789 



HYDROSTATICS. 



HYDKOSTATICS. 



790 



tube F E is inserted in an orifice near the bottom, and through this 

 tube water is poured into the cylinder, till the boards A B and c D are 

 at any distance asunder within the limits allowed by the leathern 

 sides. Then, if any weight be placed on the board c D, it will cause 

 the water to rise in the tube E P to a certain height, suppose a ; and 

 the weight of the small column a 6 of water may be considered as 

 holding in equilibrio the weight applied on c D ; which will, in fact, be 

 found to be equal to that of a cylinder of water whose base is the area 

 of the board c D, and whose height is equal to a b. 



If the tube E F were made- to decline from the vertical so as to take 

 any oblique position E v' ; it would follow, since the pressure of a fluid 

 by gravity depends on the vertical height only of the column, that the 

 fluid in the tube, from the same pressure on c D, would rise till its 

 upper surface is in a horizontal plane a a' passing through a : and 

 the weight of the column of fluid must be estimated by the area of the 

 horizontal section at 6' multiplied by the vertical height of a' above b'. 

 Hence, also, any fluid in a bent tube A CB (Jig. 3) will stand in each 



Fig. 3. 



branch, the tube being open at both ends, at the same vertical height 

 above c, the lowest point. Thus water, which is conveyed in pipes 

 from a reservoir, will occupy all the bends of the pipes, and rise at the 

 further extremity up to a horizontal plane passing through the surface 

 of the water in the reservoir, provided no vertical bend be higher than 

 that level. 



The power produced by the HYDRAULIC PBESS depends on the prin- 

 ciple exhibited in the above experiment ; and this experiment is, at the 

 same time, the proof of that equality of pressure which it has been 

 said that the particles of a fluid exert in every direction. 



The pressure exerted by a fluid against the whole side of a vessel 

 containing it, or against a surface immersed in it, whether that side or 

 surface be plane or curved, is equal to the weight of a column of the 

 fluid having the surface pressed for a base, and the distance of the 

 tipper surface of the fluid from the centre of gravity of the former 

 surface for its altitude. For let DB (ti</. 1) be the position of the 

 surface pressed, and let an indefinitely small area at c on that surface 

 be represented by m, and be pressed by the weight of the filament cd 

 of fluid above it ; then, since every part of the indefinitely small area 

 may be supposed to be at the same vertical depth, which may be repre- 

 sented by n, it follows that the pressure on c will be proportional to 

 m n. And the same thing will hold good with respect to every point in 

 the surface D B. Therefore this surface may be conceived to be pressed 

 by an infinite number of parallel forces, whose points of application are 

 on the same surface, and whose intensities are represented by the 

 products of the elementary areas into the distances of those areas from 

 the upper surface c D of the fluid. But, by the theory of parallel 

 forces in mechanics, the resultant of all those forces is a force whose 

 intensity is represented by the sum of all the elementary areas (that is, 

 the area of the surface pressed) multiplied into the distance of its point 

 of application, that is, of the centre of gravity of the surface, from the 

 same surface c D. By this theorem the pressure of water against the 

 walls ot reservoirs, lock-gates, ate., may be determined. 



The pressure against one side of a cubical vessel filled with a fluid is 

 equal to half the pressure upon the base ; for the areas of the base and 

 of-each side are equal to one another, but the centre of gravity of the 

 former is at a distance from the tipper surface equal to the whole 

 depth, and that of the latter at a distance equal to the half depth. It 

 is shown moreover in treatises on hydrostatics, that if a hollow cone 

 'standing on its base be filled with a Huicl, the pressure on the base will 

 be equal to three times the weight of the fluid; that the pressure 

 against the interior surface of a hollow sphere filled with a fluid is also 

 three times the weight of the fluid. Again, if a vessel of any figure be 

 full of a fluid, and have over every |>art of the sides and bottom a 

 vertical filament of the fluid reaching to the upper surface, the whole 

 pressure in a vertical direction on the bottom and sides of the vessel 

 will }# pqii.il ., the weight of all the fluid. Lastly, the pressure 

 exerted on tbu Hides of a vessel, estimated perpendicularly to the base, ix 

 equal to the weight of a rectangular prism of the fluid whose height in 

 equal to that of the fluid, and whose base is a parallelogram, one side 

 of which is equal to the height of the fluid, and the other to half the 

 perimeter of the vvoel. (Vince's ' Hydrostatics ;' Gregory's ' Me. 

 chanic*/ Ac.) 



It is of importance to determine the place of the centre of pressure 

 against the side of a vessel filled with a fluid, or against a surface which 

 is immersed in it; that is, to find the situation of a point in that 

 surface, at which a force being applied in a contrary direction to that 

 in which the fluid presses, the surface will be kept in equilibrio. 



Let, for simplicity, the side or surface pressed be rectangular, and ill 

 a vertical position ; let, also, 6 represent the breadth, and a the altitude 

 of the surface, or depth of the fluid : then 4 a will be the depth of the 

 centre of gravity below the upper surface of the fluid. Now if x be 

 the distance of any elementary area of the side below the same 

 upper surface, such elementary area will be expressed by 6 d x ; and 

 the pressure of the fluid against it being proportional to the depth, 

 will = bxdx. Then the tendency of that pressure to turn the side of 

 the vessel round, about its upper extremity, which is supposed to be a 

 horizontal line, will be bz?dx; consequently the whole tendency of 

 the fluid to turn the side round in that manner will be expressed by 

 fbx-dx, which between the limits x = o at the top, and x = a at the 

 bottom, is equal to \ b a?. But, if P be the required place of the centre 

 of pressure, and its distance from the upper surface of the fluid be 

 represented by p, the tendency of the same pressure applied at p to 

 turn the side about its upper extremi ty, will be ^ a 2 6 p ( J a 2 6 being the 

 horizontal pressure of the fluid against that side). Therefore we have 

 | a 3 b = 4 * bfi or P a ! * aa ' ' 8 > * ne centre of pressure is at a 

 distance from the upper surface equal to two-thirds of the depth of 

 the vessel or fluid. And, by writers on hydrostatics, it is proved that, 

 in all cases, when the surface pressed is symmetrical on each side of a 

 line joining the centres of gravity and pressure, the latter coincides 

 with the centre of percussion in mechanics. 



When a triangle in a vertical position is immersed in a fluid so that 

 its vertex coincides with the upper surface of the fluid and its base is 

 horizontal, the distance of the centre of pressure from the vertex is 

 equal to three-fourths of the perpendicular of the triangle. And when 

 a circle is so placed in a fluid with its upper part just touching the 

 surface, the distance of the centre of pressure from that part is equal 

 to five-eighths of the diameter. 



The equality of the pressures in every direction, at any point in a 

 fluid mass, is the cause that, if a solid body be plunged in a fluid, the 

 pressure of the fluid immediately under it will tend to raise the body 

 upwards with a force equal to the weight of the fluid displaced. But 

 the weight of the body is a force acting vertically from above down- 

 wards ; and, consequently, in an opposite direction to that caused by 

 the reaction of the water. Since therefore the volumes of the body 

 and of the displaced water are equal to one another ; if their weights 

 or densities should be equal, the body would remain in equilibrio 

 in whatever situation it were placed in the fluid. But should these 

 weights or densities be unequal, the body would make an effort to 

 ascend or descend, according as its density is less or greater than that 

 of the fluid ; and, in order to counteract these tendencies, it would be 

 necessary to use a force equal to the difference between this weight of 

 the body and of the displaced fluid. Hence, if a solid body be weighed 

 in a fluid, it will be found that its weight, compared with that of the 

 same body in vacuo, will be less than in the latter case by the 

 weight of an equal volume of the fluid ; and, consequently, when a 

 body is weighed in a fluid, as water or air, the true weight, or that 

 which would be obtained in vacuo, will be found by adding to the 

 observed weight that of an equal volume of the fluid. 



When a body floats in a fluid, in order to bring its upper surface to 

 coincide with that of the fluid, it must evidently be loaded with a 

 weight equal to the difference between the weight of the body or of 

 the displaced fluid, and the weight of a volume of the fluid equal to 

 that of the whole body. The weight which a floating body will thus 

 bear is denominated the buoyancy of the body ; and on the principle 

 here stated depend the common rules for finding the buoyancy of 

 rafts, vessels, &c. 



If a solid body float in equilibrio in a fluid, the centres of gravity of 

 the body and of the displaced fluid must evidently be in one vertical 

 line ; otherwise the upward action of the fluid below, which neces- 

 sarily haa its resultant in a vertical line passing through the centre of 

 gravity of the place occupied by the body, would produce in the latter 

 a rotatory motion contrary to the hypothesis. This circumstance has 

 given rise to three denominations respecting the equilibrium of 

 floating bodies. First, if the centre of gravity of the body should be 

 below that of the displaced fluid, the body is said to possess a stable 

 or firm equilibrium ; s<> that if any derangement should take place 

 from accidental causes, the body would, after a few oscillations, recover 

 its former position. If the centre of gravity is above that of the 

 displaced fluid, the body is in circumstances similar to those of a cone 

 when placed on its vertex, that is, it is liable to be immediately over- 

 turned ; and hence the body is said to float with a tottering or unstable 

 equilibrium. And if the said centres should exactly coincide, ttie 

 body would float in any position whatever : this is denominated an 

 equilibrium of indifference. The first case is that of a cylinder whose 

 axis is less than the diameter of its base ; the second is that of a 

 cylinder whose axis is greater ; and the last is that of a homogeneous 

 sphere. 



The absolute weight of a given volume of any solid or fluid body is 

 called its specific gravity. In this country, for convenience, it is 

 customary to consider one cubic foot as the given volume, and to 



