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CANADIAN COP'hiiiHt/^DniQfi.M-r{\ (Authorixtd by the Council of Public Instructicn fofUhifirio. • » ^** f-Di ^r at:cn» c %^o 1] i TORONTO: ADAM MILLER & CO., 1870. m \^5 ivately !l|eachers ^ the in seived i theNc I am tiscellan <|^ziclusioi * Ifith whic tftry Matl «lid Mast Cambri PREFACE Phe Elements oi Hydrostatics seem capable of being jiresented in a simpler form than that in which they ipear in all the works on the subject with which I &iu ^(cquainted. I have therefore attempted to give a simple Ikplanation of the Mathematical Theory of Hydrostatic? mid the practical application of it. Prior to the publication of this work some copies were pi-ivately circulated with a view to obtain opinions from leachers of experience as to the suflBciency and accuracy of the information contained in it. A few suggestions received in consequence of this arrangement will be found in the Notes at the end of the volume. I am indebted to several friends for the collection of Miscellaneous Examples given in Chapter viii. In conclusion I have to express my thanks for the favour with which my attempts^ to simplify the course of Elemen- tary Mathematics have been received by College Tutors and Masters in Schools. -• J. HAMBLIN SMITH, Cambridge, 1870. li I ( ( I On Fh On the On Sph< On the ACTI On the ] On the i On the 1 , MlBCELLAM Answe CONTENTS CHAPTER I. Om Fluid Pbessdbe . PIQB • •••••• CHAPTER II. On the Pressure of a Fluid acted on by Graviti 11 CHAPTER III. On Spboific Gravity 90 CHAPTER IV. On the Conditions of Equilibrium of Bodies undkb th« Action of Fluids 30 CHAPTER V. On thb Properties of Air . . . • • .50 CHAPTER VI. M On the Application of Am . . • • • • 66 CHAPTER VII. On the Thermometer • • • • • • 77 CHAPTER VIII. , liUCXLLANEOUS EXAMPLES 68 Ambweba • •••••••• 99 I i 1. science bodies which , tended propert: 2. . I imprcss< Ithcmseh 3. 1 [which th force, ho )article8 ttion of 4. A ioniposiu friction, v Ter anotl For ej fiction b( more c Jiter the ions, and sheet S.H. HYDROSTATICS. CHAPTER I. On Fluid Pressure. 1. Hydrostatics wjis originally, as the name imports, the science which treated of the Equilibrium of Fluids, or of bodies in equilibrium under the action of forces some of which are produced by the action of fluids. It is now ex- tended so as to include many other theorems relating to the properties of fluids. 2. A fluid is a substance whose parts yield to any force impressed on it, and by yielding are easily moved among themselves. 3. This definition separates fluids from rigid bodies, in which the particles cannot be moved among each other by any force, however great, but it does not separate fluids from owdersy such as flour, in which we have a collection of articles which can bo moved among themselves by the appli- ition of a slight force. 4. A fluid diff'ers from a jwwder in this way : the particles joniposiug a powder do not move among themselves without riction, whereas the particles that ntake up a fluid move one USQT another without any friction. For example, if you empty a mug of flour on a table the riction between the particles will soon bring the flour to rest more or less of a heap: whercjw if you empty a mug of rater the particles, moving without friction, run in all diroc- lons, and the whole body of water is spread out into a very in sheet i.H. 1 ' f;--v; !1 9 ON FLUID PRESSURE. 5 To distinguish fluids from powders we must therefore make an addition to Art. 2, and wo give the following as a complete definition of a fluid. Def. a fluid is a substance whose parts yield to any force impressed on ity and by yielding are easily moeed among themselves tcithout friction., and also act without friction on any surface ^cith which tJiey are in contact. This definition includes not only the bodies to which in ordinary conversation we apply the terms "fluid" ami "liquid," such as water, oil, and mercury, but also sucli bodies as air, gas and steam. 6. Fluids may be conveniently divided into two classes, liquid and gaseous. By the term liquid we miderstand an incompressible and inelastic fluid. In reality all fluids with which wo are acquainted are compressible, that is, a given volume of fluid can by pressure be reduced in volume. Still so gi'cat a force is required to compress to any appreciable extent such fluids as water and mercury, that we may regard them as incompressible in treating of the elements of the subject. 7. The inelastic fluids with which wo are practically acquainted approach more or less to a state of perfect fluidity, but in all there is a tendency, greater or loss, of atljacent particles to cohere with each other. This tendency is stronger in such fluids as oil, varnish and melted glass, than in such as water and mercury. Hence the former are called imperfect or viscous fluids. 8. Tlie air which wo breathe and gases are compressible fluids, and are endowed with a perfect elasticity, so that they can change their shape and volume by compression, and when he compression ceases they can return to their former Hhajjo .nd volume. 9. Vaix)urs, as steam, are elastic fluids, but with this peculiarity: at a given temperature in a given npace only a certain quantity of vapour can be contJiiniMl, and if the space or the temperature be then diminished, a lu/rtion of the vapour becomes liquid, or even in some cases a solid. 10. Before proceeding further with our subject we nmst explain the meaning of some technical terms which wo shall have to employ frequently. ■I 1 , ON FLUID PRESSURE, ist therefore [lowing as a to any force ' weed among U friction on i to which in 'aiul^Uquid," bodies as air, two classes, luidcrstand an all fluids with lat is, a given volume. Still ny appreciable we may regard ements of the are practically iierfcct fluidity, ,s, of ac\ja(;ent ncy is stronger ban in such as lied imperfect \o compressible [,y, so that they ^ion, and when (r former shape but with this In npace only a Ll if the space fcu;riion of the (solid. Lbject we nmst I which wo shall 11. A Piston is a short cylinder of wood or metal, which fits exactly the cavity of another cylinder, and works up and down alternately. 12. A Valve is a closed lid affixed to the end of a tube or hole in a piston, open- ing into or out of a vessel, by means of a hinge or other sort of nwveablc joint, in such a manner that it can be opened only in one direction. 13. A Prism is a solid figure, the ends of which are parallel equal and similar plane figures, and the sides which connect the ends are parallelograms. The figure represents a rectangular pri«in, in which each of the lines bounding tlic surfaces of the prism is at right angles to each of the four lines which it meets. \ \i 14. "Wo shall often have to use the expression Horizontal Section of a tube or hollow cylinder, and we may explain the moaning of the expression by the following example: Suppose a gun-barrel to be i)laced in a vertical position : suppose a wad to bo part of the way down the barrel with ita upper surface exactly parallel to tl»e top of the barrel : then suppose the barrel to bo cut away so as just to leave the upper surface of the wad exposed : the area of this surface of the wad is called the liorizontal section of the barrel. 15. The in 'liomaticnl theory of Hydrostatics Ih founded ^ on two luv/8, wh'cli wo shall now oxpluin. 1—2 ON FLUID PRESSURE. 16. Law 1. The force exerted by ajluidonanymrface^ with which it is in contact, is perpendicular to thai surface. 17. This law is merely a repetition of the definition of a flaid given in Art 5, and we can best explain its meaning and application by an example. If ABhQfi cylinder immersed in a fluid the pressures of the fluid on the curved surface are all perpendicular to the rr^— r-^^ -^ _______ — -i J /- ~ — — -.^^ .1 1 L :_ T — -.-T^^ — -- r--— -y / /-— — ^- - ' ■ — — ^^—^^ 1 I l~^^ ■'- -.■ . . T- ^--:~^l 1 /— ^ r , -^L ~l / Lj^=^^.-^ - -r— / 1 -~ ~J / / ^ / / / : — ■- ■" 1 j 1 • .^r;S?T~"' - ^^ ~l / /.=r=.-.-.r - -^ r - / A-^-.-— :^-:^-.-/ 1 / pB-"~ """:.. ^~'"^- ": ^-H IL^m--^^ crv^— ms^i^^-m axis of the cylinder, and the pressures of the fluid on the Jlat ends are all panillol to the axis. Now it is a law of Statics that a force has no tendency to produce inotiou in a direction perpendicular to its own direction. Ilcnco the pressures on the curved surface have no tend- ency to produce motion in the direction of the axk., and tliu pru8»'ires on the iliit ends have no tendency to produce motion in a direction periiendiculur to the axis. 18. Law 11. Any pressure communicated to tfie surface (if a fluid is equally Iransmitled through the ichole fluid in every direction, _ w 19. A characteristic property of fluids which distinguishes tlicm from solid IxKlicH is tiiis Hu-ulty which they posscsH of trauHniittin;.' 0(|UiillY in all dircclions the pressures applied to their BurfaccH. ON FLUID PRESSURE. mrfacey ur/ace. ion of a ting and isures of ,r ta the It is of groat importanco to form a correct notion of the principle of "the equal transmission of pressure," a principle which is applicable to all fluids, inasmuch as it depends upon a property which is essential to all fluids and is not an acci- dental property, as weight, colour, and others. 20. Suppose then we' take a yessel ABCD, in the form of a hollow rectangular prism, and place it on a horizontal table. Place a block of wood, cut to fit the vessel, so that it rests on the base BC and roaches up to the level EF, \\ i\\QJlat Indency to 1 direction. no teud- und the :o motion turface. Jluid in linguishos l)088C8H of Ippliod to Then if wo place a woigiit P on the top of the block an additional pressure P will be imposed on the base of the prism. Now suppose the block to bo removed and the vessel filled with an incompressible fluid up to the level o{ EF. Suiipose a piston exactly fitting the vessel to be inserted and a prossuro P applied by means of it to the surface of the fluid at EF. In this case the pressure P is transmitted by means of the fluid not only to the base BC^ but also to the sides of the vessel, and if wo take a unit of area, ns a square inch, in the side FO^ and a unit of area in the base BCy the same additional prcs* 9ure will bo convoyed to each* B ON FLUID PRESSURE. m 21. That jlicuU transmit pressure equally in all direC' Horn mayle shewn experimentally in the following manner: ABC is a vessel of any shape filled with fluid. Make openings of equal area tit A^ B, G. Close the openings by pistons, kept at rest by such a force as may bo roquiretl in each case. Tlien it will be found that if any additional force P be applied to the piston at A, the same force P must be applic ' to each of the pistons at B and C to prevent them from being thrust out. If the area of the base of one of the pistons, as By be larger than the area of the base of the piston Ay it is found that the pressure which nuist be applied to B to keep it at rest bears the same relation to the pressure applied to A that the area of the base of B bears to the area of the base of A, 22. From the preceding article it is clear that if a body of fluid, supposed to bo without weight, be confined in a closed vessel, tiie pressure communicated to the fluid by any area in any part of the vessel will be transmitted cqiinliy to every equal area in any other part of the vessel. It is owing to this fact that the use of a Safety Valve can be depended on. hi ON FLUID PRESSURE. dl direC' manner: Thus, if the vessel A be full rf steam and the pressure of the steam be required to bo kept down to 200 fbs. on the square inch, if a valve B, whoso area is a square inch, bo placed at any part of the vessel, and bo so loaded that it will require a force of 200 lbs. to raise it, then if the steam acquire an increase of pressure above 200 lbs. on the square inch, tho valve >vill open, and mil remain open till the pressure of the steam is just equal to 200 lbs. on the square inch. 23. Amj force, however small, may by the transniissioit. of its pressure through a Jluid, be made to support any weight, however large. ich a force und that if at A, tho is at D and 3, be larger nd that tho rest bears it the area f a body of in a closed jany area in ly to every Valve can Suppose DE and FIl to bo two vortical cylinders, con- nected by a pipe EH, and supi>ose FH to have a horizontal section much larger thau the horizontal section of DE: for instance, let tho area of a horizontal section of FH bo 400 square inches, and the area of a horizontal section of DE bo 1 square inch. Now if water be poured into tho cylinders, and pistons A and B be applied to the surface at D and /', whatever force we apply to A will be trunsmitted to each portion of tli(} biMo of tho pistou li which is equal in urea to the base of tho piston A, Hence a pressure of 1 lb. applied to tlio piston A will pro- duce a pressure of 4(M) lbs. on tlie base of the inston //, and will therefore support a weight of 400 lbs. placed on tho piston B. This effect of pressure by tho modiuni of a fluid is often called The Hydrostatic Paradox. 8 ON FLUID PRESSURE, Examples. — ^I. ! (1) In the experiment described in Art. 23, if the horizontal section of tbe snuill cylinder be \\ square inches, and that of the luryfcr cylinder G4 sq. in.^ find the weight supported under a pressure of 1 ton exerted on the piston of the small cylinder. (2) If the horizontal section of the small cylinder be I]| square inches, and that of the large cylinder 240 sq. in., find , the weight supported by a pressure of 3 cwt. applied to the piston of the small cylinder. (3) If the pistons are circular, the diameters being Ij inch and 50 inches, find the weight supported by a pressure of 15 lbs. applied to the smaller piston. (N.B. The areas of circles are as the squares of their diameters.) (4) A closed vessel full of fluid, with its upper surface horizontal, has a weak part in its upper surface not capable of bearing a pressure of more than 4^ pounds on the square foot. If a piston, the area of which is 2 square inches, be fitted into an a[)erturo in the upper surface, what pressure applied to it will burst the vessel ? (5) A closed vessel full of fluid, with its upper surface horizontal, has a weak part in its upper surface not capable of bearing a pressure of more than 9 lbs. upon the square foot. If a piston, the area of which is one square inch, be fitted into an apertiire in the ui)per surface, what pressure applied to it will burst the vessel \ (6) If the horizontal section of the small cylinder be \\ square inches, and the diumotor of the large piston 20 inches, find the lifting power <>f the niach'no under a pressure of 1 ton exerted on the piston of tho snuiU tube. (N.I3. The area of 22 a circle is — times the square of the radius nearly.) . ON FLUID PRESSURE. 24. « The pressure cd any point in any direction in a fluid is a conventional expression used to denote the pressure on a unit of area imagined us containing the point, and perpendicu- lar to the direction in question. For example, if the whole pressure of a fluid on the l»ottom of a vessel is 2000 lbs., and the pressure is uniform throughout, then if we take a square inch as the unit of area, and the area of the bottom ot the vessel is 40 square inches, the pressure at a point in the base is -— - lbs. cr 50 lbs. 25. The student must carefully observe the distinction between the expressions "pressure on a point'* and "pressure at a point" : the former is zero, because a point has no magnitude. 26. If a mass of fluid is at rest, any portion of it may be supposed to become rigid without, afiecting the conditions of equilibrium. Thus if we consider any portion A of the fluid in a closed vessel, we may suppose the fluid in A to become solid, while the rest of the fluid remains in a fluid state, or we may suppose ihe fluid round A to become solid, while the fluid in J remains in a fluid state. 27. The importance of the principle laid down in the pre ceding article may bo seen from the following considerations. The laws of Statics are proved only in the case of forces acting on rigid bodies. Now since the supposition of any part of a fluid becoming solid docs not affect the action of the forces acting upon it, and sinee wo can in that case obtain the effect of those forces by the laws of Statics, we shall know their effect on the fluid. iji i mfju. lO aV FLUID PRESSURE. 28. If a body of fluid, supposed to be without weight, be confined in a closed vessel, so as to exactly fill the vessel, an equal pressure will bo exerted on the fluid by every equal urea in thu sides of the vessel (Art 22), and we proceed to shew that the pressure is tlio sumo in all directions at every point of the fluid. For let bo any point in the fluid, and AB, CI) two plane surfaces, each representing a unit of area, passing through O li : 1 and parallel to two sides of tho vessel EF^ GH, Then drawing straight linos at right angles to AB^ CD from tho extremities oS AB, CD to the sides of the vessel, we may imagine all tho fluid except that contained in the prism ABNM to become solid. Then the pressure exerted on the fluid by the area MN will be transmitted \X) AB. Again, if wo suppose all the fluid except that contained in the prism CDSR to become solid, tho pressure exerted on the fluid by the area RS will be transmitted to CD. Now the pressures exerted on the fluid by the areas MN, RS are equal, and consequently the pressures on AD^ CD yAW be equal, that is, tho pressure at the point is the same in all directions. Also since the distniico of the point from tho sides of the ressel is not involved in the preceding considerations, it follows that the pressure is the same at every point. 31 8umo to fall there which tioii 01 suppo! CHAPTER IL On the Pressure of a Fluid acted on by Gravity. 29. In the preceding chapter wo considered the conse- quences tliat result from the peculiar property, essential to all fluids, of transmitting equally in all directions the pressures applied to their surfaces. We have now to consider the cflfects produced by the action of gravity upon the substance of a fluid. 30. The student must mark carefully the distinction be- tween force applied to a surface and force applied to each of the particles composing a body. As an example of these distinct forces consider the case of a book resting on a table. Force is applied to the surface of the book by the table, and thus is counterbalanced the force of gravity which acts upon each particle of which the book is composed. 31. All fluids are subject to the action of gravity in the Banio way as solid bodies. Each particle of a fluid has a tendency to fall to the surface of the earth, and in a mass of fluid at rest tlicro \i a particular point, called the centre of gravity, at which the resultant of all the forces exercised by the attrac- tion of the Earth on the particles composing the fluid may be supposed to act 32. The term density is applied to fluids, as it is to solid bodies, lo denote the degree of closeness with which the parti cles are packed. vertically to meet the surface in C, D. AC=BD, also, ^ C is parallel to BD ; .-. CD is parallel to AB (Eucl. i. 33) '. • . . CD is horizontal. Similarly any other point in the surface may bo proved to be in the sanio horizontal plane with G or D ; .*. the surface is horizontal. 37. The proposition thai the surface of a fluid at rest is horizontal is only true when a very moderate extent of surface is taken. Large surfaces of water assume, in consequence ^ of th9 attraction exercised by the earth, a spherical form. i6 ON THE PRESSURE OF A I^LUID iiWM I I j The following practical results are worthy of notice: (1) All fluids find their level. If tubes of various shapes, some large and some bmall, some straight and others bent, be placed in a closed vessel full of water, and water be then poured into one of the tubes, the fluid will rise to a uniform height in it and all the other tubes. (2) If pipes be laid down from a reservoir to any distance, the fluid will mount to the same height as that to which it is raised in the reservoir. (3) The -surface of a fluid at rest furnishes a means ot observing objects at a distance in the same horizontal plane with a mark at the place of observation. 38. We have seen that in an inelastic fluid at rest the pressure at any point depends on the depth of that point below the surface of the fluid, that is, on the length of the vertical line drawn from the point to meet a horizontal line drawn through tlio highest point in the fluid. Thus if ABC be a conical vessel with a horizontal base, standing on a table, and filled with fluid, the pressure at any point P is determined in the following manner. B A From A^ the highest point of the fluid, draw a vertical line meeting the horizontal plane passing through P in the point Q. Then the pressure at P = pressure at Q, because P and Q are in the same horizontal plane. But pressure at Q depends on the length of AQ: therefore pressure at P depends on the length of PR^ a Imo drawn vertically to meet the horizontal lino AR, j*>*'> ACTED ON BY f volnnie, and secondly, on the supposition that 1 lb. avoirdupois is taken as the unit of weight, so that the student may see that the same result must follow from both suppositions, and that such a choice may bo made as to the units as may bo suitable to any particular case. 50. To measure any quantity we fix upon some definite quan- tity of the same kind for our standard, or unit., atid then any quantity of that kind is measured by finding how many times it contains this Unit, and this number of times is called the measure of the quantity. For example, if one pound avoirdupois be the unit of weight, the measure of 16 lbs is 16. Or, to put our calculations in a tabular form, we may give the following Examples : 1 Unit. Quantity. Measure. 1 1 lb. avoird. 8 lbs. 8. 1 1 lb. avoird. 4 oz. 1 4* «, 1 1 lb. avoird. 1 cub. ft. 1 lb. troy. 6.^ cub. ft. 5760 7000 • 6'5. mpcrature 9 ;li calcula- * ^k )f water is fl 1 cub. ft. 1000 oz. av. 3 cub. in. 14 lbs. av. 3 1728 • 14xlG 1000 * and puro 9 ho unit of 1 •016 cub. ft. cub. in. 5 1728 X 016 • ir unit of H 1 f specific V )t, that io m ! !! ■ ' ! ^ ii i! * .1 ': i 1 ; , J i: 'i! ■, ! ^ ■ i- 1 j 1 1 ill 24 ON SPECIFIC GRA VITY. 61. First, when 1 cubic foot is taken us the unit of volume, and consequently 1000 oz. as the unit of weight, to solve the following examples : Ex. (1) The specific gravity of lead is 11 4, find the weight of 720 cubic inches of lead. Here r=^|g, ^=11-4. Weight required = VS (unit of weight) = ( -^1^ X 1 1 -4 ) times 1 000 oz. \1728 = 4750 oz. = 296 1 lbs. o Ex. (2) If 5 cubic feet of a substance weigh 240 lbs., what is its specific gravity % ,j „, 240x16 „ _ Here VV ^ —ttttztt- -^ F=5. 1000 W Sp. gr. required- '-— (unit of specific gravity) 2 40 X 1 6 " 1000 5 240 X 16 (unit of specific gravity) (unit of specific gravity) 1000 X 5 = "708 (unit of specific gi'avity). Ex. (3) What is the volume of a substance whose specific gravity is 9'o and whose weight is 4200 lbs. ? Herefr=~'«,6-=9-6. w Volume required = -— (unit of volume) 4200x16 1000 7 cub. ft. cub. ft. ol 01 lit of volume, to solve the 4, find the )Z. !40 11)8., what ivity) ivity) lose specific ON SPECIFIC GRA VITY. «5 62. Secondly, when 1 lb, avoirdupoi"? is taken as the unit of weight, and consequently 016 cub. ft. as the unit of volume, our examples will stand thus : fix. (1) Hero V=' 720 1728 X 016 ,^=11-4. Weight required = VS (unit of weight) < (20 X 1728 -016 X 11*4 ) times lib. = 296 lbs. Ex. (2) Here w 6 •016 Sp. gr. required = -y (unit of specific gravity) 240 - -'' (unit of specific gravity) •016 240 X 016 (unit of specific gravity = '768 (unit of specific gravity). Ex. (3) Here Jr= 4200, ^=9-6. Volume required -- -^ (unit of volume) --^.^^.^ times '016 cub. ft 4200 X 16 9-6 X 1000 7 cub. ft. cub. ft. 1 i,M l6 ON SPECIFIC GRAVITY. 53. If a number of substances be put together to form a mixture, we shall generally have the following relations : (1) sum of measures of weights of compounds » measure of weight of mixture. (2) sum of measures of volumes of compounds = measure of volume of mixture. Thus if f^i, ?/72, w^, be the measures of the weights, ^'n ^2) ^3) volumes, *i» *2> Hi specific gra- vities of the compounds, and w^ V, s t!ie measures of the weight, volume and specific gravity of the mixture, we shall have Wi + 1Ci + W3+ =w, «>i+r2 + 2?3 + ^v; and therefore Z?,.fi + r2. } V r i III 1 CHAPTER IV. On the Conditions of Equilibrium of Bodies under the Action of Fluids. 66. When a body is wholly or partially immersed in a fluid, it is a general principle of Hydrostatics that the resultant pressure ofthejiuid on the surface if the body is cqmd to the weight of tJie fluid displaced. This principle wo shall prove for two cases in Articles 57 and 61. (1) "When the body is wholly immersed in tiio fluid : (2) When the body is partially inmicrsod in the fluid. 57. To find the resultant Pressure of a Fluid on a body wholly immersed and floating in a fluid. Lot A be a body floating in a fluid and wholly innnerscd in iu Th (1 centre (2 (»r whi< tluid d aid on a body ON THE CONDITIONS OF EQUILIBRIUM ox'c. 31 Imagine the body removed and the vacimt space filled with tliiid of the same kind as tliat in which the body floated. Tlien suppose this substituted fluid to become solid. Tho pressure at each point of its surface will still be the same as it was at the same point of the surface of A The solidified fluid is kept at rest by (1) The attractions exercisod by the earth on every par- ;irie of its mass : (•2) The pressures exercised by the fluid at the different :ui Ills of its surface. Hence the resultants of these two sets of forces must be t'qual in magnitude and opposite in their lines of action. Now the resultant of set (1) is called the weight of the solidified fluid and acts vertically downwards through its centre of gravity. Hence the resultant of set (2) is equal in magnitude to tho weight of the solidified fluid and acts vertically upwardn through its centre of gravity. Now since tho pressures on the solidified fluid are tho same ns on the body Ay we see that tho resultant pressure of the fluid on A is equal to the weight of tho fluid displaced by A and acts vertically upwards through tho centre of gravity ol this di8])laced fluid. This principle wo shall now apply to tho following Ex- amples in Statics. • 'ft. 58. Ex. 1 . Find the conditions of equilibrium of a budj/ ioating in ajiuid and wholly immersed in it. m The body A (see diagram in Art. r>7) is kei)t at rest by (1) Its weight, acting vertically downwards through its centre of gravity : (2) The pressures of the fluid on its surliice, the resultant ( r which is equal to the weight of the fluid displaced by A and .ais vertically ui)wards through the centre of gravity of tht» tluid displaced. i 32 ON THE CONDITIONS OF EQUILIBRIUM Henco (1) Weight of ^ = weight of fluid displaced by A : (2) The centres of gravity of A and of the fluid displaced are in the same vertical line. These are the conditions of equilibrium. Note. A difficulty often occurs with beginners in conceiving how a solid body can be in equilibrium in tlie midst of a fluid, neither rising to the surface nor sinking to the bottom. It may however be proved by experiment that a hollow ball of copper, such as is used for a ball- tap, may be constructed of such a weight relatively to its size that when placed in water it will remain where it is placed, just as the body A is re- presented in the diagram. 69. Ex. II. Find tlie condltioiis of equilibrium for a body of uniform density wholly immersed in a fluid and in part supported by a string. h \ Let a body the measure of whose volume is Kbo suspended by a string fk'om the fixed point A so as to float below the Bur- Aico of a fluid. The body is kept at rest by (1) its weight, (2) the pressures of the fluid on its surface, (8) the tension of the string. 1728 BODIES UNDER THE ACTION OF FLUIDS, 33 JSfow (1) is equivalent to a single resultant acting vertically downwards through the centre of gravity of the bodyi (2) is equivalent (by Art. 57) to a single resultant, equal to the weight of fluid displaced and acting vertically upwards through the centre of gravity of the fluid displaced : (and these two centres of gravity coinciding) therefore (3) must act (see Statics, Art. 52) upwards in the ver- tical line through this common centre of gravity, and (1) must be equal to the sum of (2) and (3). Hence, if • S bo the measure of the specific gravity of the body, ,S" of the fluid, T of the tension of the string, there is equilibrium when VS--=VS'-\-T or T=V{S-S^, I Ex. A piece of metal, whose specific gravity is 7'3 ana Toiume 24 cubic inches, is suspended by a string so as to b« wholly immersed in water. Find the tension of the string. Taking 1 cubic inch as the unit of volume, and consoqueutt; Y^QQ oz. as the unit of weight, tension of string = 24 (7*3 - 1) x _ oz. 24 X 6-3 X 10 00 1728 01, >876oz. IJL '^ !i ! i! 34 ON- THE CONDITIONS OF EQUILIBRIUM OF 60. Ex. (3) If a body of unif''r\ . dermty be iminer%ed in a fluid and be prevented from rising by a string attached to the bottom, of the vessel containing the fluids flnd the tension of ths string. Let a body, the measure of whose volume is F, be kept under the surface of a- fluid by a string fastened to ^, a point iu the base of the vessel. The body is kept at rest by (1) its weight, acting vertically downwards, (2) tlio tension of the string, acting vertically dow n wards, (3) the resultant of fluid pressures on the body, acting' vertically upwards. Ilonce, if T bo the measure of the tension of the string, S specific gravity of the body, /S* specific gravity of the flirid, Binco there is equilibrium, BODIES UNDER THE ACTION OF FLUIDS. 35 61. To find the resultant pressure of a fluid on a body partially immersed and floating in the fluid. Let ABGD bo a body partially immersed and floatiug in a fluid, the part BCD being bolow the surface of the fluid. Imagine the bf ly removed and the vacant space BCD filled with fluid of the same kind as that in which the body floated. Then suppose this substituted fluid to become solid. The pressure at each point of its surface will still bo the same as it was at the same point of BCD, The solidified fluid is kept at rest by (1) the attractions exercised by the Earth on every particle of its mass, (2) the pressures exercised by tlio fluid at the diff'crcnt points of its surface*. Ilonco the resultants of those two sets of forces mut.t })o equal in magnitude and opposite in their lines qf action. * Thrvugliuut tliia chapter the spikco occupied by tho air is aupposcd tu bu n vftcuum. 3—2 I i ON" THE CONDITIONS OF EQUILIBRIUM OF Now the resultant of set (1) is called the weight of the solidified fluid, and acts vertically downwards through its centre of gravity. Hence the resultant of set (2) is equal in magnitude to the weight of the solidified fluid, and acts vertically upwards through its centre of gravity. Now since the pressures on the solidified fluid are the same as on the surface BCD, we see that the resultant pressure of the fluid on the floating body is equal to the weight of the fluid displaced, and iicts vertically upwards through the centre of gravity of the displaced fluid. This principle we shall now apply to the following examples in Static^. N 62. Ex. I. Find the conditions of equilibrium of a body floating and partially immersed in a fluid of uniform density. The body ABiW (see diagram in Art. (>3) is kept at rest by (1) its weight acting vertically downwards through its centre of gravity, (2) the pressures of the fluid on the surface BCD, the resultant of which is equal to the weight of fluid displaced by the body, and acts vertically upwards through the centre ol gravity of the fluid displaced. Hence (1 ) weight of the body = weight of fluid displaced ; (2) the centres of gravity of the body and of the fluid displaced are in the same vertical line. These are the conditions of equilibriimi. eiffht of the BODIES UNDER THE ACTION OF FLUIDS, 37 63. Ex. II. Wlien a body of uniform density Jloati in a fluid, the volume of the part immersed is to tlie volume of the whole body as the specific gravity of the body is to the specific gravity of the fluid. Let r be the measure of the volume of the whole body ABGD, V' „ part mmiersed BCD, S specific gravity of the body, aS" specific gravity of the fluid. Then since, Art. 62, weight of floating body = weight of displaced fluid, .'. P : F :: &' : S\ Ex. A solid, whoso si.ocific gravity is '4, floats in a fluid whose specific gravity is 1-2. What part of the solid is below ihe surface ? Lpt £c be the measure of tho part immersed, m the measure of the whole body. Then « : m = *4 : If?; . « "4 4 \ i^il ''I ill! liili 38 ON THE CONDITIONS OF EQUILIBRIUM OF ^~~^' ■ ■ ' ■ — — 64. The Hydrottatic Balance. The Hydrostatic Balance is a common balance with a hook attached to the bottom of one of the scales from which a solid may be suspended and weighed successively (1) in air and (2) when immersed in a fluid. Call the scale to which the hook is attached A and the other scale B. Then by the weight of the solid in air we mean the weight which when placed in B balances the solid suspend- ed in air from A. And by the weight of the solid in the fluid we mean tiic weight which when placed in B balances the solid suspended from A so as to be immersed in the fluid. The difference between these weights is caused by the pressures of the fluid on the surface of the solid, the resultant of those pressures being a force acting vertically upwards and equivalent to the weight of the fluid displaced by the solid. Now if V be the measure of the volume of the solid, 8' specific gravity of the fluid, measure of weight of -fluid displaced by the solid = VS', 65. To compare the specific gravities of a solid find a Huid by means of the Hydrostatic Balance. Let V be the measure of the volume of , the solid, S : specific gravity of the solid, S' specific gravity of the flui'', W.. weight of the solid in air. s BODIES UNDER THE ACTION OF FLUIDS. 39 Case I. When the solid is of greater specific gravity than ike fluid. Let W be the measure of the weight of the solid in the fluid, then W- rF'= the measure of the weight of fluid displaced by the solid, Also ^ W^ VS ; . ^^ - ^ "VS'~ W-W"\ or, s'~W-W" and thus S and aS" may be compared. Case II. When the solid is of less specific gravity ih/^r. the fluid. Attach to the solid some heavy substance, called the sinker, which will make the solid sink with it in the fluid. Let w be the measure of the weight of the two bodies in air, X in the fluid, y sinker in air, z in the fluid. Then tr- a? = measure of weight of fluid displaced by the two bodies, y-z= the sinker. Subtracting, ?^ _ a?— y + ;2?= measure of weight of fluid displaced by the 8oli n I ^ i'ii 46 ON THE CONDITIONS OF EQUILIBRIUM OF (37) A body, whoso specific gravity is 27 and weiglit iu vacuo 3 lbs., when immersed in a fluid weighs 2 lbs. ; find the specific gravity of the fluid. (38) The specific gravity of mercury is 13'6 and that of aluminium is 2*6 ; how deep will a cubic inch of aluminium sink in a vessel of mercury ? (39) If a body floats on a fluid two-thirds immersed, and it requires a pressure equivalent to 2 lbs. just to immerse it totally, what is the weight of the body ? (40) If a body weighing 3 lbs. floats on a fluid one-half immersed, what pressure will sink it completely ? (41) A piece of cork (g. o. = -24) contaiuuig 2 cubic feet is kept below water by means of a string fastened to the bottom of a vessel ; find the tension of the string. (42) Two bodies whose weights are Wj and w^ in air, weigh each MJ in water ; compare their specific gravities. (43) The cavity in a conical rifle bullet is usually filled with a plug of some light wood. If the bullet bo held in the hand beneath the surface of the water, and the plug be then removed, will the apparent weight of the bullet be increased or diminished ? ( sinkj of S] (^ (s. 0.1 13-5) pounj a stri force 7-8? i (44) A body, whose weight in air is G lbs., weighs 3 lbs. and 4 lbs. respectively in two different fluids ; compare the specific gravities of the fluids. (45) A body whose specific gravity is 7'7 and weight in vacuo 7 lbs., when immersed in a fluid iveighs 6 lbs. ; find the specific gravity of the fluid. (4G) A solid sphere floats in a fluid with throe-fourths of its bulk above the surface : when another sphere half as largo again is attached to the first by a string, the two spheres float i- at rest below the surface of the fluid ; show that the si)ocifio gravity of one sphere is G times greater than 'hut of tho other. BODIES UNDER THE ACTION OF FLUIDS 47 (47) A piece of copper (s. g. = 8'85) weighs 887 grains in water, and 910 gi'ains in alcohol ; find the specific gravity of the alcohol. (48) A uniform cylinder, when floating vertically in water, sinks a depth of 4 inches ; to what depth will it sink in alcohol of specific gravity 079 ? (49) A compound of silver (s. G. = 104) and aluminium (s. a. = 2*6) floats half immersed in a vessel of mercury (a. o. = 13*6). What weight of silver is there in 10 lbs. of the com- pound? (50) A.n iron rod weighing 10 lbs. is supported by means of a string, one-half of the rod being immersed in water. What force is exerted by the string, the specific gravity of iron being 7-8? (51) A piece of silver weighing 1 oz. in air weighs '905 oz. in water, what is its specific gravity ? (52) Two bodies weighing in aii 1 and 2 lbs. respectively \'\v are attached to a string passing over a smooth pulley ; the bodies rest in equilibrium when they are completely immersed in water. If the specific gravity of the first body be twice that of water, find the specific gravity of the second. *U»e body, that of lead being 11. (66) A substance weighs 10 oz. in water and 15 oz. in alco- liol, the specific gravity of whit'h is '7947 times that of water : find the numbur of cubic inches in tlio substance, taking the weight of a cubic foot of water as 1 000 oz. il I ■■:':■ •I \ 48 OA' 7-^^ CONDITIONS OF EQUILIBRIUM OF (67) A block of ice, the yolume of which is a cubic yard, is observed to float with ^ths of its volume above the surface, and a small piece of granite is seen embedded in the ice ; find the size of the stone, the specific gravities of ice and granite being respectively '918 and 2'65. (68) A cubical blcck of wood weighs 12 lbs. ; the same bulk of water weighs 320 oz. ; what part of the wood will be below the surface when it floats in water ? (69) A board 3 inches thick sinks 2| inches in water : what will a cubic foot of the same wood weigh, if a cubic foot of water weigh 1000 oz. ? (60) The specific gravity of beech-wood is 'Se. W bat por- tion of a cubic foot of that wood will be immersed in sea water whose specific gravity is 1*03 ? (61) A cubical iceberg is 100 feet above the level of the sea, its sides being vertical. Given the specific gravity of sea water =1 '0263 and of ice = '9214, find the dimensions of the iceberg. • (62) If a body of weight W float with three-quarters of its volume immersed in fluid, what will bo the prep""*"^ on a hand wliich just keeps it totally immersed ? (63) Two hydrometers of the same size and shape float in two diffisrent fluids with equal portions above the surfaces ; and the weight of one hydrometei : that of the other : : m : n ; com- pare the specific gravities of the fluids. (64) A hydrometer, loaded with 40 grains, sinks 4 inches lower when floating in a fluid whose specific gravity is '3 than in water ; without the weight it rises in the water one-twelfth of an inch higher : find the weight of the hydrometer. (66) If tlie volume between two successive graduations on the stem of a hydrometer be Trfnrj^^* P*^*"^ ^^^ ^^-^ whole bulk, and it floats in distilled water with 20 divisions, and in sea water with 46 divisions, above the surface ; find the specific gravity of sea water. (66) A piece of lead is found to weigh 13 lbs. in water, and when a block of wood weighing filbs. in attached to it the two together woigh 8 lbs. iu water. Find the specific gravity of the wood. W OF 3ubic yard, ;he surface, e ice ; find knd granite the same )od will be kter: what jic foot of Whatpor- sea water BODIES UNDER THE ACTION OF FLUIDS. 49 (67) What is the weight of a hydrometer which sinks as deep in rectified spirits, specific gravity '866, as it sinks in water when loaded with 67 grains ? (68) The weight of a body A in water of specific gravity = 1 is 10 oz., of another body B in air whose specific gravity = •0013 is 15 oz.; while A and B connected together weigh 11 oz. ifc water: shew that the specific gravity of B is 10713. (69) A substance weighs 20 oz. in water and 25 oz. in alco- hol, the specific gravity of which is "7947 times that of water ; find the number of cubic inches in the substance, taking the weight of a cubic foot of water as 1000 oz. . , ^el of the ity of sea }ns of the uarters of "^I'-A on a pe float in aces ; and : n ; com- 3 4 inches is '3 than le-twelfth * lations on bulk, and }ea water ic gravity ^ator, and t the two gravity of ■.ir. CHAPTER V. On the Properties cf Air, 69. The thin and transparent fluid which surrounds us on all sides, and which we call the Air or the Atmosphere, is a material body which possesses weight and resists compression. We can prove by experiment that even a small mass of air has an appreciable weight, by exhausting the air from a glass vessel (by a process which we shall describe in the next article). We then find that the vessel weighs less than it weighed before the air was taken out of it. That the air resists compression is evident from the force required to drive down the piston of a syringe when the open end is closed. Every body exposed to the atmosphere is subject to a pressure of nearly 15 pounds on each square inch of its surface. We feel no inconvenience from this great pressure, because the solid parts of our bodies are furnished with incom- pressible fluids, capable of supporiing great pressures, while the hollow parts are filled with air like that which surroniids us. Also, since the atmosplioro acts equally on all parts of our bodies, we have no diflSculty in moving. -ki ON THE PROPERTIES OF A IK. SI 70. Hawkf^ee^s or the common Air Pump. unds us on phere, is a mpression. lass of air om a glass the next ss than it I the force i the open [ect to a zh. of its pressure, th incom- res, while lurrouiidfl rts of our -45 anc DE are two pistons with valves opening upwards, which are \rorked up and down two eylindrical barrels by means of the toothed wheel W in such a way that one piston descends as the other ascends. The barrels com- municate, by means of valves at G and F opening upwards, with a pipe leading into a strong glass vessel V caliod the receiver. Suppose B to bo at its lowest position and therefore E at its highest position. Then as B ascends the valve at B closes, and the air in the receiver and pipe opens G and expands itself in the barrel. As soon as B begins to ascend E begins to descend, the valve at E opens, the valve at F remains closed. The air which before occupied the receiver and pipe, now occupies the receiver, the pipe, and one of the barrels^ and is therefore rarefied. Now let the wheel be turned back : then as E ascends the valve at E closes and F is opened, and meanwhile B is opened as it descends, and G being closed, a quantity of the rarefied air is taken from the receiver and pipe. This procoHS may bo continued till the air in the receiver is so rarefied that it cannot lift the valves at G and F, and thou the action of the instrument must cease. 4—2 1 i| !■ i i ON THE PROPERTIES OF AIR. 71. SmeatovUs Air Pump, A B D -4(7 is a cylindrical barrel communicating with a strong vessel D called the receiver. At A and C, the ends of the barrel, are valves opening upwards. A piston with a valve B opening upwards works up and down the barrel. Suppose the piston to be in its lowest position. Then aa the piston ascends, the pressure of the air being removed from the upper surface of the valve at (7, the air in DC opens C and expands into the barrel, while the valve at B is closed by the pressure of the atmosphere. Thus a quantity of air is drawn away from the receiver. As soon as the piston begins to descend, the valve at A is closed, B opens and C is closed, and no external air comes into the barrel or receiver. When the piston again ascends the air in the barrel io again drawn out. ON THE PROPERTIES OF AIR. 53 The only limit to the exhaustion of the air by this pump arises from the difficulty in making the piston come into close contact with the valves at A and (7. Note. The advantage of Smeaton's Air Pump is that since the valve at A closes as soon as the piston begins to descend it relieves ^ from the pressure of the atmosphere, and the valve at B is opened by a very slight pressure from the air beneath. Hence this pump is capable of producing a greater degree of exhaustion than Hawksbee's. 72. To find the density of the air in the receiver of Smeaton's Air Pump after n ascents of the piston. Let the measures of the capacities of the receiver and the barrel be respectively x and y. Then the air which occupied the space whose measure Is x when the piston was at C, will occupy the space whose measure is 0? + 2/ when the piston comes to A^ • density &fter one ascent _ x density at first x + y^ fi tjfj :. density after one ascent = . (density at first). Similarly, density after second ascent = . (density after one ascent) XAry = f — -j . (density at first), , and so on ; j . (density at first). The same formula is applicable to Ilawksbce's Air Pump, If* represenc the measure of the capacity of the receiver and pipe, and y »the measure of the capacity ot each of tho barrels. \ ■ ;| i ■t: '11'! I i 54 ON- THE PROPERTIES OF AIR, 73. The Barometer. Ac The Barometer is an instrument for measuring the pressure of the atmosphere. If we take a glass tube about 32 inches long, open at A and closed at B, and fill it with mer cury : if we then close the end A, invert the tube, place it in a vessel full of mercury, called the basin, and then remove the stoppage from A, the mercury in the tube will sink a little, leaving a vacuum in BC^ and resting when the height of the column CD, that is, the distance of the surface of the mercury in the tube from the surface of the mercury in the basin, is from 28 to 31 inches. That the column CD is supported by the pressure of the atmosphere may be shewn by placing the instrument under the receiver of an air pump. Then as the air is exhausted? the mercury will sink in the tube, and if all the air could bo pumped out the mercury would sink entirely into the vessel. This experiment proves that the pressure of the air on the exposed surface of the mercury in the basi^^ sustains the column of mercury in the tube. ON THE PROPERTIES OF AIR. 55 74. To shew that the pressure of tlie atmosphere is ac- curately represented by the weight of the column of mercury in the Barometei. e pressure >pen at A close the mercury, m ^, the m in BG, at is, the from the nches. •e of the it under hausted; oould be 3 vessel, on the » column Take in the surface of the mercury in the basin an area M equal to the area of the horizontal section of the tube at D. Then area M= area of the base of the column of mercury in the tube, and since these areas are equ3,l and in the same horizontal plane, the pressures on them are equal. Now pressure downwards on J/ = atmospheric pressure on area M, and pressure downwards at i> - weight of column of mercury CD. Therefore the atmospheric pressure on area M is equal to the weight of the column of mercury CD. It follows then that the atmospheric pressure on any area is equal to the weight of the column of mercury iu the barometer, having the same area tor its base. Consequently the weight of the column of mercury in the barometer is the proper representative of the pressure of the atmosphere on a given surface. \\ ill i ! 56 OJV THE PROPERTIES OF AIR. 75. Hence it follows that the height of the column of mercury in the barometer is pro- portional to the pressure of the atmo- sphere. If then we have a vortical tube of uni- form bore filled up to the level D with mercury, if D be exposed to the atmo- spheric pressure and if M be some other level in the tube, and if h be the height of the barometric column, pressure at Z) _ weight of a column of mercury of height h pressure at jW ~ weight of a col. of mercury of height {h + DM) ' h J} H — = 5 1 h + DM' 76. To find the Atmospheric Pressure on a Square Inch. The pressure of the atmosphere on a square inch is deter- mined by finding the weight of a column of mercury whoso base is a square inch and whose height is the same as the height of the column of mercury in the barometer. Taking the specific gravity of mercury as 13'6, the weight of a cubic foot of distilled water as 1000 oz., and the height of the barometric column at the level of the sea as 30 inches, we have pressure of atmosphere on a square inch = (30xlxlx:-5??xl3 1728 ■•) ounces, ■ -»■■; 30 X 1000 X 13 6 1728 X 10 236 J ounces, = 14}Jflbs. ounces. ilR, ON THE PROPERTIES OF AIR, 57 the column Uf f heights mare Inch, ih is deter- !ury whoso me as the he weight height of inches, we 77. In estimatinsf the pressure at a point in the interior of fluid exposed to the atmospheric pressure, we must add to the pressure on a unit of area containing the point the atmospheric pressure on a unit of area. Suppose for instance we have to find the pressure at a depth of 100 feet in a lake, (1) neglecting atmospheric pressure, (2) taking the atmospheric pressure into account. Take a square inch as the unit of area : then (1) Pressure at depth of 100 feet on a square inch — weight of a column 'of water 100 feet in height, resting on a base of a square inch = weight of a column of water whose cubic content is (100 X 12 X 1 X 1) cubic inches /1200 ,^^A lbs. _ 1200x1000 1728x16 = 43 i^ lbs. 72 (2) Pressure at depth of 100 feet on a square inch = (43 -- + 15 J lbs. nearly, 29 = 58 =^ lbs. nearly. 74- 78. The Atmosphere is most dense at the surface of the Earth, and its density diminishes with its height. Hence as one ascends a mountain the weight of the incumbent air is diminished, and the mercury in the barometer sinks. Thus the barometer furnishes a means of ascertaining approximately the height of a mountain. 79. A Barometer might be formed with any fluid, but mercury is preferred to other fluids because of its great density. A Water-barometer must have a tube of great length, since the atmosphere supports a column of water m ore than 13 times as high as the column of mercury supported In the mercurial barometer. - ,VS8 ON THE PROPERTIES OF AIR. W Ay i 80. Tlie pressure of a given quantity of air^ at a given ^ y temperature, varies inversely a^ the space it occupies. ' W The following proof by oxperimont establishes the truth of this law. ABC is a bent tube, cylindrical, uniform and vertical. The . branch AB is much longer than the branch BG. The ends are open. ^ Mercury is poured drop b} drop into the end A till the surface of the mercury in the two branches stands at the same level at P and Q. The end C is then closed. Then the pressure of air in t7Q=the atmospheric pressure. 4q J) Let mercury bo again poured in at ^, (the effect of which is to compress the air in CQ,) till the surface of the mercury in the shorter branch stands at /?, halfway between C and Q. It is then found that the mercury in the longer branch will stand at a point 2>, such that the length of the column of mercury DM {M being level with II) is exactly equal to the height of the barometer at the time of making the experiment. B was tl « — ON THE PROPERTIES OF AIR, 59 Now pressure at M= pressure at R. s the truth of But pressure at M= weight of column of mercury DM + pressure of atmosphere at 2>, = atmospheric pressure + atmospheric pressure = twice the atmospheric pressure ; >nc pressure. .•. pressure of the air in CR - twice the atmospheric pressure. Hence the pressure of the air in CR is twice as great as was the pressure of the air in CQ. That is, when the given quantity of air in CQ has been compressed into halfi\\Q space, the pressure of the compressed air is twice as great as it was at first. 81. The proof given in the preceding Article may be put in a more general form. R being any point between G and Q, thus :— Let mercury be again poured in at A till the surface of the mercury stands at D and R in the branches, and lot M be level with R, Then it is found that if the spaces CQ, CR successively occupied by the air bo measured, and if h be the height of the barometer at the time of porf 'timing the experiment, space C'Q _/* + 7)-^ apace CR ~ h * Now it is clear by Art. 7C, pressure supporting air in CQ h pressure supporting air in CR ^ h + DM' pressure of air in CQ CR pressure of air in CR CQ' 6o ON THE PROPERTIES OF AIR. iil' I '!■ l^B ! H Cor. Hence r/e can shew that the elastic force of air Taries as its dehsity. For since the same quantity of air is confined in CQ and CR » density of air in CR : density of air in CQ :: CQ : CR :: pressure of air in CR : pressure of air in CQ. 82. The Condenser. AG is a cylindrical barrel with a valve at the bottom, C, opening downwards into a vessel B^ called the receiver. A piston with a valve A, opening downwards, works in the barrel. Suppose the piston to be at the top of the barrel. When the piston descends, the air in the barrel being condensed closes the valve at A, and opens the valve at C. Thus the air which was contained in the barrel is forced into the receiver. When the piston is raised again, the denser air in B keeps the valve at C closed, while the pressure of the atmosphere opens A^ and the barrel is refilled with at- mospheric air, which is forced into the receiver at the next descent of the piston. The process may be continued till the required quantity of air has been forced into B. ic force of air onfined in CQ pressure of air 017 THE PROPERTIES OF AIR. 6i 83. To find the density of the air after n descents of the piston. Let X and y be the measures of capacities of the receiver and barrel respectively. Then the air which occupied the space whose measure is x + Vt when the piston was at the top of the barrel, will occupy the space whose measure is x when the piston comes to the bottom of the barrel ; density of air ii^ receiver after one descent x-\-y density of air at first X X -\-'U .-. density of air after one descent = — - . (density of air at first). X Similarly, density after second descent = or r . (density of air at first) and so on ; \ density after wth descent = . (density of air at first). X Examples. — V. (1) If the capacity of the receiver in Smoaton's Air Pump be ton times that of the barrel, what will be the exhaustion produced by six strokes of the piston 1 (2) Find the pressure of the air in the receiver of an Air Pump after two strokes of the piston, the volume of the receiver being eight times that of the barrel. (3) Find the ratio of the volume of the receiver to that of the barrel in the Air Pump, if at the end of the third stroke tlio density of the air in the receiver : the original density :: 729 : 1000. li '' 62 ON THE PROPERTIES OF AIR. 'I ii. J If (4) Is it necessary that the section of the tube through which the mercury rises in the barometer should be the same throughout ? (6) Assuming that a cubic foot of water weighs 1000 02. and a cubic inch of mercury weighs 7| oz., find the pressure on a square inch at a depth of 90 feet below the surface of the sea, when the barometer stands at 30 inches. (6) If the area of a section of the basin of a barometer be 10 times that of a section of the tube, and the mercury fall 1^ inches in the tube, find the true variation in the height of the mercury, and draw a figure representing the instrument. (7) If a hole were made in the tube of a barometer, what would be the effect? (8) If the weight of the column of mercury which is above the exposed surface in a barometer be an ounce, and the area of the transverse section of the tube r^ of a square inch, what is the pressure of the atmosphere on a square inch? (9) Wh'^n the mercurial barometer stands at 30 inches, what will bo the height of the column in a barometer filled with a fluid of specific gravity 3*4, the specific gravity of mer- cury being 13'6 ? (10) If a piece of iron float in the mercury contained in the tube of a barometer, will it have any effect on the indica- tion of the instrument ? (11) If a body wore floating on a fluid, with which the air was in contact, and the air were suddenly romovod, would the body rise or sink in the i!uid ? (12) What would be tho effect of admitting .% little air into the upper part of the tube of the Barometer ? (13) A pipe carries rain water from the top of a house to a largo tank, the surplus water in which escapes through a valve in the top which rises freely. A weight of 21 lbs is I)lacod on it, and it is found that the water rises in tho pipe to the height of 20 feet before the vulvo opens. Fuid its area, assuming that the height of thu Water-Barometer is 34 feet And the atmospheric pressure 15 lbs. on the square inch. ■■■ ON THE PROPERTIES OF AIR. 63 (14) A cylinder filled with atmospheric air, and closed by an air-tight piston, is sunk to the depth of 500 fathoms in the sea; required the compression of the air, assuming the specific gi'avity of sea- water to be 1027, the specific gravity of mercury 13*67, and the height of the barometer 30 inches. (16) A barometer is sunk to the depth of 20 feet in a lake: find the consequent rise in the mercurial column, the specific gravity of mercury being 13*57. (16) If a body, exposed to the pressure of the air, float in water, prove that it will rise very slightly out of the water as the barometer rises, and sink a little deeper as the barometer falls. (17) Water floats on mercury to the depth of 17 feet ; compare the atmospheric pressure with the pressure at a point 16 inches below the surface of the mercury, taking into ac- count the atmospheric pressure on the surface of the water, having given that the heights of the mercurial and water barometers are 30 inches and 34 feet respectively. (18) Explain clearly why a balloon ascenda. (19) Explain how it is that a bladder filled with air, will, if conveyed deep enough in the sea, sink to the bottom. (20) What would be the height of the column of mercury (s. 0.= 13*56) corresponding to a pressure of 14 lbs. 2oz. on the square inch '\ a little air (21) A cubical vessel full of air, whose edge equals inches, is closed by a weightless piston. Find the number of pounds which must be placed on the piston in order that it may rest in equilibrium at a distance of 2 inches from the bottom of the vessel : the pressure of the atmosphere being 15 lbs. on a square inch. (22) The lower valve of a pump is 30 feet 4 inches above the surface of the water t» be raised : find the height of the barometer when the pump ceuscs to work, the specific gravity of mercury being 13*6. ml m ON THE PROPERTIES OF AIR, (23) It is foirnd that the cork of a bottle is just driven out when the pressure of the air within is double that without ; the bottle is then filled with mercury and inverted, and it is again found that the cork is just driven out. Given that the barometer was standing at 30 inches at the time, find the height of the bottle. \ li: (24) Find the ratio of the volume of the receiver to that of the barrel in a Condenser, if at the end of the third stroke the density of the air in the receiver : its original density (25) A hollow cylinder closed at the upper end and open at the lower is depressed from the atmosphere into water, its axis being kept vertical, and is found to float with its upper end in the surface of the water. What will be the oflfect on the cylinder of an increase of atmospheric pressure ? (26) If the volume of the cylinder in a Condenser be one- fifth the volume of the receiver, find the pressure at any point of the latter after 20 strokes. (27) The pressure at the bottom of a well is double that at the depth of a footj what is the depth of the well if the pressure of the atmosphere be equivalent to 30 feet of water ? (28) A cubic foot of water weighs 1000 oz. ; w^iat will bo the pressure on each square inch of the base of a cube tvhose edges are 10 inches, when filled with water ] (29) A cubic foot of water weighs 1000 ounces, and the pressure of the air on a square inch is 236 ounces ; find thfj pressure on 16 square inches at a depth of 9 feet below the surface of a pond. (30) If -4, j5, (7, bo throe points in a uniform fluid at ro'.t, the three points being in the same vertical line, and the dif forence of the pressures at A and B : difference of the pres- sures at A and C/ as / : 1 1 If a glass bo inverted, and with its mouth horizontal be pressed down into a basin of water, it will bo seen that though some portion of water ascends into the glass, the greater part of the glass is without water." This is caused by the compression of the air, Avhich prevents the water from rising in the glass. The Diving Bell works on the same principle. A heavy Iron chest BCEJ), open at DE, is suspended from a rope A^ ajid lowcjvd into the water, with its open end downwards. The water will then rise till the air in tlio chest is sufficiently compressed to prevent the water from ri.sing beyond a certain height 3fiV. Air is pumped in occasionally through a pipe /*, and the impure air is allowed to escape through another pipe Q. ON' THE APPLICATION OF AIR. 6^ 85. The Common or Suction Pump. AB is a cylindrical barrel in which a piston P, wi. avalvo opening upwards, is worked up and down by the iiandlo R. liCia a pipe, communicating witii the barrel by a valve, open- ing upwards. The end (7, which is pierced with a number of small holes, is placed under the surface of the water which is to be raised. Suppose the piston to be at the bottom of the barrel. Then when the piston is raised the vjilve P is closed by the pressure of the air on its upper surface, and tlierc bcin^' little or no air in PB, the valve B is opened by the action of the air in BG, and as it continues open during the whole as- cent of the piston, the air in BH, the part of the suction-pipe above the surface of the water, expands into the barrel, and becomes less dense than the air which presses on the water outside the suction-pipe. The water is consequently forced up the pipe by the pressure of the atmosphere, till the pressure downwards at // is equal to the atmospheric pressure. When the piston descends the valve B closes, and tlie air m PB, being condensed, opens the valve P. This process being continued, the water will at length rise through the valve B, and at the next ascent of the jiiston a mass of wat'^r wHl bo lifted and discharged through tho spout Z>. 6—2 !^l 68 ON THE APPLICATION^ OF AIR. The distance BH must be less than the height of a column ot water which the atmospheric pressure can sustain, that is, less than 32 feet. 86. The Forcing Pump. E D (O^ P -J'' JJ n mf^ ' pipe I t AB is a cylindrical barrel in which a solid piston P is worked up and down the space AF. BC is a suction-pipe of whtch the end G is placed under the surface of the water. DE is a pipe communicating with the barrel. At B and D are valves opening upwards. Suppose the piston to be at the bottom of its range in the barrel. TheJi when the piston is raised the valve at D remains ON THE APPLICATION OP AIR. 69 r a column in, that is, I i -on P is d under re in the remains closed, the air in DBF expands as the piston rises, and the air in BH opens the valve B and expands into the barrel. The water is therefore forced up tlie suction-pipe by tlie pressure of the atmosphere. ^ When the piston descends the air in PFBD is condensed, closes the valve B^ opens the valve i>, and escapes through D. When the piston ascends again the water rises higher in BG, and this process is continued till the water rises through B. Then the i)iston on its descent forces the water up the pipe DE. b7. In order to produce a continuous stream through the pipe at Ey the pipe is introduced into an air tight vessel DH into which the valve D opens. When the water has been forced into this vessel till it rises above O, the lower end of the pipe, the air which lies between the surface of the water in the vessel and the top of the vessel is suddenly condensed at each stroke of the piston, and by its reaction on the water forces it through the pipe OE in a con- tinuous stream. inir |! J 3( '■ fo ON THE APPLICATION OF AIR. 88. The Fire Engine. This machine consists of a double forcing-pftmp, both pumps communicating with the same air- vessel M. The pipe T descends into a reservoir of water. The valves opening upwards are at F", V and /2, R', ^ is a fixed beam round which the piston-rods work. The water is discharged through the pipe H* ON THE APPLICA TION OF AIR. amp, botb vork. 4 I 71 89. The Lifting Pump. JDc:<3 — ■ L 1 u A ra: K « AB ia^ cylindrical barrel in which a piston with a valve J/ opening upwards works, the piston rod passing through an air- tight collar at A. BG is the suction-pipe of which the end C is placed under the surface of the water. DE is a pipe up which the water is to be raised. At D and B arc valves opening upwards. The water will be brought within reach of the piston by a process similar to ' hat which h: 3 been described in the case of tlio other pumps. When the piston ascends lifting water the valve at D opens, and the water is discharged into the pipe DE. Wheii the piston descends, the valve at D closes, and prevents the return of the water in DE into the barrel. Each stroke of the piston increases the quantity of water in DE, and thus the water may be raised to any height, provided tliat the barrel AB, the pipe ED, and the piston rod be strong enough to bear the pressure of the superincumbent column of , pressure of atmosphere at 6" in direction (7/? = pressure on area />, .", pressure of atmosphere at //in direction ///? = pressure of atmosphere at C in direction CB. Now pressure of atmosphere at // is diminished by the weight of column of fluid /i//, and pressure of atmosphere at C is diminished by the weight of column of fluid BC^ and since I -^ re, having with fluid, in a vessel led. ;hes of the tment, the vessel is le highest surface of k^hich the /), equal ic! surface ON THE APPLICATION OF AIR. 73 the column BG is greater tlian column /?//, the effective pros- sure of atmosphere in direction HB is greater than the effective pressure of atmosphere in direction GB^ and therefore the fluid will be driven by the eff'ective atmospheric pressure in a con- tinuous stream in the direction HBG. 91. On intermitting Springs. Intermitting Springs are springs which run for a time, then stop for a time, and then begin to run again. This phenomenon is explained by the principle of tlio Siphon. Let A bo a reservoir in a hill in which water is gradually collected through fissures, as B, C, 2>, communicating with tho external air. ossuro essuro essiiro id by tho sphere at and since i Now suppose a channel MNR to run from A, first ascend- ing to N and then descending to R, a place lower than the reservoir. As tho water collects in A it gradually rises in the channel t(» N. and then flows along NR, and by the princii»le of the Siphon it will continue to How till A is completely drained. Then the flow ceases till tho watering has collected sufllcient- ly to roach N, 24 "" ■p, i M Jh, 74 OAT THE APPLICA TION OF AIR. 92. Brainah^s Press. The Hydrostatic Press, generally called Bramah's Press, is a machine by which an enormous pressure is obtained by means of water, the only assignable limits to its power being the strength of the materials of which it is formed. ^(7 is a forcing-pump, by the action of which water is forced into a tube BD, which has a valve B opening inwards. ^ is a strong cylindrical piston, with a base many times larger than the base of the piston Ay working in a water-tight collar at ilf, N. W ^ K H V D Between the top of the piston E and a fixed beam FG^ a bale of goods, such as paper, cotton or wool, is placed. Suppose the area of the base of E to bo 200 times tliat of the banc of yi. Then if a pressure of 100 lbs. bo applied to yl, a pressure of (200 X 100)11)8. or 20,000 lbs. will be conveyed to the base of E. Thus any amount of pressure may bo applied to W^ either by increasing the i)rossuro applied to A^ or by making the ba«o of E largor in comparison with the base of A. yy ON THE APPLICA TION OF AIR. 75 ih's Press, is icd by means r being the iter is forced irds. many times \ water-tight icam FG^ a 3od. inus that of prodsiiro of D baao of E, ) W^ either iii;^thobiwe AA' Examples. — VI. (1) What will be the effect of making a small aperture in the barrel of a Forcing Pump ? If the piston work uniformly up and down the length of the barrel, and a small aperture bo made one- third of the way up the barrel, how much more time than before will be consumed in filling a tank '/ (2) If the upward motion of the piston of a Common Pump be stopped, when the water has risen to the height of 16 feet in the supply pipe, but has not yet reached the piston, find the tension of the piston-rod, the area of the piston being 4 square inches, and the atinosi>heric pressure 15 lbs. on the square inch. (3) What would be the effect of opening a small hole at any point in the Siphon, first above, secondly below the surface of the fluid in the vessel ? (4) What is the greatest height above the surface of ;i spring over which its water may be carried by means of a siphon-tube, when the barometer stands at 29 inches, the specific gravity of mercury being 1 3*57 ? (6) Wliat would take place in a siphon at work if the pressure of the atmosphere were removed ] (6) Will the siphon act better at the top or the bottom of a momitain ? (7) CouM a siphon bo employed to pump water out of the hold of a ship floating in a harbour ? (8) What is the greatest height over which water can be carried by means of a siphon when the mercurial barometer stands at 30 inches ? (9) If the ends of a siphon were immerse 1 in two fluids of the same kind and the air were removed, describe what would take place. (10) A hollow tube is introduce I into the bottom of a cylindrical vessel through an aT-tight collar ; and a large tube, of which the top is closed, suspended over it, so as not quite to touch the bottom : consider the eflect of gradually pouring water into the cylinder, until it reaches the level of tiio top kA the inverted tube. 76 ON THE APPLICATION OF AIR. (11) A siphon is placed with one end in a vessel full of water, and the other in a similar empty one, both of wli*c «^»*ti on the plate of an n >*-punip. As soon as the water ha::' cover- ed ihe bwor end oi' the siphon; a receiver is put on, and ectively, and we obtain C R 100 'ho' a R Hence the three scales are thus connected, c F-^n _ 11 4" » "" i* 8o ON THE THERMOMETER. 98. The following examples will shew how to find t .. number of degrees marked on any one of the three scales wlc n the number marked on one of the other scales is giv^en. Ex. (1) What reading on the Centigrade scale coni.- aponds to 66® Fahrenheit ? Since - = 5 " 9 ' and i^'^ 56, C 56-32 5 ■ 9 ' .-. 9(7- : 5 X 24, •.VC7= 7 = 33J. .*. the reading on the Centigrade scale is 13^ degi'ees, Ex. (2) What reading on the Fahrenheit scale corresponds to 14° Centigrade ? Since C= 14, 14_ i^-32 5~ 9 ' .-. 126 = 5i^-160, .'.67^=286, that is, the reading on the Fahrenheit scale is ti*l\\ Ex. (3) If the sum of the readings on a Centigrade \\\u\ w Reaumur be 90, what is the reading on each ? Hero wo have two equations, from which we can fimt C and Ry 5 4 ^^^' C+ 72 = 90 (2); .-. 4C=572 4C+4i2 = 360 .•.4/2 = 360-522, .-.9/2=360, and HO /2=:40 uud (7=50, s ) find r'..' sales wltrn en. ale cono ON THE THERMOMETER, ^l EXAMPLKa— VII. ji*ees. Trespomls idu MDii a an find (' (1) Give tlio number of degrees in the Centigrade and Reaumur's scale respectively that correspond to the following readings on Fahrenheit's scale, (1) 30", (2) 45«, (3) 560, (4) 0^ (6) -7^ (6) -45". (2) Give the number of degrees in the Centigrade and Fahrenheit's scale respectively that correspond to the following readings on Reaumur's scale, (1) 5«, (2) 20^ (3) 0^ (4) -1S», (5) -64", (6) 120«. (3) Give the number of degrees in Fahrenheit's and Reaumur's scales respectively tliat correspond to the following readings on the Centigrade scale, (1) 160, (2) 45", (3) 11 0«, (4) 0«, (5) -45», (6) -24» (4) Is it necessary that the section of tlio tube throug^i which the mercury rises in the Thermometer should be the same throughout ? (5) If the sum of the readings on a Centigrade and Fahren- heit be 60, what is the reading on each ? (G) At what temperature will the degrees on Fahrenheit be fi\e times as great as the corresponding degrees on the Centigrade ? (7) At what point do Fahrenheit and the Centigrade mark the same number of degrees ? (8) Show how to graduate a Thermometer on whose scale 20° shsill denote the freezing point, and whose 80th degree shall indicate the same temperature as 80" Fahrenheit. (9) What will bo the reading on the Centigrade when Fahrenheit stands at 78** ? (10) The sum of the number of degrees indicating the same te:^perature on the Centigrade and Fahrenheit is 88, find the number of degrees on each. (11) What reading on tho Centigrade corresponds to 49® Fahrenheit '\ 8. H. i 82 ON THE THERMOMETER, (12) What would be the inconvenience of having the bore of the Tliermometer la^'ge ? (13) At what temperature will the degrees on Fahrenheit be 3 times as great as the corresponding degrees Centigrade ? (14) The numbers of degrees indicated at the same instant by a Centigrade and a Fahrenheit's thermometer are as 5 : 17 ; determine the temperature. (15) What is the temperature when the number of degrees on the Centigrade is as much below zero, as that on Fahren- heit's is above zero ? (16) One Thermometer marks two temperatures by 9^ and 10" ; another Thermometer by 12^ and 14*^; what will the latter mark, when the former marks 1 5*^ % (17) One Thermometer marks two temperatures by 8" and 10® ; another Thermometer by IP and 14^; what will the latter mark when the former marks 16"? f (18) If the difference of the readings on Fahrenheit and Reaumur be 47, what are the readings ? If the difference in- crease by a given number of degrees, find how much each of the thermometers has risen. ^ ng the bore Fahrenheit entigrade ? ame instant •eas 5 : 17; r of degrees on Fahren- CHAPTER VIII. 3S by 9* and 11 the latter )s by 8*^ and 11 the latter •enheit and ference in- ich each of Miscellaneous Examples. 99. We shall now give a series of examples to illustrate more fully the principlos explained in the preceding Chapters. The important law of pressure in the case of compressed air, of which we treated in Arts. 80, 81, will be referred to as Marriotte's Law*, Examples worked out. 1. Water is 770 times as heavy a^ air. At what depth in a lake would a bubble of air he compressed to tlie density of water f supposing Marriotte^s law to hold good throughout for compression? At the surface the density = that of atmosphere, and 3'i feet of water are equivalent to one atmosphere ; /. at depth of 33 ft. the density = twice atmospheric pressure. (2 X 33) ft = three times (769x33)ft = 770timcs .'. the density will bo equal to that of water at a depth of (769 X 33) ft. -i.^., 25377 ft. I* J. 1 i. J • It was proved by the independent researches of Maniotte, a irrench Physician, and Boylo, the EngJMf^ Philosopher. 6—2 ^, o 7a /j y /^ IMAGE EVALUATION TEST TARGET (MT-S) 1.0 ■U Uii 12.2 I.I lU u 110 I 2.0 1.25 ^ IIIIIJ4 Hiotographic Sciences Corpordtion ^: <^ <^^^ \^ aa WIST MAIN STRUT WIISTIR.N.Y. I4SI0 (716) •7a-4S03 2. A body weighs in air 1000 grs., in water 300 grs.^ and in anotJier liquid ^20 grs, : what is the specif gravity of the latter liquid ? In water the body loses (1000 -300) grs., i.e. 700 grs., in other liquid (1000 -420) grs., i.e. 580 grs.; .'. equal volumes of water and of the other liquid weigh re- Bpectively 700 grs. and 580 grs. 680 .*. measure of specific gravity of other liquid = 700 •8^85714. 3. Taking account of atmospheric pressure^ and taking 3'3/eet as the height of the water barometer j at what depth in a lake is the pressure twice what it is at a depth of one yard / Pressure at the surface = weight of column of water 33 ft. high, Pressure at 3ft. depth = weight of column of water 36ft. high; .'. for a double pressure we must take 36 feet lower, that is, 36 feet lower than 3 feet, or 39 feet from the surface. 4. A Jlat piece ofiron^ weighing 3 Ibs.^ floats in mercury; f> and if another piece cfiron of like density weighing 2 — lbs. is placed upon ity the flat piece is just immersed. Compare the specify gravities of iron and mercury » Total weight of iron = ^ 3 + 2 ^ \ lbs. = 6 ^ lbs. The volumes of the part immersed and of the whole will bo ft as the weights, that is, as 3 : 5 ^ , or as 78 : 135. .*. 8p.gr. of iron : sp. gr. of mercury = 78 : 136, = 26 : 46. 5. Air is cor{/ined in a cylinder surmounted by a piston without weighty whose area is a square foot. What weight must be placed on tJie piston that the volume qf air may be reduced to fiaJf its dimensions? By Marriotto's law tlio air when reduced to half its volume will have double its original prossiiro. llcnco taking 16 lbs. per square inch as the original atmospheric pressure, it be* Miscellaneous examples. ) gr».i and mty of the grs.; weigh re- •8286714. md taking chat depth ipth of one 33 ft. high, 36ft. high; voTf that is, je. I mercury; Compare bs. hole will bo 136, 46. by a piston Itat weight air may be f its volume king 1 6 lbs. tsuro, it bi»> Js comes 30 lbs. per square inch below the piston. But the at- mosphere still exerts a pressure of 15 lbs. por square inch above the piston. Therefore a pressure of 15 lbs. ftore per square inch is required to keep the piston at rest. .'. weight required = (15 x 144) lbs. = 2 1 60 lbs. 6. If the capacity of the receiver of an air-pump be 10 times that qf the barrel, shew that, qfter 3 strokes qfthe piston, the air in the receiver tcill have lost nearly one-fourth of its density. By the formula of Art. 72, if p^ and p„ be the densities originally and after the n^ stroke, and R and B bo the capa- cities of the receiver and barrel, Pi •• po Vio-f i; .*. density lost = II- 1000 . 1331' 13317 ^° 331 1 , 1337'***'^ ^Po nearly. 7. A block of wood ( s. o. r« j weighing 156 lbs. is float- ing in fresh water. What weight placed on it will sink it to tlie level of the water f Let d?= the weight in lbs. Then » + 156 "weight in lbs. of water displaced by volume of wood alone, = j|xl6C. = 169; /. jr = (169-166)lbs. = 13lb8, 8. In a mixture of two fluids, of which the speciflc gra- vities are 3 and 5 respectively, a body, whose 8. o. is 8, loses half its weight. Compare the volumes mixed. Weight lost = weight of fluid displaced, B weight of body whose s. a. is 8^ 2 .*. 8. 0. of the mixture is 4. •?: • S6 MISCELLANEOUS EXAMPLES. And since the separate specific gravities are 3 and 5, while the sp. gr. of the mixture is ',\ the fluids must be mixed in equal volume?. 9. A vessel of water has for its horizontal section a rect- angle 6 feet hy 2 feet. A substance weighing 650 lbs. is im- mersed in it, and the water rises 8 inches. Find the specijk ravity of the svlbstance. Sectional area =12 square feet. Volume of substance = f 12 X -^ cub. ft. = 8 cubic feet ; /. 8 cubic feet of the substance weigh 550 lbs. ; 650 .'. 1 cub. ft. 8 lbs., or 68*75 lbs. Also, a cubic foot of water weighs 62*6 lbs., .*. sp. gr. of substances- -r^i:- = ri. 10. A cylinder floats in a fluid A with one-third cf its axis immersed, and in another B with three fourths cf its axis immersed. How deep will it float in a fluid which is a mixture of equal volumes cf A and B f 3 1 Sp. gr. of ^ : sp. gr. of i? -= ^ : ^ , * 9 : 4; 9 + 4 /. sp. gr. of mixture of equal volumes = -^j— =6*6. 2 1 If therefore the body has - of* its axis immersed in a fluid o of s. o. 9, whon^it is immersed in a fluid of s. 0. 6*6 the part immersed is obtained from the following relation, where x is the part immersed, 6i -9=^ ®2 • ^8 X. :. a= 9x. 13 >, while the ed in equal MISCELLANEOUS EXAMPLES, 87 100. We shall now give a set of easy Examples to be worked by the student by way of practice. iion a rect- ) lbs. is im- the specific 68*75 lbs. hird qf its rtJis qf its which is a = 6-6. d in a fluid '6 the part where x is Examples.— VIII. I. An iceberg (s. a. -925) floats in sea-water {d. o. 1-025). Find the ratio of the part out of the water to the part im- mersed. J. A body floats in a fluid (s. o. -9) with as much of its volume out of the fluid as would be immersed if it floated in a fluid (8. G. ri). Find the specific gravity of the body. 3. Find the Fahrenheit Temperatures corresponding to - 40« and + SSO** Centigrade. 4. The capacities of the barrel and receiver in a Smea- ton*s air-pump are as 1 : 3. A barometer enclosed in the receiver stands at 28 inches. What will be the height after three upward strokes of the piston ? 6, Two hydrometers of the same size and shape float in two different fluids with equal portions above the surfaces, and the weight of one hydrometer : that of the other ~1 : p. Compare the specific gravities of the fluids. 6. A man weighing 10 stone 10 oz. floats with the water up to his chin whc^i he has a bladder under each arm equal in size to his head and without weight. I** his head be one- twelfth of his whole bulk, find his specific gravity. 7. At ^ iiat height does the water barometer stand when the mercuritti barometer stands at 28 inches (s. o. of mercury ^13'6)? 8. What degree Centigrade corresponds to 27" Fahren- heit? 9. A man 6 feet high dives vertically downwards with his hands stretched 18 inches beyond his head. What depth has he reached when the pressure at his fingers' ends in \^ that at his feet] ^' 1^^ 88 MISCELLANEOUS EXAMPLES. 10. A string will bear a strain of 10 lbs. 7 oz. Determine the size of the largest piece of cork (s. o. '24) which it can keep below the surface of mercury (s.o. 13'6). 11. In De Lisle's TherniOmeter the freezing point is 150" and the boiling point zero. What degree of this thermometer corresponds to 47*^ Fahrenheit? 12. Cork would float in n atmospheres. Find n (s.o. of air and cork being '0013 and *24). 13. An elastic body of s. o. '5 is compressed to 20 + « 20 + 4?j ^ its natural size by immersion n feet in water. At what depth will it rest? . i 14. If the body in Question 13 weigh 10 lbs., what are the magnitudes and directions of the forces which will keep it ii equilibrium at depths (a) 5 feet, and Q3) 30 feet? 15. At what depths will the force required to keep tho body in Questions 13 and 14 at rest bo 1 lb. ? 16« At what temperature are the readings on Reaumur, Centigrade and Fahrenheit proportional to 4, 5, 25 ? 17. At what temperature is the sum of tho readings on Reaumur, Centigrade and Fahrenheit 212 ? 18. A body (s. o. 2 '6) weighs 22 lbs. in vacuo and another body (s. o. 7*8) weighs n lbs. in vacuo ; and their apparent weights in water are equal. Find n. 19. Find the specific gravity of the fluid in which the apparent weights of 1 lb. of one substance (s. a. 3) and 3 lbs. of another substance (s. o. 2'25) are equal. 20. Equal volumes of two substances (s. o. 2'7 and 6*1) are immersed in water and Balance on a straight lever 71 inches ong. Find the position of the fulcrum. 101. We proceed with some examples of somewhat greater difficulty than those already given. KoTE. We shall assume that the volume of a sphere is = irr*f r being tlie radius. 8 MISCELLANEOUS EXAMPLES. 89 >eteruiiiie ; can keep int is 150" rmometer n (s.a. 0! 20 + » . 20 + 4» hat depth \t are the keep it ii keep the ileaumur, idings on another apparent rhich the I 3 lbs. of 6*1) are 71 inches b greater phere is Examples worked out, 1. Shew hoto tfie depth qf the descent in a Diving Sell can he determined from observations on the barometer. Let AB be the surface of the water, CD the water level in the bell at the end of the descent. Now pressure at CD is equal to pressure throughout the upper part of the bell, and is therefore equal to the pressure due to atmosphere + weight of column of water {,x + y) ft. high. Hence if S bo the measure of the specific gravity of mercury, and hy h' the measures of the heights of the mercu- rial column at surface of the water and at the bottom, measure of pressure at CD = A* + (a? + y) x 1 . But measure of pressure at CD - h's ; .*. h8+a!+y = h'St ,'. x={h'-h)s-'y. Now, by Marriotte's law, if a be tiie measure of the height of the bell. }i t y h a ;'» h or, y^j:,a\ ge^{h'—h)s- Y d' go MISCELLANEOUS EXAMPLES. 2. What must be the least size in cubic feet of an inflated balloon, that it may rise from the earth when filed toith gas whose specific gravity compared toith that qf air is '08, the weight of a cubic foot of air being '3 grains, and the collapsed balloon car and contents weighing altogether 550 lbs, F Taking 1 as the measure of the specific gravity of air, and V oftheyolume of the inflated balloon^ weight of inflated balloon, | ^ xVx'3) grs. neglecting weight of envelope,/ weight of air displaced =('Fx 1) grs. « Fgra. Now 1 cubic ft. of air weighs to tho regular action ot M, and renders the expljumtion more simple. 8. H. 7 98 NOTES, It U also Btatoil on pogo 72 that the longer branch muH b$ outsit tho yosBol. This is Dot neceitary, for tho instru- ment wili work with the shorter branch outside, provided that the extremity of that branch be below the surface of tho fluid. 8. To the Thermometers it might be well to add that which is called Do Lisle's. This is much used in Hussian Bcien title operations. In it the boiling point is marked 0®, and the f reeling point 160^ 9. It should be carefully observed that the freezing point of a Thenuometer is found by placing the instrument not in frnzing water, but in nulHng ics. ich mini instru- provided iirface of Eidd that Husnian d 0^ and ing point it not in ANSWERS. ExAifPLEa I. (pago 8.) 1. 66} tons. 2. 30 tons. 3. 29629-d2d lbs. 4. 1 oz. 5. 1 oz. 6. Tho aroa of a circle whoso radius is r is nr\ and tak ing ^ as an approximate value of tt, tho answer is G587}Kcwt ExAMPLBS II. (pago 18.) 1. 20 lbs. 2. 37 A lbs. 3. 7:6. 4. 9:8. 5. 10 feet. 6. 12 lbs. 7. 9lbi. 8. 1 ton 7 cwt. 3 qrs. nibs. 9. 11 lbs. 12foz. 10. 22500lbs. 11. 1125^3 lbs. 12. }of its height 13. Since the external pressure on the cork in crease* with the depth, while the internal pressure is conttantf the cork will be forced in when the former exceeds the latter. 14. 12itons. ' 16. 18 feet. Examples III. (page 27.) 1. 166 lbs. 2. 18:1. 3. 7f}oz. •016n 4. -8. 6. 6oz. 6. Ifjoz. 7. m 8. 7776. 9. 116. 10. 844. 11. 14. 12. -cub. in.; r cub. in. 4 4 13. Volumes as 1 : 2, weights as 1 ; 4. 14. 2:1. 16. 2JJ. 16. 2. 17. 9*325. lOO ANSWERS. 18. If £?|, day d^ be the measures of the densities of the fluids, and d bo the measure of the density of the mixture, 19. 8-241... 20. -802... 21. 18*41. 22. 1'61... 23. 3* 13. 24. 8*6... oz. 25. The volumes are as 57 ' 1, the weights as 2223 : 97. Examples IV. (page 42.) 1. 3*5. 2. 607870 tons. 3. three-fourths. 4. 4 dwts. 20jf grs. 5. 4. 6. 3 times weight of tub. 7. two-thirds. 8. r^oz. 07 9. 432 oz. 10. 3 oz. 11. 42 OZ. 12. 3gOZ. 15. O CO GO t> 16. % 1 20. 16 19' 21 ^^^2 ^^' 2731 • 6 13. ^Ibs. 14. 3-5. b 17. ^. 18. 17f. 19. 2f. 22. 2. 23. 47Ubs. 24. 12 feet. 25. '66. 26. Because the specific gravity of salt water is greater than that of fresh water. 27. ^ inches. bo 28. 1728. 29. 7 lbs. d^oi. 30. Edge of cube is 2 feet. 31. 6ff 32. height of triangle n/3 33. 1:4 when vertex is downwards ; S : 4 when vertex is upwards. 34. 16 lbs. 35. -76. - -• ' - - .. 26 . 36. 2 inches. 37. '9. 38. inch. 135 39. 4 lbs. 40. 3 lbs. 41. 95 lbs. 42. tri(«r,— tr) : tCg («?i-tc). 43. Increased, if the wood bo lighter than water. 44. 3 : 2. 45. 1*1. 14129 47. r^rrr or '8 nearly. 48. Scinches. 49. 84? lbs. 60. 9i|lbs. 61. 10}g. 62. 13. 63. 6 inches. 66. 2|. .^ 86400 , , ««• -2063 '^^•*"- 67. ogg of a cubic yard. t ANSWERS. lOI 68. - of Yolome. o 59. 750 oz. 61. 936302451-687 cub. ft. 60. ^ of a cub. ft 62. f . 63. in : n. 64. 900 grains. 65. 1 4^3^ or 1*0272 nearly. 66. 'Si 67. 433 grains. 69. 42^V!r cub.in. i • if the 13. „ Examples V. (page 60.) I. Density = ( — j times original density. 2. —times original pressure. 3. 9 : 1. 4. No : because the pressure varies with the depth alone; so that if the section varied there would still be equal vertical increments of space for equal increments of pressure. 5. 53J lbs. 6. 1 Jg inches. 7. The mercury would fall to the level of th** jurface in the cup. 8. 14'625 lbs. 9. 10 feet. 10. No : because a volume of mercury equal to that displaced by the iron will descend and allow the iron to take its place without disturbing the general upper surface. II. Sink: see answer to (16). 12. The mercury would descend a little. 13. 238 square inches. 14. '0109 of original volume. 15. 1ft. 5^^Vi°- 16- When the floating body is partially immersed, both air and water are displaced : but the absolute weight of floating body = weight of displaced fluids, which must therefore be constant : there- foro when the barometer rises, there must be a less water displacement, i.e. the body rises : while any decrease in the atmospheric pressure (when the barometer falls) will necessi- tate an increased water displacement, and therefore the body then sinks a little. 17. 1 : 2. 20. 28*8 inches. 21. 1080 lbs. 22. 26 tf inches. 23. 5 feet. 24. 6 : 1. 26. The air will be compressed inside, and so displace less water : and since it floated originally, it will now sink, because the weight of displaced fluid is now less than the weight of the body. 26. 6 times original pressure. 27. 32 ft. 28. 6^ffOZ. 29. 4776 oz. oO. AB : BG-p : q-p- 32. The space between zero I02 ANSWERS, point and any graduation ought to be less than the space indicated by the number placed against that graduation in the ratio of 17 : 18. 33. 14*935 lbs. nearly. 3i. x!f of an iach. 35. Low. 36. 4? inches. Examples VI. (page 75.) 1. It will increase the time of filling the receirer, since the only effective work would be done by the descending piston, after passing the hole. It will fill the tank in 3 times the original time. ' 2. 27} lbs. 3. (a) If the hole be below the leyel of short end, no effect. 0) If above this level but still in the long branch, all the fluid in this branch below the hole will descend, and all above in the same branch will ascond causing the remainder of the fluid to flow through the short branch, till the siphon is emptied. iy) If in the short branch, all the fluid below the hole is this branch will descend ; all above in the same branch will ascend and flow through the long branch, emptying the siphon. (d) If at the top of the siphon, the fluid will descend in each branch and empty the siphon. 4. 32 ft. 9-63 in. or 3279416 feet. 5. The fluid would descend in each branch and the siphon be emptied. 6. Equally well at both, if the siphon be not too high. 7. No: because the hold is lower than the surface in harbour. 8. 33 ft. Ill in. 9. If the air be removed from the siphon, the fluids would first ascend in each branch and after- wards flow as usual. 10. The water would rise in the in< verted tube as high as the top of the inserted tube and afterwards flow out of it. 11. First, the watar would soon cease to flow. Secondly, it would rise in each branch, and afterwards flow. 12. (a) The water will flow into the lower vessel. 03) The water will descend in each branch till it stands at 34 feet above each surface, (y) The same as (a). 13. Each branch 2 feet. ANSWERS. 103 3 space btioD in ly. 3r, smce cending 3 times end, no inch, all , and all mainder ) siphon the hole ,nch will ing the descend would ligh. face in )m the after- the in- )e and Id soon p, and Ito the ich tiU as(a)i Examples VII. (page 81.) 1. (1) -!*•; -r. (2) 7f^ by>. (3) 13i»; 10S». (4) -17F; -14f^ (6) -21§'>; -17J0. (6) -42i»; -ZA^\ 2. (l)6i«;43i* (2)25^77^ (S)0'»;32». (4) -22i»; -8J». (6) -80*'; -1120. (6) 160«; 302°. 3. (1) 60^»; 12f. (2) 113°; 36°. (3) 2300; 88«. (4) 32"; 0». (6) 6»; -12». (6) -Hi"; -19^®. 4. Yes: if the graduations are to be uniform. 5. 10" Cent, and 60° Fah. 6. 10° Cent, and 50°Fah. 7. -400. 8. Make each degree -=■ ths that on o Fahrenheit. 8. 25|o. 10. 20° Cent., 68° Fah. 11. ^\ 12. The graduations would be inconveniently small. 13. 800Fah. 14. 20® Cent., 68® Fah. 15. -1 If Cent, llfFah. 16. 240. 17. 23*. 18. 69® Fab., 12® Reaum.; 9d 4d if (2 be the number of degrees, Fah. rises — and Reaum. -?-. Examples Till, (page 87.) 1. 4:37. 2. -496. 3. -40»and6620. 4. 11-8126 inches. 6. I : p. 6. 1085. 7. 31ft. 8-8 in. 8. -2j°. . 9. 22^ ft 10. ^ cub. ft 11. 13740. 12. 2400 13 13. 10 feet (/3) 2 r lbs. upwards. 1 14. (a) 2 r lbs. downwards ; 15. 7 i ft, and 13 I ft 16. 600 Fahrenheit 9 18. 15 17" end. 17. 1220 Fahrenheit 19. 2. 20. 17 j inches from one - •' EzAifPLES IX. (page 94.) 1. (1) 97 : 62. (2) 92 : 47. (3) 69 : 90. In (3) fyilcnim is at one end, and gold between fulcrum and silver. 104 ANSWERS. 2. 3:4. _ 10x112x16 ^^ ^' r20 ^*' 4L -of the gas has been expelled, and - of the whole weight thrown out 6. Oas to preserve equilibrium of internal and extemai pressures on the balloon. Ballast to preserve eqoilbrium of vertical pressures on the balloon. 2 4 6. Sp. gr. = - . Height immersed = 5 = inches. 7. No change will take place till the stone falls from the ice, it will then displace less water than before, and the sur- face will consequently sink. 9. Taking a cubic foot of water to weigh 1000 oz., the resultant pressure is 30000 lbs. The pressure would be the same inside as outside. 10. 102 fathoms. 11. 125 oz. 12. The fluid surface. » If) 2 ,„ ^(2A + c-N/c» + 4A2) + a(«^c«+4A«-c) JIU. 7' ^' 2AI 19. 10 lbs. 20. 10 and 8. 21. 11 inches. 22. 33 : & 4. - of the gas t thrown out **n. each $0 25 tt 40 (( 1 00 >« 25 «( 30 ii 90 i( 75 75* 16 15 1 00 15 40 i« 1 00 30 30 25 15 it Vr .bl; i :•. ^ >l PRICE, $1.00. FlEMIN' f lege Grami and Bciontii I am ploaBi Bome difficiL I consider tL . of its merits, an. f Albert College, Maj i .T«7 . During th., 1 it two r th- yeartj Analysis in the Uppei Fc;j d c ,.C. Collets factory results. It is the best work of the 1 poses with which I v^ ^ 'uj ^nted. , as bee : 1- of English Qramtn& t Oaf.ilAtB^aLwaiga. ' > ^wards ,g junp^|K. ^^ J have t t. ^BH^^*'^' vHrciseswh^j 9 CiCar ♦he jd .tains ¥ HUGHES, . Inspector, Toronto. ♦ • * Fo; Aiiua Mchool purposes and for use in the higher classes of our Public Schools, Mr. Fleming's book is unques- tionably the very best at present available.— Gk)&6.