; but as there are 4 of them to bear A, they may be but half the diameter, and then their velocity will be to that of A, as I is to 1 2 ; or A might be fet on one wheel, as at B, with fupporters to keep it on; and, if friftion-wheels are added to fridion- wheciS, the friftion may be reduced to almoft no- thing by that means.
cl'jcie aj^ain that the friftion i"> more than i-rooo part. By which it ap- pears, that the friction ofthe gudgeons, •weil fet on good ftores or brafs boxe<!, i<^, not in miiin v.^orthy of the expence of evading. It bears but a fmall proportion to the friftion or refiflance of the air, e(]3ecially where the velo- !^ity is great. See art. 9, and 9th law of ialiing bodies.
hap. X. MECHANIC S. 47
Late Invention to reduce Friclion. ^**^^'^- 33*
WHEEL-CARRI AGES, pullies, and fuch Rollers appii- ^heels as have large axles in proportion to their l^-^l^.^H^ "''* lameters, have much friftion. There has been latedifcovery, ii) England, oFappiying the prin- iple of the roller to them ; which may be io done s almoll totally to deftroy the friction.
The eafleft method poiTibie, of moving heavy odics horizontally, is tiie roller.
Let A B, fig. 15, reprefent a body of 100 tons Fig. 15- k^eight (with the underiide perfc(!:lly ImootU nd even) fet on two rollers, perfectly hard, mooth, and round, rolling on the horizontal aain C D, perfectly hard, fmooth, and even ; f^iJ^^^*"" '" t is evident that this body is lupported by two ines perfeftly perpendicular, and, if globes were fed inftead of rollers, the leafl force would move t in any horizontal diredion ; even a fpider's web /ould be fufficient, giving it time to overcome he vilinertia of the body : But as perfeil hard- efs, fmoothnefs, Sec, are not attainable, a little fiction will ftill remain.
This principle is, or may be, applied to wheel- iarriages, in the following manner ; j Let the outlide ring BCD, fig. 16, reprefent Fig.r^. jhe box of a carriage-wheel, the infide circle A jhe axle, the circles a a a a a a the rollers round he axle between it and the box, and ti.!e inner ing a thin plate for the pivots of the rollers to un in, to keep them at a proper diflance from ach other. When the wheel turns the rollers -afs round on the axle, and on the infide of the ox, and we may fay without friftion, becauie
here is no rubbing of the parts paft one rmother.'*
I
* To explain this, let \\i fuppofs the rollers aaaaaa to have cogs, and the aft; A, and b«x t* have cogi aUb, the rollers £'.n»rii)j^int4( tUe Ih-^tt iiid 'vat*
^;?>^ VC7-) "^-^S-i '<i>i ' ^^ t:i>5 (.£;>:) t<j?^ t<;>i ^i:>5 (<::>2 •<5>^c^^
CHAPTER XL
— 1^.l
Art. 34. Of Ma.xIxMUms, or the greatest Effects ot
ANY Machine.
Effeft of a
machine,
■wha.t.
Old theory of Tnaximum, motion, and load of en- gines.
HE efFeft of a machine, is the diftance whici it raoves or the velocity with which i] moves any body to which it is applied to give mo, tion, in a given time ; and the v/eight of the bod] multiplied into its diftance moved, or into its ve; locity, fnews the effeft.
The theory publillied by pliilofophers, and r ccived and tauglit as true, for feveral centuriesi, paft, is, that any machine will wOrk with itS; greateft perfection when it is charged with juft
The princi- the infide of tlie box. >Jow it is evident, that if the box will turn round the pies of the ap- axle, it m.uft be without any Aiding of parts ; (and in faft, the prominent parts plication of of the rollers, axle and box, will aft as cogs ) then, if the rollers and axle bC) rollers ex- all of one diameter, they will have an equal number of cogs ; and as the dia- plained. meter of the box will be 3 times the diameter of the rollers, it will have
3 times as many cogs. Now it is evident, that the axle mufl turn i i-Jf. times round, before the lame cogs of the rollers and Ihaft will meet, that Vv'ere together when it ftarted ; becaufe, in that time the rollers will have moved over 1-3 of the box : therefore the axle mult turn 3 3-3 times equal to 4times round, by the time the box is once meafured by the rollers. Then fuppofe we hold the axle at reft, and turn the box round like a carriage wheel ; then, while the box turns i 1-3 times round the axle, it will caufe the rollers, to 'move once round; and while the box or wheel turns round the axle 4 times, the rollers will run round it three times. For fuppofe we divide the ; box into 3 parts, B C andD, then beginning to turn the box from B to D, itt < is evident, that v/hile the roller a b meafures once round the axle and returnsj to the fame place, it will alfo meafure the box from B to C, ard C will havc-^i taken the place of B, and the next revolution of the roller, D will take thejj'i place of C, and the tliird revolution B returns to where it was at tirft, and ' the box has mcde 4 revolutions, while the roilers.have made 3 round the j axle, and wkhout any iliding of parts, therefore without friftion- I might go on to ihe^v, that if the axle be much larger than the rollers, they will alfi work without fading.
Ghap, XT. MECHANICS. 49
4-9 of the power that would hold it in equilibrio. Art. 34. and then its velocity will be juft 1-3 of the great- eft velocity of the moving power.
To explain this, theyfuppofe the water-wheel, fig. 17, to be of the underfhot kind, 16 feet dia- Fig. 17. meter, turned by water ilfuingfrom under a 4feet head, with a gate i foot wide, i foot high drawn ; then the force will be 25olbs. becaufe that is the "v^eight of the column of v/ater above tlie gate, and its velocity will be 16,2 feet per fecond, as jfhall be fliewn under the head of Hydraulics ; then the wheel will be moved by a power of s^olbs. and if let run empty, will move with a velocity of 16 feet per fecond ; but if v/e hang the weight W to the axle (of 2 feet diameter) with a rope, and continue to add to it until it ftops the wheel, and holds it in equilibrio, the weight will be found to be 200oIbs. by the rule, art. 19; and then the ellcd" of the machine is nothing, becaufe the velocity is nothing : But as we decreafe the weight W, the wheel begins to move, and its velocity increafes accordingly ; and then the pro- du(!l of the weight multiplied into its velocity, will increafe until the weight is decreafed to 4-9 of 2000=888,7, which, multiplied into its dil- tance moved or velocity, will produce the great- eiv: effetl, and the velocity of the v/heei will then be 1-3 of 16 feet, or 5,33 feet per fecond. So fay thofe who have treated of it.
This will appear plainer to a young learner, if Theory of lie will conceive this wheel to be applied to work maximums an elevator, as E, fig. 17, to hoift wheat, and appircadon ^of fuppoie that the buckets, when all full, contain an elevator. 9 pecks, and will hold the wheel in equilibrio, '^' ^ It IS evident it will then hoift none, becaufe it ha^ no motion ; then, in order to obtain motion, w^ ninft lefTcn the- quantity in the buckets, when th wheel will begin to move, and hoift fafter an
H
5© MECHANIC S. Chap, XL
Alt. 34. fafter until the quantity is decreafed to 4-9, or 4 pecks, and then, by the theory, the velocity of the machine will be r-3 of the greateil: velocity, when it will hoili the greatefl quantity poffible in a given time : for if we leffen the quantity in the buckets below 4 pecks, the quantity hoiftedin any given time will be lefFened.
This is the theory efliabliihed, for demonftra- tion of which, fee Martin's Philofophy, vol. I, page 185—187.
Art. 2,S* ^^^ Theory mveftigated, .
inveftio-ation ■''• "^ Order to invcfligate this theory, and thel of the old better to underftand what has been laid, let uSi theory. confidcr as follov/s, viz.
1. That the velocity of fpouting v/ater, under^ 4 feet head, is r6 feet per fecond, nearly.
2. The feclion or area of the gate drawn, in feet, multiplied by the height or the head in feet,* gives the cubic feet in the whole column, which multiplied by 62,5 (the weight of a cubic foot of water j gives the weight or force of the wholes column preiling on the wheel.
3. That the radius of the wheel, multiplied by the force, and that product divided by the radius of the axle, gives the weight that will hold the wheel in equilibrio.
4. That the abfblnte velocity of the wheel, fubtradled from the abfolute velocity of the wa- ter, leaves the relative velocity with which the water fbrikes the wheel in motion.
5. That as the radius of the wheel is to the ra- dius of the axle, fo is the velocity of the wheel to the velocity of the weight hoifted on the axle.
i Chap, XL MECHANICS. 51,
6. That the effe-fts offpouting fluids are as the Art. 35. fquares of their velocities (fee art. 45, law 6)
but the inflant force of ftriking fluids, are as their velocities fimply. See art. 8.
7. Thflt the weight hoifted, multiplied into its perpendicular afcent, givers the effeft.
8. That the weight of water expended, multi- plied into its perpendicular defcent, gives the power ufed per fecond.
On thefe pinciples I have calculated the follow- ing fcale ; firfl fuppofing the force of llriking fluids to be as the fquare of their ftriking or rela- tive velocity, which brings out the maximum a- greeably to the old theory, viz.
When the load, at equilibrio, is 2COO, then the maximum load is 888,7 = 1 of 2000, when the eifeft is at its greateft, viz. ^91, 98, as appears in the 6th column, and then the velocity of the wheel is 5,333 feet per fecond, equal to 1-3 of 16, the velocity of the M^ater, as appears in *
the 5th line of the fcale: but as there is an evi- old theory dent eiror in the firfl principle of this theory, by <io"bted. counting the inftant force of the water on the wheel to be as the fquare of its flriking velocity, therefore it cannot be true. See art. 41.
I then calculate upon this principle, viz. That the inflant force of flriking fluids is as their velo- city fimply, then the load that the machine will carry, with its different velocities, will be as the velocity fimply, as appears in the 7th column, and the load, at a maximum, is iooolb~'r of 2000, the load at equ;]ibrio, when the velocity of the wheel is 8 feet = f- of 16 the velocity of the v/ater per fecond ; and then the effedl is at its greateil, asfhewn in the 8th column, viz. loco, as appears in the 4tiiline of the fcale.
This I call the new theory, (becaufe I found New theory, that William Waring had alfo, about the fame tim.e, eftabllflied it, fee art. 38) viz. That when
f
52 M E C R A N I C S. Chap, XI,
Art. 55, any machine is charged with juft 1-2 of the load that will hold it in eqnilibrio, its velocity will be juft 1-2 of the natural velocity of the moving power, and then its effed will be at a maximum, or greateft poffible.
This appears to be the way by which this great error has been fo long overlooked by philofophers, and which has rendered the theory of no ufe in practice, but led many into expenfive errors, thereby bringing great difcredit upon philofb- phy.
For demonftrations of the old theory, fee Mar- tin's Phil. vol. I, page 185 — 187.
Ratio of the pov/er and elFeft at a maximum, the
power being 4000 in
each cafe.
EfEedt, by new theory.
Weight hoiiled, accord- ing to new theory.
EffeA, by the old theory
Weight hoiiled, accord- ing to the old theory.
Maxi- I-" mum by q new the- '*-' cry - - 'Nt-
^- Maximum •"by old theo-
O
»^
"o^ r\ 0 r^GO Ox o <o
i-O CO O CO Tx -O "-O tN. txo>0 C>coco r^co
O O O O CM '-C o o o O -''> O 'O CO t\ o 10 o >-(0j>.O CM cOco'-Ot^O
JO
^
^' 00 "
Tx >-i O ^'i — O (M 0}
O CO "^^ O 00 C\ CNSO 00
■^ CO '-'^ ^J^/ "-C ^ -O (V^
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*~r; >-* O '-'00 to'O'-" O
<sco OOOCO tVc^ coo
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M M C)
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Velocity of the weight afccnding. |
4-1 |
10 |
woo c^ |
WO |
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*V «S, JN f> |
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l-i W W-l |
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Velocity with which the |
NO |
||||
|
Avater (Irikes the wheel |
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NO |
|||
|
in motion, or relative |
(U |
■ f> |
|||
|
Telocity. |
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0 |
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0 0 H4 M l-H l-H (-1 l-H |
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Velocity of the wheel |
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CO |
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per feeond, byfuppo- |
4-1 0 |
CO |
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fition. |
ti^ |
<o |
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54 M E C H A N I C S. Chap, XL
i Art. sS. New Theory doubted,
BUT although I know that the velocity of the wheel, by this new theory is much nearer prac- tice than the old, (tho' rather flow) yet I am led to doubt the theory, for the following- reaions, viz.
When I confider that there are i6 cubic feet oJ water, equal looolbs. expended in a lecond, whichj multiplied by its perpendicular defcent, 4 feet,prG-| duces the power 4000. The ratio of the power: and effect by the old theory is as 1 o to i ,47, and by the new as 4 to i ; as appears in the 9th co-:j lumn of the fcale ; which is a proof that the old ^ theory is a great error, and fufEcient caufe olj
doubt that there is yet fome error in the new, Ancii as the fubjeft is of the greateft confequence in prac- tical mechanics. Therefore I proceed, to endea-.| vour to difcover a true theory, and will Ihew myi work in order, that if I eftablilli a theory it ma)i be the eafier underftood, if right ; or detected, i:!l wrong.
Attempts made to difcover a true Theory,
In the fearch, I conflruded Fig 18, pi. II. whicl reprefents a fimple wheel with a rope pafling ove: it and the weight P, of 100 lbs. at one end to a^ by its gravity, as a power to produce efredls, b] hoifting the weight w at the other end.
This feems to be on the principles of the lever and overfhot wheel ; but with this exception, thai the quantity ofdcfcending matter, ailing as po\y* er, will fiill be the fame, although the Telocit; will be accelerated, whereas in overiliot wheels the power on the wheel is inverfely, as the velo city of the wheel.
Here we muft confider,
£. That the perpendicular defcent of pov^er PJ
hap, XL MECHANICS. ^^
er fecond, multiplied into its weight, ftiews the An. 36. ower.
2. That the weight w when multiplied into ;s perpendicular afcent gives the efFedt.
3. That the natural velocity of the falling bo- y P, is t6 feet the firfl fecond, and the diflance t has to fall 16 feet.
4. That we do fuppofe that the weight w, or eliftance will occupy its proportional part of the elocity. That is if w be = 4 P? the velocity
[vith which P will then defcend, will be 4 16 = 8 et per fecond,
5. If w be =: P, there can be no velocity, con- sequently no effeft ; and if w — o then P will de- tend 16 feet in a fecond, but produces no effect; jecaufe, the power, although 1600 per fecond, Is applied to hoift nothing.
; Upon thcfe principles I have calculated the fol- lowing fcale.
LB
A SCALE for determining the Maximum Charge, ani Velocity of loolbs. defcending by its Gravity.
■.^-
|
o i »i o i ET' fli n |
^25 o % "MS ►a < o m ^ o* fD O •-! p^* ►*!"< ^ 1-. rt Ti n> ft X -^ • •a' n o a J2- |
|
lb. lOO |
fect. i6 |
|
cfq' |
^ 1-1 |
Propo and |
^.1 |
rt^? |
pi 9' |
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Is- |
i\ |
5"^ |
0 "1 |
||
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2. 5> |
© 51 |
rt ►0 |
|||
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•n |
<! |
^ w |
u* |
0 |
|
|
■ O' |
0 rt |
<! rt |
rt rt |
rt |
|
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r» |
. <! rt |
2. S- |
'^ ^ |
0 rt |
|
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J-. 0 |
rt'-S |
rt n, |
rt-«T3 |
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rt % p 3 |
09 O >— 0 13 |
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rt |
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lb. |
feet. |
feet. |
i |
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0 |
1600 |
10:0 |
|||
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,16 |
15.84 |
15,84 |
1584 |
10 : ,01 |
|
lO |
1,6 |
14:4 |
144 |
1440 |
10 : I |
|
20 |
3^2 |
12,8 |
256 |
1280 |
10 : 2 |
|
30 |
4>8 |
11,2 |
336 |
I I 20 |
10 : 3 |
|
40 |
6,4 |
9,6 |
384 |
960 |
10 : 4 |
|
50 |
,8 |
8, |
400 |
Soo |
10: 5 |
|
60 |
9,6 |
6,4 |
384 |
640 |
10 : 6 |
|
70 |
11,2 |
4.8 |
336 |
480 |
10 : 7 |
|
, 80 |
12,8 |
3^2 |
256 |
320 |
10 : 8 |
|
90 |
14,4 |
1,6 |
144 |
160 |
10 : 9 |
|
99 |
15,84 |
,I0 |
15,8 |
16 |
10 : 9,9 |
|
100 |
16, |
0. |
0 |
0 |
maximum.
by new theory
%-=
Chap, Xi, MECHANICS. 57
By this fcale it ap|)ears, that when the weight Art. 36. w isr=50-=:f- P the power ; the effect is at a max- imum, viz. 400, as appears in the 6th column, when the velocity is half the natural velocity, v\l. 8 feet per fecond ; and then the ratio of the power to the elFefl is as iq to 5, as appears in th& Sth line.
f By this fcale it appears, that all engines that Theory for kre moved by one conflant power,which is equably andbadofen- kccelerated in their velocity (if any fuch there ginesmovedg foe) as appears to be the cafe here muft be charg- Jhorri^tlon ed with weight or refiftance equal to half the is equably ^c- bloving power, in order to produce the greateft "^^^'^ • pfFe£t in a given time ; but if time be not regard- ed, then the greater the charge, fo as to leave any velocity, the greater the eifedt, as appears by Ithe 8th column. So that it appears, that an over- |hot wheel, if it be made immenfely capacious, and to move very flow, may produce effedts in the ratio of 9,9 to lo of the power.
Scale of Experiments, Krt. 27.
THE following fcale of adtual experiments were made to prove whether the refiftance occu- pies its proportion of the velocity, in order that I might judge whether the foregoing fcale was foun- ded on true principles ; the experiments were not very accurately performed, but often repeated, and proved always nearly the fame. See plate H, fig. 18.
SCALE
OF
EXPERIMENTS.
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3,33x4 4X3'5
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23,31 28
31,08
42 46,2
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n 2.^ ^'P; 3
So'?;'
< g rt
rt 5 o
^ C
>-• p
3 !^ CW «
S-o
i^ 2. f*
a' n
5 2.
rt rt O^
24
33,8
44,35 max.newtheo.
59,14
72 maximum.
-^2
Chap, XI. MECHANICS. 59
By this fcale it appears, that when the power Art. 37. P falls freely without any load, it defcends 40 feet in five equal parts of time, but, when charged with3,5lbs.=:4 P, which was 7lbs.it then took up 10 of thofe parts of time to defcend the fame diftance ; which feems to fhew, that the charge occupies its proportional part of the whole velocity, which w^as wanted to be known, and the maximum ap- pear as in the lafl fcale.* It alfo fhews, that the effect is not as the weight multiplied into the fquare of its afcending velocity, this being the meafure of the effed that would be produced by jthe ftroke on a non-elaftic body. i This experiment partly confirmed me in what ll have called the New Theory ; but ftill doubt- iing, and after I had formed the foregoing tables, I called on the late ingenius and worthy friend, iWilliam Waring, teacher in the Friends* Acade- jmy, Philadelphia, for his affiftancc. He told me Jhe had difcovered the error in the old theory, [and corrected it in a paper which he had laid be- jfore the Philofophical Society of Philadelphia, [wherein he had fhewn that the velocity of the 'underlhot water-wheel, to produce a maximum effedt, mufl be juft one half the velocity of the jwater.
JVilliam JVarmg''s Theory, Art. 38.
I The following are extra6ls from the above wniiamwar- jtnentioned paper, publifhed in the third volume ing's theory. •of the Tranfaftions of the American Philofophical jSociety, held at Philadelphia, p. 144.
After his learned and modeft introdudlion, in which he fliews the neceffity of corredling fo great in error as the old theory, he begins with thefe svords, viz.
* Since writing the above, I have feen At^vood's Treatife on Motion, vherein he gives a fet of accurate experiments, to prove fbeyond doubt) hat the conclufion I have drawn is right, viz. That the charge occupies ts proportional part of the whole velocity. See the American Encyclopae- lia, Vol. X. p. 786.
do
Art. 38.
MECHANICS.
Chap, XI ^
Difinition.
Demonftra- tion.
** But, to come to the point, I would juft pre- mife thefe
DEFINITIONS,
If a ftream of water impinge againft a wheel ia motion, there are three different velocities to be confidered appertaining thereto, viz.
Firfl, The abfolute velocity of the water.
Second, The abfolute velocity of the wheel.
Third, The relative velocity of the water t( that of the wheel ; /. e. the difference of the ab-sl folute velocities, or the velocity with which the water overtakes or flrikes the wheel.
Now the miftake confifts in fuppoiing the mo- mentum, or force of the water againfl: the wheel, to be in the duplicate r^tio of the relative veloci- ty ; Whereas J
p a o P.
I.
The force of an invariable ftream, impinging againft a mill- Wheel in motion, is in the limple proportion of the relative velocity.
For, if the relative velocity of a fluid againft a, lingle plain, be varied, either by the motion of the plain, or of the fluid from a given aperture, or both, then the number of particles afting on the plain, in a given time, and likewife the mo- mentum of each particle being refpe£lively as the relative velocity, the force, on both thefe ac- counts, muft be in the duplicate ratio of the rela- tive velocity, agreeable to the common theory, with refped to this fingle plain ; but the number of thefe plains, or parts of the wheel, adled on in a given time, will be as the velocity of the wheel, or inverfely as the relative velocity ; therefore the moving force of the wheel muft be as the fim- ple ratio of the relative velocity Q^E. D.
'^imp. XL MECHANICS. ^I
t- Gr the propofition is manifeft from this confi- Art. 5«. iieration, that while the ftrcam is invariable, jvvhatever be the velocity of the wheel, the fame [lumber of particles, or quantity of the fluid, mufl flrike it fomewhere or other in a. given time ; con- fequently, the variation of th'C force is only on account of the varied impingent velocity of the ilame body, occaiioned by a change of motion m. [the wheel ; th^t js, the momentum is as the rela- tive velocity,
j Now this true principle, fubllituted for the er^ roneous one in ufe, will bring the theory to agree f-cmarkably with the notable experiments of the Ingenius Smeaton, publifhed in the Philofbphical pTranfa^tions of the lloyal Society of London, for the year J 751, vol. LI; for which the honorary annual medal was adjudged by the fociety, and Iprefented to the author by their prelident. I An inflance or tv/o of the importance of this borreftion, may be adduced, as follows :
PROP. II.
i The velocity of a wheel, moved by the impaj^: of a ftream, mud be half the velocity of the fluid, to produce the greateft effeft poffiblc.
/'V = the velocity, M = the momentum, of the
-| fluid.
(^v = the velocit)^, ?:;= the power, of the wheel.
Then V — v = their relative velocity, by defini-^ tion qd.
o'
M
And, as V:V— v:;M;— xV— v == P, (Prop. I)
M — ^
whichxvzrP, v=— -xVv — v^=a maximum; hence
Vv — v^=:a maximum and its flia£lion (v being a variable quantity=:Vv — 2vv=o ; therefore— 4V ; that is, the velocity of the wheel = half that of the fluid, at the place of impad-, when theeffea: is .a maximum, Q^ E, D,
62
Art. 38.
MECHANICS.
C/iap. Xh
Part omitted.
Art. 39.
Further ex- traftsfromW. Waring's pub- lication CQn- ceming his «ew theory.
The effedl of underfliot wheels as the fquares of the velocities of the water.
The ufual theory gives v=7V, where the er- ror is not lefs than one fixth of the true velo- city !"
wm. waring.
Philadelphia, 7 th) 9th mo. 1790.)
Note, I omit quoting prop. Ill, as it is in alge- bra, and refers to a figure, becaale i am not writ- ing fo particularly to men of fcience, as to practi- cal Mechanics,
— ■«®>=^<s>>-. —
Extract from a further paper, read in the phi^ lofophical fociety, April 5th, 1793.
" Since the philofophical fociety were pleafed to! favour my crude obfervations on the theory of mills, with a publication in their tranfadions, I am apprehenlive fome part thereof may be mifap- plied, it being therein demonllrated, that ' the force of an invariable ftream, impinging againfb a mill-wheel in motion, is in the fimple direct ratio of the relative velocity.' Some may fuppofe, that the eifeft produced, fhould be in the fame proportion, and either fall into an error, or find- ing by experiment, the effedt to be as the fquare of the velocity, conclude the new theory to be not well founded ; I therefore wilh there had been a little added, to prevent fuch mifapplication, be- fore the fociety had been troubled with the read- ing of my paper on that fubjedl : perhaps fome- thing like the following.
The maximum eifeCt of an underfhot wheel, produced by a given quantity of water, in a giv- en time, is in the duplicate ratio, of the velocity of the water ; for the effect mull be as the impe- tus adting on the wheel, multiplied into the ve- locity thereof: but this impetus is demonllrated to be limply as the relative velocity, prop. I.
Zhap, XL MECHANICS. 6%
incl the velocity of the wheel, producin'g a max- Art. 3^. mum, beiRg half of the water by prop. II. is [ikewife as the velocity of the water ; hence the power ading on the wheel, multiplied into the velocity of the wheel, or the eifed: produc- ed, muft be in the duplicate ratio of the velocity Df the water. (^ E. D.
CoROL. Hence the effedof a given quantity of water, in a given time, will be as the height of the head, becaufe this height is as the iquare af the velocity. This alfo agrees with experi- ment.
If the force, acting on the wheel, were in fluplicate ratio of the water's velocity, as ufual- ly afTerted, then the effed would be as the cube thereof, when the quantity of water and time arc ^iven, which is contrary to the refult of experi- Bient."
Waring^ s Theory doubted, * 1^
From the time I firfl called on William Waring, -wr. warmg's intil I read his publication on the fubjeft (after theory douUt- lis death) I had refted partly fatisfied, with the lew theory, as I have called it, with refpeft to he velocity of the wheel, at leafl ; but finding ihat he had not determined the charge, as well as he velocity, by which we might have compared he ratio of the pov/er and the effed produced, and jtnotagree- hat he had affigned reafons fomewhat different for ing fuUy witk i he error ; and having found the motion to be rather P^^*^^"' 00 flow to agree with practice, I began to fufpeft he whole, and refumed the fearch for a true theo- y, thinking that perhaps no perfon had ever yet itinfidered every thing that affects the calcala- io.n, I therefore premifed the following
«i^ M E e H A N i c i^r. Chap, m
Art. ^
POSTULATES.
I. A given quantity of perfect elaftie 6r folici
Rtatter, impinging on a fixed obftacle, its effeC"*
tive force is as the fquares of its different velocii
ties, although its inftant force may be as its veioci'.
ties fimply, by annotation, art. 8.*
; 2.. Ah equal quantity of elaftic matter, imping-'
V ing on a fixt obftacle with a double velocity, pro*
; ■ duces a quadruple effect, art. 8 ; i.e. their effects
are as the fquares of their velocities. Confc-
quently,
3. A double quantity of faid matter, impinging with a Rouble velocity, produces an octuble effect, or their eifects are as the cubes of their velocities, art, 47 Sc 6y,
4. If the impiaging matter be non-elaftic, fuck as fluids, then the inftant force will be but half in each, cafe, but the ratio will be the fame in each cafe.
5. A double velocity, through a given aper- I ture, gives a double quantity to ftrike the obfta- cle or wheel, therefore the effects, by poftulate
^ 3, will be as the cubes of the velocity. See art.
47-
6. But a double relative velocity cannot inereafe
the quantity that is to act on the wheel, there- fore the effect can only be as the fquare of the ve-* locity, by poftulate a.
7. Although the inftant force and effects of ftriking fluids, on fixt obftacles, are only as their, fimple velocities, yet tbeir effects, on moving (
i wheels, are as the iquares of their velocities ; be-
caufe, I, a double ftriking velocity gives a double! inftant force, which bears a double load on the ! wheel; and 2, a double velocity moves the load
* Becaiife tlie diftance it will recede after the ftroke through any refiliins medium, will be as the fquares of its impingirj vejociti&s.
Chap, XT. MECHANICS. 65
a double diftance in an equal time, and a double Art. 40. load moved a double diftancc, is a quadruple ef-
fea.
— ■<^c<^<s»- —
Search for a true Theory^ commenced on a new Plan, Art. 41.
I T appears, that we have applied wrong prin- ciples in our fearch after a true theory of the max- imum velocity and load of underfhot water-wheels, or other engines moved by a conflant power, that Idoes not increafe or decreafe in quantity on the bngine, as on an overfliot water-wheel, as the ve- locity varies.
I Let us fuppofe water to ilfue from under a head pf 16 feet, on an underfhot w^ater-wheel ; then, if jthe wheel moves freely with the water, its velo- city will be 32,4 feet per fecond, but will bear no load.
L Again, fuppofe we load it. To as to reduce its otion to be equal the velocity of water fpouting 'from under 15 feet ; it appears evident that the load will then be juft equal to that i foot of the head, !the velocity of which is checked ; and this load multiplied into the velocity of the wheel, viz. 131,34X1 — 31,34 for the effedl. ! This appears to be the true principle, from which we mufi: feek the maximum velocity and load, for fuch engines as are moved by one conflant ipower ; and on this principle I have calcaiated ;the following fcalc.
K
A SCALE FOR DETERMINING fUE TRUE MAXIMUM VELOCITY AND LOAD For undershot WHEELS,
|
H o |
P5 <• a. |
Veloc feco ty he a |
3aB |
loci edb |
|
n |
» S — |
3^ 0 -1 ft "^ |
||
|
o |
15 |
3 !» Cf=! (T> |
0 S. 0 P ft 0 |
|
|
1 3' p |
2 3 |
o" ft ,« -< |
• ^. 0 ^ fO ft ro n P ''^■ |
ft cr c a- |
|
o 3 |
ft n> "> |
5=r a. ft ' 5^ P |
||
|
o |
Q 1 i-r |
. ' ' 1 |
||
|
feet. |
feet. |
feet. |
• |
|
|
i6 |
i6 |
32.4 |
0 |
0 |
|
15 |
31^34 |
I , |
31' '34 |
|
|
14 |
30^2 |
2, |
60,4 |
|
|
12 |
28, |
4' |
112 |
|
|
10 |
25.54 |
6 |
153,24 |
|
|
8 |
22,8^ |
8 |
182,4 |
|
|
7 |
21,43 |
9 |
192,87 |
|
|
6 |
19, &4 |
10 |
198,4 |
|
|
5,66 |
i9>27 |
io»33 . |
1 98^95 |
|
|
5^33 |
18,71 |
ro , 66 |
199,44 |
|
|
5 |
18, |
ir |
198 |
|
|
4 |
16 ,2 |
12 |
194,4 |
|
|
3 |
14 |
13 |
172 |
|
|
2 |
11,4 |
14 |
159,6 |
|
|
I |
8,1 |
T5 |
120, |
|
|
0 |
0 |
16 |
0 |
Maximum motion load.
^-^'
Zhap. XL MECHANICS. ^^
i In this fcale, let us fiippofe the aperture of the Art. 41. |j:ate to be a fquare foot ; t-hen the greatefl load l:hat will balance the head, will be 16 cubic feet )f water, and the different loads will be lliewn in ubic feet of water.
And then it appears, by this fcale, that v/hcn :he wheel is loaded with ic, 66 cubic feet of wa- ter, jufl 2-3 of the greateft load, its velocity will be 18.71 feet per fecond, juft ,577 parts of the velo- city of the water, and the clfeft produced is at a iTiaxi:riirni, or the greateft poffible, viz. 199,44.
To make this more plain, let us fuppofe A B, plate II, fig. 19, to be a fall of water 16 feet. Fig. 19. which we wifh to apply to produce the greateft Icffed poilible, by hoifting w^ater on its fide dppo- fite to the power applied ; firfi, on the underfliot principle, where the v/ater a6ls by its impulfe on- ly. Now let us fuppofe the water to ilrike the [wheel at I, then, if we let the wheel move free- lly without any load, it wilj move with the velo- city of the water, viz. 32,4 feet per fecond, but will produce no cffe6t, if the water iifue at C ; al- though there be 32,4 cubic feet of water expend- ed, under 16 feet perpendicular defcent. Let the v/eight of a cubic foot of water be reprefented by unity or i, for eafe in counting; then 32,4,Xi6 will iliew the pov/cr expended, per fecond, viz. 518,4 ; and the water it hoifls multiplied into its perpendicular afcent, or height hoiiled, v/ill fnew the effedi. Then, in order to obtain efi'eil from the power, v/e load the wheel; the fimpleft way of doing which, is, to caufe the tube of vv'aterC D to ad on the back of the bucket at I ; then, if CD be equal to AB, the wheel will be held in equili- brio ; this is the greateft load, and the whole of the fail AB is balanced, and no part left to give the wheel velocity ; therefore the effc<ft=:o. But if v\^e make CD=i2 feet of A B, then from 4 to A=4 feet, is left unbalanced, to give velocity to the wiicel, which is now loaded with 12 feet, and
68 MECHANICS. Chap, XL
Art. 41. exaftly balanced by 12 on the other fide, and per- ,
feftJy free to move either way by the leafl; force- applied : Therefore it is evident, that the whole' prejGTure or force of 4 feet of A B will ad: to give velocity to the wheel, and, as there is no refift- ance to oppofe the preffure of thefe 4 feet, the velocity will be the fame that water will fpout from under 4 feet head, viz. 16,2 feet per fecond, which is fliewn by the horizontal line 4=16,2,, and the perpendicular line 12=12 reprelents the*! load of the wheel ; the reftangle or produ6l of thefei two lines, form a parallelogram, the area of which; is a true reprefentation of the effieft, viz. the load; 12 multiplied into 16,2 the diftance it moves pef; fecond=i 94,4, the effecb. In like manner v/e may; try the eifeft of different loads ; the lefs the load^j the greater will be the velocity. The horizontal lines all {hew the velocity of the wheel, produced by the refpeftive heads left unbalanced, and the per- pendicular lines {h^w the load on the wheel : and we find, that when the load is io,66=|- 16, the load at equilibrio, the velocity of the wheel will be 18,71 feet per fecond ; which is -l^a parts, or a little lefs than 6 tenths, or 4- the velocity of P7wer&ndef- the Water, and the eifed: is 199,44, themaximiun •--as3t<)2 (3p greateft pofiible : and if the aperture of the
on overihot >, ^ , . -niO
V. heels. gate be i root, the quantity will be 10,71 cubic
feet per fecond. The power being 18,71 cubic feet expended per fecond, multiplied by 16 feet the perpendicular defcent, produces 299,36, and the ratio of the power and efiecl being i o to (y~^ or as 3 : 2 ; but this is fuppofing none of the force loft by non-elafticity.
This may appear plainer, if we fuppofe the v.'a- ter to defcend the tube A B, and, by its prcirnre, to raife the v/ater in the tnbe C D ; now it is evi- dent, that if we raife the v/atcr to D, v.-e liave' • no velocity, therefore efted — o. Then agsin, if we open the gate at C, we have 32,4 feet per fe- cond v^elocity, but becaufe w-e do not hcifz tiie
Chap. XL M E C H A N I C S.
water any diftance, effect :=o. Therefore the Art. 41- maximum is fomewhcre between C and D. Then fuppofe we open gates of i foot area, at different heights, the velocity will fliew the quantity of cubic feet raifed ; which multiplied by the per- pendicular height of the gate from C, or height raifed, gives the effect as before, and the maxi- mum as before. But here we muft confider, that in both thefe cafes, the water acts as a perfect definite quantity, which will produce effects equal to elaftic bodies, or equal to its gravity (fee art. /^9) which is impracticable in practice : Whereas when it acts by percuffion only, it communicates onh^ half of its original force, on account of its non-elafticity,theotherhalf beingfpentinfplalhing about (fee art. 8) ; therefore the true effect will be^^V? (^ little more than 1-3) of the moving pow- er ; becaufe nearly 1-3 is loft to obtain velocity, and half of the remaining 2-3 is loft by non-elafti- city. Thefe are the realons, why the effects pro- duced by an under&ot wheel is only half of that produced by an overfhot v/heei, the perpendicu- lar defcent and quantity of v/ater being equal. And this agrees wnth Smeaton's experiments (fee art. 68) ; but if we fuppofe the velocity of the wheel to be 1-3 that of the water=io,8, and the load to be 4-9 or 1 6, the greateft load at equilibrio ; which is=7,i 1 1, as by old theory, then the effect will be 1 0,8X4 •(;) of 16=76,79 for the effect, v/hich is quite too little, the moving pov/er be- ' iiig 32,4 cubic feet of u^ater, multiplied by 16 feet dercent=5i 8,4, the effect by this tlieory be- ing lefs than .^Vo- of the povv^er, about half equal to the effect by experiment, v»'^hich effect is fet on the outiide of the dotted circle in the iig. ( 1 9.) The dotted lines join the corner of the parallelo- grams, formed by the lines tliat reprefent the loads and velocities, in eacii experiment or lup- poiition, the areas of which truly reprefent the effect, and the dotted line Aa d x, meeting the
7° MECHANICS. Chap, XI,
Art. 4i. perpendicular line xE in the point x, forming tiie parallelogram ABCx, truly reprefents the power = 5 18 ,4.
Again, if we fuppofe the wheel to move with half the velocity of the water, \iz. 16,2 feet per lecond, and be loaded with half the greateft load = 8, according to Waring'.s theory, then the ef- fect will be 1652x8 = 129,6 for the effect, about -rVo- of the pov/er, which is ftiil lefs than by ex- periment. All this feems to confirm the maxi- mum brouglit out on the new principles.
But, if we luppofe according to the nev.^ prin- ciple, that, when the wheel moves with the ve- locity of 16-2 feet perfecond, which is the velo- f;ity of a 4 feet head, that jt vvill then bear as a load the remaining. 12 feet^, then the effect vv-ill be 16,2X12=194,4, which nearh/ agrees v/ith praftice : but as moil mills in pradtice move faf- ter, rather than flower, than what I call the true maximum, fliew^s it to be nearefc the truth, the true maximum velocity being ,577 of the veloci- ty of the water, and the mills in pradlice moving with 2-3, and generally quicker.'*
* The reafon why the wheel beS|i,-5 fo great a load at a maximum, ap- pears to be as ibllo-ws, viz.
A 16 feet head of water over a gate of i foot, iiTues 32,4 cubic feet of 'vater in a fecor.d, to llrike the wheel in the fame time, that a heavy Ijocly will take up in falling through the height of the head. Nov.^ if 16 cubic feet of elaftic matter, was to fall 16 feet, and ftrike an elallic plain, it would rife hy the force of the Itroke, to the height from whence it fell ; or, in other words, it will have force fumcient, to bear a load of 16 cu- bic feet.
Again, if 32 cubic feet of non-ekftic matter, moving with the fame ve- locity, (with which the 16 feet of elaftic matter ftruck the plain) flrike a wheel ia the lame time, although it communicate only half the force, that gave it inotion ; yet, becaufe there is a double quantity flriking in tiie farr.e time, the eifefts will be eqnai, that is, it will bear a load of 16 cubic feet, or the whole column to hold it in equilibrio.
Again, to check the whole velocity, requires the whole column, that produces the velocity, confeqiiently, to check any partoftlie velocity, will require fuch apart of the column that pro luces the part checked; and we find by art. 41, that, to check the velocity of the wheel, to be ,577 of the velocity of the v.^ater, it requires 2-3 of the whole column, and this is t'-ie maximum load. V/hen the velocity of the Avheel, is multi- plied by 2-3 of the column, it produces the etfecl:, which will be to the pov/er, as 38 to lOo ; or as 3,8 to 10, fomewhat more than 1-3, and the friffion and refiiiance of the air may reduce it to 1-3.
k
\phap: XI, MECHANICS. 71
I
I This. Tcale alfo eftablifhes a true maximum Art. 41.
charge for an overfliot wheel, when the cafe is
fuch, that the power or quantity of water on the Maximum
,1 • t \ r 1^1 1 charge of
[wheel at once, is always the lame, even although overfhot Ithe velocity vary, which would be the cafe, if wheels, fup-
I J J ^ ' DOllllP' the
the buckets were kept always full : for, fuppofe fame quantity Ithe water to be fhot into the wheel at a, and by tobe always
.^ , , , ^ . ^, mthebwek-
fits gravity to raiie the whole water again on the ets. bppolite fide ; then, as foon' as the water riles \i\ the wheel to d, it is evident that the wheel ivillftop, and eife£lr=o ; therefore v/e muft let the water out of the wheel, before it rifes to I, which will be in cifed to loofe part of the bower to obtain velocity. If the buckets both [lefcending and afcending, carry a column of wa- er I foot iquare, then the velocity of the wheel ill fliew the quantity hoifted as before, which, kiultiplied by the perpendicular afcent, lliews the |ffe£t, and the quantity expended, multiplied by :he perpendicular defcent (hews the power ; and ve find, that when the wheel is loaded with 2-3 »f the power, the eftecl will be at a maximum, . e, the whole of the water is hoifted, 2-3 of its yhale defcent, or 2-3 of the w^ater the whole of jhe defcent, therefore the ratio of the power to :he effc(ft is as 3 to 2, double to the effeft of an jinderfhot wheel : but this is, fuppoling the quan- jity in the buckets to be always the fame ; w^hcre- 'kS, in overlbot wheels, the quantity in the buck- ets is univerfally as the velocity of the wheel,, 1. e. the flower the motion of the wheel, the I^Tcater the quantity in the buckets, and the kreater the velocity the leis the quantity : but, gain, as we are obliged to let the overfhot wheel iiove with a conliderable velocity, in order to 'ibtain a Heady, rcguh>' motion to the mill, we vill find this charge to be always nearly right ; ience I deduce the following theory.
72 MECHANICS. Chap, XL
Art. 41. THEORY.
A true theory This fcale fcems to have ftiewn, deduced. j^ That when an underfliot mill moves with
^^yq or nearly ,6 of the velocity of the water, it will then bear a charge, equal to 2-3 of the load^ that w411 hold the wheel in equilibrio, and then the effeft will be at a maximum. The ratio of the power to the efFed; will be as 3 to i, nearly.
2. That, when an overfliot wheel is charged with 2-3 of the weight of the water afting upon the wheel, then the effedl will be at a maximum, /. e. the greateft effect, that can be produced by faid'power in a given time, and the ratio of the power to the efFeft will be as 3 to 2, nearly.
3. That 1-3 of the power is neceiTarily loll to obtain velocity, or to overcome the vifiaertia of the matter, and this will hold true with all ma- chinery that requires velocity as well as povv^er. This I believe to be the true theory of water- mills, for the following reafons, viz.
1. The theory is deduced from original reafon- ing, without depending much on calculation.
2. It agrees better than any other theory, with the ingenious Smeaton's experiments.
3. It agrees beft with real practice, from the bcft of my information.
Yet I do not wifli any perfon to receive it im- plicitly, without firft informing himfelf, whether it be well founded, and agrees with practice: for this reafon I have quoted faid Smeaton's ex- j periments at full length, in this work, that th^|| reader may compare them M^ith the theory. ;«||
Theorem for finding the Ma.xhium Charge for uncler- ^ JJiot JVheels.
As the fquare of the velocity of the water or Vv^heel empty, is to the height of the head,
Chap, XL MECHANICS. 73
or prefTure, which produced that velocity, fo is Art. 42. the fquare of the velocity of the wheel, to the head, prefTure, or force, which will produce that velocity ; and this prcffure, deduded from the v/hole prefTure or force, v/ill leave the load moved by the wheel, on its periphery or verge, which load, multiplied by the velocity of the wheel, fhews the efrect.
P Pc O B L E M.
Let ¥=-32,4, the velocity of the water or wheel, P=i6, the prefTure, force or load, at equili-
brio, v=:the velocity of the wheel, fuppofed to be
16,2 feet per fecond, p=the prefTure, force or head, to produce faid
velocity, Izrthe load on the wheel. Then, to find 1, the load, we muft firft find p ; Then, by
Theorem VV:P::vv:p, And P~p=l
VVprrvvP '
vvP
P = = 4
1:=P — prr I 2, the load.
"VVhich, in words at length, is. The fquare of the velocity of the wheel, multiplied by the whole force, prefTure, or head of the water, and divided by the fquare of the velocity of the wa- ter, quotes the prefTure, force or head of water, that is left unbalanced by the load, to produce the velocity of the wheel, which prefTure, force or head, fubtrafted from tiie whole prefTure, force or head, leaves the load that is on the wheel.
MECHANIC S. Chap, XL
Theorejn for finding the Velocity of the TVheeU ivhen we have the Velocity of the IVaier^ Load at Equi- librio, and Load on the PP' heel given.
As the fquare root of the whole preflure, force or load at equilibrioj is to the velocity of the water, fo is the fquare root of the difference, be- tween the load on the wheel, and the load at equilibrio, to the velocity of the wheel.
P R O B L E M.
Let V=ve]ocity of the water=32,4,
PrrprefTurc, force, head, or load at equili-'
brio = ]6, l = the load on the wheel, fuppofe 12, v = veiocity of the wheel,
Thenbythe__ '_
Theorem ^P:V::^P— ]:v And /^PXv-y/^F— 1
A^p ' (of the wheel.
That is, in words at length, the velocity of the w^ater 32,4, multiplied by the fquare root of the difference, between the load on the wheel, 12, and the load at equilibrio 16=2=64,8, divided by the fquare root of the load at equilibrio, quotes 16,2, the velocity of the wheel.
Now, if we feek for the maximum', by either of thefe theorems, it v/ill be found as ii) the fcale, fig. 19.
Perhaps here may now appear the true caufe of the error of the old theory, art. 35, by fup- poiing the load on the wheel, to be as the fquare of the relative velocity, of the water and wheel.
And of the error of what I have called the new theory, by fuppofing the load to be in the fimple
I Chap. XI. MECHANICS. j^
ratio of the relative or ftriking velocity of the Art. 42. water, art. 38 ; whereas it is to be found by nei- ther of theie proportions.
Neither the old nor new theories agree with praiiHce ; tiierefore we may fufped: they are founded on error.
But if, what I call the true theory, flioiild con- tinue to agree with pratlice, the praftitloner need not care on what it is founded.
Of tJie Maxhmun velocity for Over/hot TVheels^ or Art, 43. thofe that are moued by the vjtight of the Wa- ter,
BEFORE I difmifs the fubjedl of maximums, I J think it befl to conlider, whether this dodlrine will apply to the motion of the overfhct wheels. It feems to be the general opinion of thofe, who I conhder the matter, that it will not ; but, that I the flower the wheel moves, provided it be ca- pacious enough to hold all the water, without lofing any until it be delivered at 'the bottom of ;the wheel, the greater will be the effect, v/hich I appears to be the cafe in theory (fee art. 36) ; but hovv^ far this theory will hold good in praclicc, is to be confidered. Having met with the ingeni- ous James Smeaton's experiments, where he Ihews, that, wiien the circumference of his lit- jtle wheel, of 24 inches diameter, (head 6 inches) moved with about 3,1 feet per fecond (although the greatcft effeii; was diminiflied about -^ of the I whole) he obtained the bell eifeft, with a fteady, regular motion. Hence he concludes about 3 feet to te the beft velocity for the circumference of oveifh^t mills. See art. 68. I undertook to compare this theory of his, with the beft mills
7^ MECHANICS. Chap, XI,
Art. 43- in practice, and, finding that thofe of about 17
feet diameter, generally moved about 9 feet per oJInioTofthe ^e^o'id. being treble the velocity alfigned by Smea- proper veioci- ton, I began to doubt the theory, which led me cumferenc?"^" to inquire into the principle, that moves an over- of overfliot fhot wheel, and this I found to be a body defcend- rot'atr'ee''' ^"^.^7 its gravity, and fubjeft to all the laws of withpradice. falling bodies, (art. 9) or of bodies defccnding in- Theprincipie cliued plains, and curved furfaces (art. 10, 11,) of the power the uiotion being equably accelerated in the whole
that moves r- • i r • i • i • i c
overfhot or its dcicent, its velocity being as the Iquare root
wheels is of the diflance defcended through, and that the
ingbody. diameter of the v/licel was the diflance the water
Their veioei- <^efcended through. From thence I concluded,
ties vary, and that the vclocity of the circumfcrence of the
fquar'Troot^of ovcrfiiot whecls, was, as the fquare root of their
their diame- diameters, and of the diflance the water has to
^"' dcfcend, if it be a breafl or pitch-back wheel :
then, taking Smeaton's experiments, with his
w^heel of 2 feet diameter, for a foundation, I
fay, As the fquare root of the diameter of Smea-
ton's wheel, is to its maximum velocity, fo is the
fquare root of the diameter of any other wheel,
to its maximum velocity. Upon thefe principles
This rule I have calculated the following table ; and, hav-
foundtoagree ing Compared it \\dth at leafl 50 mills in pracftice,
withpraftice. ^^^^^^ jt to agree fo nearly with ail the befl con-
flru6ted ones, that I have reafon to believe it is
founded on true principles.
Their veioci- If au overfhot v/hcel moves freely v/ithout re-
tieswiUbea {jjlance, it will acquire a mean velocity, between
mean be- •'■' - 7 ^ ^ j •>
tween the that of the v/atcr coming on the wheel, and t^e leaitand p-reatcfl veiocitv it v/ould acquire, by falling free- body falling ij through its wliolc defceut : therefore this d^^?^"e^t.!! ^"^^ mean velocity will be greater, than the velocity of the water coming on the wheel ; confequent- ly the backs of the buckets v/ili overtake the wa- ter, and drive a great part of it out of the wheel. But, the velocity of the water being accelerated hy its gravity, overtakes the wheel, perhaps half
:hap. XL MECHANICS. yy
Way down, and preffes on the buckets, until it ^rt. 43. eaves the wlieel : therefore the water prefTes ,„
I 1 T 1 ^ • 1 I V • Water prefTes
iiarder upon the buckets in the lower, than m harder on the the uDper quarter of the wheel. Hence appears ^^^^^^t'*^"
, ■'r»i r 1 ini- upper quarter
the realon why lome wheels cait their water, of the wheie. ivhich is always the cafe, when the head is not aliicient to give it velocity enough to enter the puckets. But this depends alfo much on the po- rtion of the buckets, and diredtion of the .fliute Into them. It, however, appears evident that ^he head of water above the wheel, Ihould pe nicely adjufted, to fuit the velocity of the kheci. Here v/e may confider, that the head Ibove the wheel adts by perculTion, or on the fame [principles with the underlhot wheel, and, as we lave fhewn (art. 41) that the underfliot wheel hould move with nearly 2-3 of the velocity of the Abater, it appears, that we Ihould allow a head pver the wheel, that will give fuch velocity to ^he water, as will be to that of the wheel as 3 f:o2. Thus the whole defcent of the water of a '^'
mill-feat Ihould be nicely divided, between head The whole md fail, to fuit each other, in order to obtain ^efcentmuft :he belt eiiecb, and a fleady-moving mill. Firft videdbe- iind the velocity that the wheel will move with, andl^//^^'^ 3y the weight of the water, for any diameter y^ou may fuppofe you will take for the wheel, md divide faid velocity into two parts ; ihen ;ry if your head is fuch, as will caufe the water :o come on with a velocity of 3 fuch parts, mak- ng due allowances for the fridion of the water, iccording to the aperture. See art. 55. Then, f the buckets and direction of the fhute be right, :he wheel will receive the water vv^ell, and move ;o the bed advantage, keeping a fceady, regular notion when at work, loaded or charged witli a 'efiflance equal to 2-3 of its power, (art. 41. 42.)
A TABLE OF VELOCITIES of the CIRCUMFERENCE Of undershot WHEELS,
Suitable to their Diameters, or rather to the Fall, after the Water ftrikes ■ the Wheel ; ,and of the head of Water above the Wheel, fuitable to faid "Velocities, alfo of the Number of Revolutions the Wheel will Per- form in a Minute, when rightly charged.
*
|
o |
<! |
|
M- O |
|
|
9 |
ss o |
|
M^ O |
|
|
S>5 |
|
|
<D |
|
|
^ o |
|
|
O |
P H^ |
|
CL „. |
|
|
1-1- o |
|
|
n |
n |
|
5" |
^n' |
|
n H |
|
|
»> |
o m |
|
ft |
0 a |
|
2 |
3.1 |
|
3 |
3.78 |
|
4 |
4,38 |
|
'J |
4,88 |
|
6 |
5.3^ |
|
7 |
5, B |
|
8 |
6,19 |
|
9 |
<^>57 |
|
lO |
6,92 |
|
II |
7; 24 |
|
12 |
7>57 |
|
13 |
7,86 |
|
14 |
8,19 |
|
M |
8,47 |
|
16 |
8,76 |
|
17 |
9. |
|
18 |
9;28 |
|
19 |
9> 5 |
|
2Q |
9.78 |
|
21 |
10, |
|
22 |
10,28 |
|
23 |
10, 5 |
|
24 |
ID, 7 |
|
2-^ |
10,95 |
|
26 |
II, 16 |
|
27 |
11,36 |
|
28 |
11,54 |
|
29 |
11,78 |
|
30 |
11,99 |
|
5! |
^ |
. H |
^ |
|
r^ r^ 9- p- 0 g ft 0- |
3 |
•13 ? •^ 0 |
|
|
3: ^.0 |
8 tf^S" |
f» |
2.S |
|
0 "> ^ |
ft n: >— ' |
p |
C 0 |
|
-"^1 |
C 3 (t |
0 |
ft e |
|
S'Sp |
See fn w, 0, |
^ |
0' |
|
*+) S.' °^ fv "■ 0 ft ^ < |
0 rt- ,-► |
p ft |
0 |
|
rt p fB |
*^ rD |
^ |
►^ |
|
■ ^ 2 |
r-f |
||
|
J3 t_^ ^ |
'o 5 |
rr\ |
|
|
CL ft |
ft 5 |
||
|
•n 0 ^ |
?^?1 |
^ |
|
|
So s |
f^ 2 |
||
|
• i^ >— |
|||
|
1,41 |
,1 |
1,51 |
14,3 |
|
1,64 |
,1 |
1,74 |
'3' |
|
1,84 |
;I |
1,94 |
12,6 |
|
2, |
.2 |
2,2 |
12 |
|
2,17 |
.3 |
2,47 |
11,54 |
|
2,34 |
,4 |
2,74 |
11,17 |
|
2,49 |
,5 |
2,99 |
10,78 |
|
2,68 |
,6 |
3,28 |
10,4 • |
|
2, 8 |
,7 |
3.5 |
10,1 |
|
3. |
,8 |
3.8 |
9,8 |
|
3.13 |
;9 |
4.03 |
9,54 |
|
3.34 |
ij |
4,34 |
9,3 |
|
3,49 |
1,05 |
4'54 |
9>i |
|
3,76 |
I, I |
4,86 |
8,9 |
|
3^84 |
i'i5 |
4,99 |
8,7 |
|
4,°7 |
1,2 |
s^-^i |
8,5 |
|
4, 2 |
1,25 |
5,45 |
8,3 |
|
4,27 |
1-3 |
567 |
8,19 |
|
4,42 |
' .^'35 |
5^77 |
8,03 |
|
4,56 |
3,4 * |
5>9^ |
7,93 |
|
4, 7 |
1,45 |
6,15 |
7,75 |
|
4, 9 |
1,5 |
6,4 |
7,63 |
=^-
'•r
\hap. XL MECHANICS. 7p
■THIS dodlrine of maximums is very interefi- Art. 44. g, and is to be met with in many occurrences
irOUgh life. AppHcatioa
1 . It has been fhewn, that there is a maximum of maxl*-^^""* ad and velocity for all engines, to fuit the pow- mums.
• and velocity of the moving power.
2. There is alfo a maximum fize, velocity and ed for mill-flones, to fuit the power; and velo- ty for rolling-fcreens, and bolting-reels, by hich the greateft work can be done in the beft anner, in a given time.
3. A maximum degree of perfeAion and clofe- bfs, with which grain is to be manufaftured in- ) flour, fo as to yield the greateft profit by the ill in a day or week, and this maximum is con- nually changing with the prices in the market, \ that what would be the greateft profit at one me, will fink money at another. See art. 113.
4. A maximum weight for mallets, axesj bdges, Sec, according to the ftrength of thofe lat ufe them.
A true attention to the principles of maxi- mums, will prevent us from running into many rrors.
<<?^ «<S=^ ^<p^ ^!5>^ t<5>> t<:?^ (.j;?^ ti^ «.^>5 1{5>> t<s?^ Vi?^ t<;>i v:^
CHAPTER Xil.
H 2^ D R A U L I C S.
—..^^^ —
UNDER the head of Hydraulics we fhall on^| ly confider fuch parts of this fcicnce, as im^l mediately relate to our purpofCj viz. fuch as niay^ lead to the better undcrflanding of the principles and powers of water, afting on mill-wheels, and conveying water to them.
Art. 45. Of Spouting Fluids,
SPOUTING Fluids obferve the followingi
laws :
1. Their velocities and powers, under equal preffures, or equal perpendicular heights, andi equal apertures, are equal in all cafes.*
2. Their velocities under different preffures or perpendicular heights, are as the fquare roots of j thole prelTures or heights ; and their perpendicu-
* It makes no difference whether the water ftands perpendicular above the ' aperture, or incliningly (fee plate III, fig. 22) providing the perpendicular j height be the fame ; or whether the quantity be great or fmall, providing it ' be fufficient to keep up the fluid to the fame height. '
Chap, XII. H Y D K A U L I C S. 8l,
lar heights or prefTures, are as the fquares of their Art. 45-
velocities.*
I 3. Their quantities expended through equal ! apertures, in equal times, under equal prelTures, »
[are as their velocities (imply. t I 4. Their preilures or heights being the fame, [their effects are as their quantities expended.^ > ^. Their quantities expended being the fame,
their effedls are as their preilure, or height of
tjieir head direclly.jj
6. Their inftant forces v/ith equal apertures, are as the fquares of their velocities, or as the height of their heads directly.
7. Their efrefts are as their quantities, multi- plied into the fquares of their velocities.^
* This law is fimilarto the 4th law of fallins bodies, their velocities be- Foundation of ing as the fquare root of their fpaces pafTed through ; and by experiment it the rule for is known, that water will fpout from under a 4 feet head 16, 2 feet per fe- finding the ve- cond, and from under a 16 feet head, 32,4 feet per fecond, which is only locity of wa- double to that of a 4 feet head, although there be a quadruple prciTure. ter iinder any Therefore by this law we can find the velocity of water fpouting from un- head, der any given head; for as the fquare root of 4 equal 2 is to 16,2 its veloci- ty, fo is the fqaare root of 16 equal 4, to 32,4 its velocity. And again, as 16,2 fquared, is to 4 its head, fo is 32,4 fquared, to 16 its head ; by v/hich ratio we can find the head that will produce any velocity.
t It is evident that a double velocity will vent a double quantity.
\ If the prefture be equal, the velocity muft be equal ; and it is evident, that double quantity with equal velocity will produce a doable eiTefl.
II That is, if we fuppofe 16 cubic feet of water to iffue from under a 4 feet head in a fecond, and an equal quantity to ilfue in the fame time from under 16 feet head, then their ehefts will be as 4 to 16. But we muft note, that the aperture in the laft cafe muft be only half of that in the iirft, as the velocity v/ill be double.
§ This is evident from this confideration, viz. that a quadruple impulfe is required to produce a double velocity, by law 2iid, where the velocities are as the fquare roots of their heads ; therefore tteir cfierts muft be as the fquares of their velocities.
DEMONSTRATION.
LET A F, (plate III, fit^. 26) reprefent ahead of water 16 feet high, and ,
fuppofe it divided into 4 dih'eren': hcado cf 4 feet each, as B C D E ; then fup- J ' ' pyie v/e draw a gate of r foot fquare at each head fucceflively, always f.nking "' ' i the v/atcr in the head, fo that it will be but 4 fiet above tl.e centre of the gtice in each cafe . '■
\ Now it is known that the velocity imder a 4 feet licad, is 16,2 feet per fecond ; fay 16 feet to avoid fracftion~, which will iffue 16 cubic feet of wa- I ter per fecond, and for fake of round numbers, let unity or i reprefent the ' quantity of a cubic foot of water i t'nen, by the ythl.iw the eiledl v/ill be as } the quantity multiplied by the iljuae of the velocity; that is, 16 multipli- ed by 16 is equal to 256, whicli multiplied by 16, the quantity, is equal to 4096, thecilecl of each 4 ft-et head ; and 4096 mviltiplied by 4 is equal to i i63l.],for tiie fuin of efiefts, of all the 4 feet heads.
i U - ■
52
Art. 46.
Vig.. 2&.
Art. 47-
Theory tiiat is eilabl^.fhed coinpared witd the efta- blifhed laws. And found to aff.ee.
HYDRAULICS. Chap. XIIl
8. Therefore their eiFe£ls or powers with equal
Then as the velocity under a 16 feet head is 32,4 feet, /ay 32 to avoid fraftions : the gate muftbe drawn to only half the lize, to vend the 16 cubic feet of water per fecond as before, (becaufe the velocity is double) then, to find the eifeft, 32 multiplied by 32, is equal to 1024 ; which muldpiied by 16 the quantity, gives the elFeft, 16384, equal the fum of all the 4 feet head ; which agrees with praftice and experience, the beft teachers. But if their effects were as their velocities fimply, then the efFeft of each 4 feet head would be, 16 multiplied by 16, equal to 256 ; which, multiplied by 4, is equal to io-24,for the fum o^the eJefts of all the 4 feet heads : and 16 mul-
tip.Ucd by 32 equal to 513, for the efieft of the 16 feet head, which is only half of the eifeftof the lame head when divided into 4 parts; which is con- trary to both experiment and reafon.
Again, let us luppo'e the body A of quantity 16, to be perfeftly elaftic, to fall 16 feet and ftrike F, aperfeftelaftic plain, it will (by laws of falling bo- dies^ ftrike with a velocity of 32 feet per fecond, and rife 16 feet to A again. _ But if it fall only to B, 4 ft;et, it will ftrike with 16 feet per fecond, and rife 4 feet to A again. Here the effeift of the 16 feet fall is 4 times the ehett of the 4 feet fall, becaufe the body rifes 4 times the height-
But it v/e count the effeftive momentum of their ftrokes to be as their ve- locities fimply, then '6 multiplied by 32 is equal to 512, the momentum of the 16 feet fall ; and r6 multiplied by 16 is equal to 256 ; wnich, multiplied by 4, is equal to 1024, for the fum of the tnomentums of the ftrokes of j6 feet divided into 4 equal falls, which is abfurd. But if we count their mo- mentums to be as the fqnares of their velocities, the effects will be equal.
Again, it is evident that whatever impulfe or force is required to give a body a velocity, the fame force or refiftance will be required to flop it; therefore, if the impulfe be as the fquare of the velocity produced, the force' or refiftance will be 25 the fquares of the velocity alfo. But the impulle is as the fquares of the velocity produced, which is evident from this confde- ration, Suppofe we place a light body at the gate B, of 4 feet head, and pref- fed with 4 feet of water ; when the gate is draWn it will fly off with a veio- cjty of r6 feet per fecond; andif we increaie the head to 16 feet, it will fly oif with 32 feet per fecond. Then, as the fquares of 16 equal to 256 is to the fquare of 32 equal to 1024, fo is 4 to 16. Q. E. D.
To compare this 7th law with the theory of uuderftiot mills, eftablifiied art. 42, where it is fliewn that the power is to the etredf as 3 to i ; then, by the 7th law, the quantity fliewn by the fcale, plate II, to be 32,4 multi- plied by 1049,-6 the fquare of the velocity, which is equal to 3401,2124, the eflea of the 16 feet head : tlien, for the eifecl of a 4 fieet head, with equal aperture quantity, by fcale, 16,2 multiplied by 262,44, the velocity fquar- ec|, is equal to 425, 1 528, the effeft of a 4 feet head ; here the ratio of the ehecVs are as 8 to i .
Then, by the theory, which Ihews that an underfliot wheel will hoift 1-3 of the water that turns it, to the whole height from which it defcended, the I 3 of 32,4 the quantity, be uig equal to 1^,8 multiplied by 16, perpendicu- lar aiceut ; which is equal to 172,8, eiiect of a 16 feet head; and 1-3 of ] 6,2 quantity, which U equal to 5,4 multiplied by 4, perpendicular afcent, is equal to 21 ,6 elfeft of 4 feet head, by the theory ; and here again the ra- tio of the efieds are as 8 to i ; and,
a- 3401,2 1 2J, the effect of j 6 feet head, ^ i .,^1. 1
is to 435, 1 52?, the etfe.it of 4 feet head, S ^^ '^^ ^^''''
fo is 172,8 taee.feaof i6teethead, ? , ^, ^,.
t.) 2 . ,6 the ehec^ of 4 feet head, 5 ^^ ^^^ ^^^"^^
The quaiitities being equal, their ellec^s arc as the height of theirheads di- rai^iy, as by ith \■^.\v, and a; the iquare.-. of their velocities as bv 7th law. Hence it appear^, that the t-trory agrees with the eftabliflied law's, which I tci.;e CO be aconfi.-jnation that it is v, i;ll founded.
Chap, XI L
HYDRAULICS.
83
appertures, are as the cubes of their veJociiies.*
9. Their velocity under any liead is equal to the velocity that a heavy body would acquire, in falling from the fame height. t
10. U'heir velocity is iucli under any head or height, as will pafs over a dillance equal to twice the heigiit of the head, in a horizontal direction, in the time that a heavy body falls the dillance of the height of the head.
1 1. Their a6tion and reaction are equal. |
12. They being non-elaflic, communicate only half their real force by impuife, in flriking obfla-
' The efFc'ils of ftriking fluids with equal apertures are as the cubes of jtheir velocities, for the following reafons, viz ift, II" an equal quantity ftrike with double velocity, the elTert is quadruple on that account by the 7th law ; and a double velocity expends a double quantity by 3d law ; there- fore, the eflecTiis amounted to the cube of the velocity. — The theory for un- iderfhot wheels agrees with this law alio.
A SCALE foHnded on the 3rd, 6th and 7th laws, fhewing the efiec'ts of ftriking Fluids, with different Velocities.
w
Art. 47.
|
> <D rt- |
1 ■5] cr |
< % o" 0 rt |
n c i |
ft Q. |
rT CI. |
0« 2. c 0 rt ftl I 0" |
n e |
rr |
0 r^ fi) < |
|
|
I jX |
I |
= |
I |
X i « = |
I |
as |
■ |
|||
|
r |
X |
2 |
=^ |
2 |
X |
4| = |
8 |
as |
8 |
|
|
I |
X |
3 |
=: |
3 |
X |
9 |
= |
27 |
as as |
27 |
|
I |
X |
4 |
4 |
X |
16 |
= |
64 |
64 |
Vi'
'1^:-.
f The falling body is afted on by the whole force of its own gravity, in i:he M'hole of its defcent through any Ipacc ; and the whole fum of this atflion ,:hat is acquired as it arrives at the loweil point of its fall, is equal to the jreflure of the whole head or perpendicular height above the illue ; there- fore their velocities are equal.
' I That is, they re-aftback againft the penftock v/ith the fame force that t iffiies againft the obftacle it ftrikes : this is the principle by which Bar- Iter's mill, and all thufc that are improvements thereon, moves.
84 HYDRAULICS. Chap, XII.
Art. AT. cles ; but by their gravity produce efFe<fts, equal to elaftic or Iblid bodies.*
Application of the La-a^s of Motion to Underfhot IV heels.
I
To give a fhort and compreiienfive detail of the ideas, I have colIe£ted from the different authors, and from the refult of my ov/n reafoning on the laws of motion, and offpouting fluids, as they ap- ply to move underfhot mills, 1 conflrn£ted fig. 44. plate V.
Let us fuppofe two large wheels, one of 1 2 feet, and the other of 24 feet radius, then the cir- cumference of the largeft, will be double that of the fmalleft: and let A 16, and C 16, be two pen- fcocks of water, of 16 feet head, each.
1 . Then, if we open a gate of i fquare foot at 4, Lav/scfmoti- to iffuc from the penflock A 16, and impinge on L'^gfltidfap." the fmall wheel at I, the water being preffed by plied in prac 4. feet head, will move 16 feet per fecond. (we ^"^'^' omit fracScions) The inllant prefTiire or force
on that gate, being 4 cubic feet of v/ater, it will require a refiftance of 4 cubic feet of water, from the head C 16 to ftop it, and hold it in equilibrio. (but we fiippofe the water cannot efcape uniefs the wheel moves, fo that no force be lofl by non-' elafticity) Here equal quantities of matter, with', equal velocities, have their momentums equfil.
2. Again, fuppofe we open a gate of i fqnare foot at A 16 under 3 6 feet head, it will flrike the lavge wheel at k, with velocity 32, its inftant- force or preiTure being 16 cubic feet of watei", it will require 16 cubic feet refiftance, from the head C 16, to ftop or balance it. in this cafe the
* when non-eJaitic bodie-, ftrike an obftacle, one half of tlieir force is fpentin a lateral diretftion, in changing their figure, or in fplaftiing about. See art. 8.
For want of due conrideration or Icnov/Iedge of this principle, many have been the errors committed by app'iv'.ng water to act by irnpulfc, when it Vi'ould have produced a double cilecl by its gravity.
Chap. XII. HYDRAULICS.
preiTure or inflant force is quadruple, and fo is ^j-t. 47. the refinance, but the velocity only double, to the firil cafe. In thefe two cafes the forces and reliftances being equal quantities, with equal ve- locities, their momentums are equal.
3. Again, fappofc the head C 16 to be raifed to E, 16 feet above 4, and a gate drawn ^ of a fquare foot, then the inftant prefTure on the float I of the iniall wheel, will be 4 cubic feet, preffing on \^ of a fquare foot, and will exadlly balance 4 cu- bic feet, pre/fing on i fquare foot, from the head A 16 ; and the wlieel will be in equilibrio, (fup- poling the v/ater cannot efcape until the wheel moves as before) although the one has power of velocity 32, and the otl:ier only 16 feet per fe- cond. Their loads at equilibrio are equal, confc- quently their loads at a maximum velocity and charge, w^iil be equal, but their velocities differ- { ent.
I Then, to try their effecls, fuppofe, iirft, the {wheel to move by the 4 feet head, its maximum ivelocity to be half the velocity of the water, which is 16, and its maximum load to be half its greatefl load, which is 4 by VVaring's theory ; then the velocity 16 | 2xby the load 4 | 2 = 16, the eftecl of the 4 feet head, with 16 cubic feet expended ; becaufc the velocity of the water is 16, and the gate i foot.
Again, fuppoic^ it to move by the 16 feet head and gate of i- of a foot ; then the velocity 32 j 2 xby the load4 | 2—32, the effed:, with but 8 cu- bic feet expended, becaufe the velocity of the water is 32, and the gate but -i of a foot.
In this cafe the inflant forces are equsl, each being 4 ; bat the one moving a body only -'- as heavy as the other, moves with velocity 32, and produces eifecl 32, while the other, moving with velocity 16, produces eifect 16. A double velo- city, with equal inflant prefTure, produces a dou- ble eiTecl, which fecms to be according to the Newtonian theory. And in this fenfe the
P
86 HYDRAULICS. C/iap, XIL
Art. 47. momentunis of bodies in motion, are as their
quantities, multiplied into their limple velocities, and this I call the inllant momentums.
But when we coniider, that in the above cafe, it was the quantity of matter put in motion, or water expended, that produced the elfe^t, we find that the quantity i6, with velocity 16, pro- duced effedt i6 ; while qu. 8, with velocity 32, produced efFed: 32. Here the eifedts are as their quantities, multiplied into the fquares of their yeiocities ; and this I call the effedlive momen- tums.
Again, if the quantity expended under each head, had been equal, their efFe£ls would have been 16 and 64, which is as the fquares of their ■ velocities, i6aHd32.
4. Again, fuppole both wheels to be on one fhaft, and let a gate of f of a fquare foot be drawn at i6C,to ftrike the wheel at k, the head being 16 feet, the inftant prelfure on the gate will be 2 cu- bic feet of water, which is half of the 4 feet head with I foot gate, from A 4 ftriking at I ; but the 16 feet head, with inllant prcffure 2, acting on the great wheel, will balance 4 feet on the fmall one, becaufe the lever is of double length, and the wheels will be in equilibrio. Then, by Wa- ring's theory, the greatell load of the 16 feet head being 2, its load at a maximum will be i, and the velocity of the water being 32, the max- imum velocity of the wheel will be 16. Now the velocity 16X1 = 16, the elFecl of the ] 6 feet head, and gate off of a foot. The greatell load of the 4 feet head being 4, its maximum load 2, the velocity of the w^ater i6,and the velocity of the wheel 8, now 8X2=16, the effed. Here the effefts are equal : and here again the effecrs are as the inflant preffures, multiplied into their limple velocities ; and the reliftances that would inftant- ly ftop them, muft be equal thereto, in the fame ratio.
Chap, XtL HYDRAULICS. ^y
But when wc conllder, that in this cafe, the 4 Art. 47. feet head expended 16 cubic feet of water, with velocity 16, and produced effect 16; while the 16 feet head expended only 4 cubic feet of water, with velocity 32, and produced efFe(il 16, we find, that the eifeds are as their quantities, multiplied into thcfquares of their velocities.
x\nd when we conllder, that the gate of 4 of a fquare foot, with velocity 32, produced effeds equal to the gate of i fquare foot, with velocity 16, it is evident, that if we make the gates equal, the eifecls will be as 8 to i ; that is, the efFefts of fpouting fluids, with equal apertures, are as the cubes of their velocities; becaufe, their inftant forces are as the fquares of their velocities by ^th law.
The Hydroflatic Paradox, Art. 48.
THE preffure of fluids is as their perpendicu- lar heights, without any regard to their quanti- ty ; and their preffure upwards is equal to their jpreirbre downwards. In fhort, their preffure is every way equal, at any equal diflance from their jfurface.'* i
I * To explain which, let A B C D, plate III, fig;. 22, be a veffel of water of HvdrwftatiF
li cubical form, with a fmall tube as H, fixed therein ; let a hole of the fame naradox
jlize of the tube be made at e, and covered with a piece of pliant leather, ujained
jiailed thereon, fo as to hold the water. Then fill the veflel with water by
|;he tube H, and it will prefs upwards againft the leather, and raife it in a
ponvex form, requiring juft as much weight to prefs it down, as will be
pqual to the weight of water in the tube H. Or if we fet a glafs tube over
|:he hole at o, and pour water therein, we will find that the water in the
|:ube o, mult be of the fame height of that in tube H, before the leather will
|\ibfide, even if the tube O be much larger than H ; which fliews, that the
preifure upwards is equal to the prefl'ure downwards ; becaufe, the water
prelled up againft the leather with the wfeole weight of the water in the
;;ube H. Again, If we fill the velfel by the tube I, it will rife to the fame
jieight in H that it is in I ; tlie preffure being the fame in every part of the
i-eif.-l as if it had been fiUad by H; and th2 preffure on the bottom of the
I'effel will be the fame, whether tiie tube H be of the whole fize of the vef-
jel, or only one quarter of an inch diameter. For fuppofe H to be I -4 of an
Inch diameter, and the whole top of the veffel of leather as at o, and we
pour water down H, it will prefs the leather up with fuch force, that it will
jequire a column of water of the v/hole fize of the vellel, and height of H,
to caufe the leather to fubfide. ^. E. D.
ex-
88
Art. 48..
HYDRAULICS.
C/iap, XI J,
In a vciTel of a cubic form, whofe fides and bot- tom are ieqiial, the preffure on each fide is jufl half the prefTiire on the bottom ; therefore the prefTure on the bottom and fides, is equal to 3 times its prefTure on the bottom.*
And in this fenfe fluids may be laid to act with three times the force of foiids. Solids act by gravity only, but fluids by gravity and prefTure jointly. "Solids act v/ith a force proportional to their quantity of matter ; but fluids act with a prefTure proportional to their altitude only.
— — •«S>'^<^>' — -
The Vy-eight of a cubic foot of water is found Art. 50. by experience, to be 1000 ounces avoirdupoife, or 62,51b. On thefe principles is founded the folio V7inp- theorem.
T
E O II E M
Theorem for The area of the bafe or bottom, or any part of preirufeofthe ^ vciTel, of whatever form, multiplied by the water on the greatefl perpendicular height of any part of the gate, &c. Hoid^ above the centre of the bafe or bottom,
Art. 49.
Water may be conveyed to the wheel of a mill anyway moit conveni- ent.
And aj^ain, Suppofe we make two holes in the veflel, one clofe to the bot- tom, and the other in the bottom, both of one fize, the water will ifiue with equal velocity out ©f each ; which may be proved by holding equal veiTels under each, which will be filled in equal time ; which fhews, that the pref- fuve on the fides and bottom are equal under equal diftances from the fur- face. And this velocity will be the fame whether the tube be filled by pipe I, or H, or by a tube the v/hole fize of the veflel, provided the perpendicu- lar height be equal in ail cafes.
From what has been faid, it appears, that it makes no difference in the power of water on mill-wheels, whether it be brought on in an open forebay and perpendicular penftock, or down an inclining one, as I C ; or imder ground in a clofe trunk, in any form