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column of quicksilver will hang in the lower end, which, when dipped in water lower than 14 times its own length, will, upon removing the finger, be suspended and pressed upwards.
3. Let a large open tube be covered at one end with a piece of bladder drawn tight : pour into the tube a quantity of coloured water sufficient to press the bladder into a convex form; then, dip the covered end of the tube slowly into a deep vessel of water ; the bladder, by the upward pressure, will become first less convex, then plane, and at last concave.
4. If the like be done with several tubes, whose covered orifices are cut obliquely at different angles, the lateral pressure will be seen to increase with the depths to which the tubes are immersed.
5. Let a
circular piece of brass, whose upper surface is covered with wet leather, be held close to one orifice of a large open tube, by means of a cord or wire fastened to the middle of the plate, and passing through the tube : let the plate, thus kept close to the orifice of the tube, be immersed with the tube into a large vessel of water: when the plate is at a greater depth than 8 times its thickness in the water, the cord or wire may be left at liberty, and the upward pressure of the fluid will keep the plate close to the tube.
6. Let a small bladder, tied closely about one end of an open tube having a large bore, be filled with coloured water till the water rises above the neck of the bladder ; upon immersing the bladder into a vessel of water, the bladder will be compressed on all sides, and the coloured water will be raised up in the tube in proportion to the depth to which the bladder is sunk.
PROP. IV. When a fluid flows through a tube which is wider in some parts than in others, the velocity of the fluid will, in every section of the tube, be inversely as the area of the section.
Let ADMN, a bended tube larger at IL than at FG, be filled with water to the height Plate 5. ADFG Let the water be forced downwards in the part ADBP, and consequently be Fig. 1. made to rise in the other part KHMN. It is manifest, that the water which is forced out of one part of the tube, is driven into the other. Hence equal quantities pass through every section of the tube in the same time : for if less, or more, water passed through the section FG than through IL in the same time, the quantity of water between FG and IL must be increased or diminished, which cannot be, since no cause is supposed which could increase or diminish it. But if equal quantities pass through unequal parts of the tube in the same time the water must run proportionally faster where the tube is narrower, and slower where it is wider. If, for example, as much water runs through the section FG, as runs in the same time through the section IL, the watepinust move as much faster at FG than it moves at IL, as the tube is narrower at FG than & IL ; that is, the velocity is inversely as the area of the section.
Cor. The momentum will be the same in every section of the tube ; for the quantity of water at each section is directly as the area of the section, and the velocity is inversely, as the area ; therefore the velocity is inversely as the quantity of matter: whence (Book II. Prop. XI.) the momentum is every where the same.
ScĦOL. Hence we may account for the suspension of the fluid in a tube the upper part of whose bore is capillary, and the lower of a much larger dimension, as was seen in the experiment, Book I. Prop. VII.
Let there be a tube consisting of two parts DR and RCK, of different diameters: DR the smaller part of the tube, is able (Book I. Prop. VIII.) to raise water higher than the other : let then the height to which the larger would raise it, be TC, and that to which it would rise in the lesser, if continued down to the surface of the fluid, be XH. If this compound tube be filled with water, and the larger orifice CK be immersed in the same fluid, the surface of the water will sink no farther than XL, the height to which the lesser part of the tube would have raised it. But if the tube be inverted, and the smaller orifice XL be immersed, the water will run out till the surface falls to TF; the height to which the larger part of the tube would have raised it.
Let the tube DR be conceived to be continued down to HI ; and let it be supposed that the fluids contained in the tube XLHI, and the compound one XLKC, are not suspended by the ring of glass at XL, but that they press upon their respective bases, HI and CK. Let it farther be supposed that these bases are each of them moveable, and that they are raised up or let down with equal velocities; then will the velocity with which XL the uppermost stratum of the fluid XLCK moves, exceed that of the same stratum, considered as the uppermost of the fluid in the tube XLHI, as much as the tube RCK is wider than DR, (by this Prop.) that is, as much as the space MNKC exceeds SLIH Consequently, the effect of the attracting ring XL, as it acts upon the fluid contained in the vessel XLCK, exceeds its effect, as it acts upon that in XLHI, in the same ratio. Since, therefore, it is able to sustain the weight of the fluid XLHI by its natural power, it is able, under this mechanical advantage, to sustain the weight of as much as would fill the space MNKC : but the pressure of the fluid XLCK is equal to that weight, as having the same base and an equal height (as will be shewn by Prop. VI.) Its pressure, therefore, or the tendency it has to descend in the tube, is equivalent to the power of the attracting ring Xi, for which reason it ought to be suspended by it.
Again, the height at which the attracting ring in the larger part of the tube is able to sustain the fluid is no greater than NF, that to which it would have raised it, had the tube been continued down to MN. For here the power of the attracting ring acts under a like mechanical disadvantage, and is thereby diminished, as much as the capacity of the tube TFNM is greater than that of HIXL ; because, if the bases of these tubes are supposed to be moved with equal velocities, the rise or fall of the surface of the fluid TEXL, would be so much less than that of TFMN. And, since the attracting ring TF is able, by its natural power, to suspend the Auid only to the height NF in the tube TEMN; it is in this case able to sustain no greater pressure than what is equal to the weight of the fluid in the space HIXL: but the pressure of the fluid TFXL, which has equal height, and the same base with it, is equal to that weight; and therefore is a balance to the attracting power.
From hence we may clearly see the reason, why a small quantity of water put into a capillary tube, which is of a conical form, and laid in an horizontal situation, will run towards the narrower end. For let AB be the tube, and CD a column of water contained Plate 5. within it; when the fluid moves, the velocity of the end D will be to that of the end C Fig. 5: reciprocally as the cavity of the tube at D, to that at C, (by this Prop.) that is, (El. XII. 2.) reciprocally as the square of the diameter at D, to the square of the diameter at C; but the attracting ring at D is to that at C, singly as the diameter at D to the diameter at C Now, since the effect of the attraction depends, as much upon the velocity of that part of the fluid where it acts, as upon its natural force, its effect at D will be greater than at C; for though the attraction at D be less in itself than at C, yet its loss of force upon that account, is more than compensated by the mechanical advantage it has arising from hence, that the velocity of the fluid in that part is more increased than the force itself is diminished at D. The fluid will therefore move towards B. See on this subject Mr. Vince's Principles of Hydrostaticks, p. 65-9.
PROP. V. In bended cylindrical tubes, fluids at rest will be at the same height on each side.
In the tube ADMN, filled with water to the height AD, the water cannot descend from Plate 5. AD, without rising towards MN. The water in each side of the vessel may therefore be Fig. 1, considered as two forces acting upon each other in contrary directions : and consequently these two masses of Auid will only be at rest when their momenta are equal, that is, (Book II. Prop. XI. Cor.) when the quantities of matter are inversely as the velocities, or (Prop. IV.) directly as the area of the section through which it flows. Thus, at the sections BP, KH, the momenta are equal, when the quantities of matter, or cylindrical masses of fluid are as the areas of the sections, that is, as the bases of the cylinders ADBP, FGHK. But cylinders are as their bases (El. XII. 11.) only when their perpendicular heights are equal. Therefore the momenta of the two cylinders of fluid will be equal, and consequently the mass will be at rest, only when the perpendicular heights of each column are equal.
Exp. 1. In a bended tube of large but unequal bore, water will rise to the same height on each side.
Let water spout upward through a pipe, having a small orifice inserted into the bottom of a deep vessel ; it will rise nearly to the height of the upper surface of the water in the vessel. The resistance of the air, and of the falling drops, prevents it from rising perfectly to the level.
Cor. If, therefore, a pipe convey a fluid from a reservoir, it can never carry it to a place higher than the surface of the fluid in the reservoir.
SCHOL. In this demonstration, we do not consider the velocity with which the two columns of Auid are moving, but the velocity with which, if they move at all, they must begin to move. And since, if their perpendicular height is the same, the velocity with which they must begin to move will be inversely as their respective quantities of matter, they cannot begin to move but with equal momenta ; and their motions must be in contrary directions, because one column cannot descend without making the other ascend: therefore those equal momenta would destroy each other. These two columns then, making a continual effort to move with equal momenta in contrary directions, counterbalance each other.
PROP. VI. The pressure of Auids is proportional to the base, and the
. perpendicular height of the fluid, whatever be the form of the vessel or quantity of the fluid.
Let the fluid be contained in a perpendicular cylindrical vessel. In such a vessel, ABCD, because the whole weight of the fluid, and no other force, presses directly upon the bottom CD, the pressure (by Prop. I.) must be as the quantity, that is, (El. XII. 11, 14.) as the base and perpendicular height of the Auid.
Plate 5. Fig. 6.
Case 2. Let the fluid be contained in a perpendicular vessel, the bottom of which is equal to that of the cylinder in the last case, but its top narrower than the bottom.
Let the vessel DBLP, have the portions of its base LA, CP, each equal to OR. From
Or thus : Because (by Prop. V.) if a tube were inserted at NT, of the diameter OR, the water being at the height DB, would rise to the level FE, there must at NT, be an upward pressure towards F sufficient to fill up the columns of fluid FELA, that is, equal to the weight of as much water as would fill the space FENT. Consequently the re-action, that is, the pressure upon the base LA must be equal to the weight of as much water as would fill FENT. But the base LA supports this re-action, and likewise the weight of the water NTLA, which are together equal to the weight of DBOR. The base LA, therefore, sustains a pressure equal to the weight of the column DBOR. And every equal portion of the base may, in the same manner, be shewn to sustain an equal pressure. Therefore, the pressure on the base is the same in vessels of the form supposed in this case, as in cylinders of equal bases, and of the same altitude with these vessels. The same may be shewn with respect to a vessel of the form of plate 5, fig. 7.
Let the vessel be of the same base and altitude, but have its top wider than the
Let the fluid of the vessel be divided into strata EF, GH, IK, &c. Let us also imagine Plate 5. the bottom of the vessel C to be moveable, that is, capable of sliding up and down the
Fig. 8. narrow part of the vessel, from C to GH. Let it further be supposed that this moveable bottom, is drawn up or let down with a given velocity, while the vessel itself is fixed and immoveable ; it is evident the lowest stratum, which is contiguous to the bottom, will be raised or let down with the same velocity, and will therefore have a momentum proportional to that velocity, and the quantity of matter it contains : but (by Prop. IV. Cor.) the rest of the strata will have the same momentum : consequently, the momentum of all taken together, that is, of the whole fluid, is the same as if the vessel had been no larger in any one part than it is at the bottom, for then the momentum of each stratum would also have been as great as that of the lowest. The pressure, therefore, or action of the fluid, with which it endeavours to force the bottom out of its place, is as the number of strata, that is, the perpendicular height of the fluid, and the magnitude of the lowest stratum, that is, the base.
In the inclined cylindrical vessel ABNI, as much as the fluid is prevented from pressing Plate 5. upon the base NI, by being in part supported by the side of the vessel AN,
Fig. 9. pressure upon the base increased by the re-action of the opposite side BI, which is equal to the action of the former, because the fluid, pressing every way alike at the same depth below the surface, exerts an equal force against both the sides. The base NI is therefore pressed with the same force with which it would be pressed, if the fluid contained in the vessel ABNI was included in the vessel EDIO, having an equal base, and the same perpendicular height with the vessel ABNI ; that is, (by the first case) the pressure is as the base NI and altitude CN.
Since then, the pressure upon the base of vessels, either wider or narrower at the top than the bottom, and likewise the pressure upon the base of vessels inclined to the horizon, is equal to that upon the base of a cylindrical vessel of the same base and height, the sides of which are perpendicular to the horizon ; and since the pressure upon the base of such a cylinder is as its base and height; the pressure upon the bottom of all vessels filled with fluid, is propotional to their base and perpendicular height.
Let two tubes of different forms be successively applied to the same moveable circular base, suspended by a wire, passing from the centre of the base through the tubes, to the beam of a balance : when the different tubes are filled to the same height, it will require the same, weight at the opposite end of the balance to keep the base from sinking. Hence any quantity of fluid, how small soever, may be made to balance and support ang quantity how great soever, which is called the bydrostatical paradox.