428 



HYDRODYNAMICS. 



Pressure rnerly, that a portion of the water is fro/en, so as to form 

 briiim 1 "!'" a tu ' >e ^ ' ce e P* wnose diameter is equal to that of the 

 Fluid*, particle p, without any change taking place in the pres- 



In this case, the particle p is ob- 



PLATE 

 CCCXlfl. 



Fig. 6. 



sure sustained by n. . t ( 



viously pressed downwards with the weight of the 

 column ep; and, consequently, the measure of this 

 pressure is the absolute weight of the column ep. But 

 as the particle is in equilibrio, it must be pressed with 

 this force in every direction. 



The proposition is also true of a particle situated at 

 m, for drawing the horizontal line mg; and supposing a 

 syphon of ice fghm to be formed, it is obvious that the 

 column of fluid in the branch mh is in .equilibrio 

 with, or balanced by, the column in gh , consequently 

 the particle of water at m is pressed with the same 

 force as the particle at g, that is, with a column of wa- 

 ter whose height is fg. 



Cor. Hence it follows, that every particle of a ves- 

 sel containing fluid is pressed with a force equal to a 

 column of fluid, whose base is the particle, and whose 

 height is the depth of the particle below the surface ; 

 for, since the particle of fluid adjacent to this particle 

 of the vessel is pressed in every direction with this 

 force, it must exert the same force against that particle 

 of the vessel. 



PROP. IV. 



The pressure exerted by a fluid upon any given por- 

 tion of the vessel which contains it, is equal to a co- 

 lumn of the fluid whose base is the area of the given 

 portion, and whose altitude is the depth of the centre 

 of gravity of the portion below the fluid surface. 



Let nin be the given portion of the vessel ABCD 

 tilled with fluid, and let us conceive this portion to be 

 occupied by any number of particles m, o, p, n, &c. then 

 the pressure sustained by each of these particles, by 



Prop. III. will be x m a-f.oxo.r-f- pXp^+Xz, 

 &c. ; but, by the property of the centre of gravity or 

 inertia, (See MECHANICS,) the sum of these products is 

 equal to the distance EF of the centre of gravity E, 

 from the surface at F, multiplied into the number of 

 particles m, n, o, p ; that is, 



Fig. 7. 



Fig. 8, 9. 



hydrostatic 

 paradox. 



Fig. 10. 



Fig. U. 



+nXz=EFx.>0,/>; consequently, since m,n,o,p 

 represents the area or the number of particles in the 

 given portion mn, the pressure upon =EF x nn. 



Cor. 1. It follows from this proposition, that the 

 pressure sustained by the bottom of the vessel is not 

 the same as the weight of the fluid contained in the 

 vessel. In the cylindrical vessel shewn in Fig. 7, or 

 in any vessel, whatever be its shape, in which the sides 

 are perpendicular to ks bottom, the pressure upon the 

 bottom is accurately measured by the weight of the water 

 which it contains ; but.in vessels of all other shapes, such 

 as Fig. 8, 9, the pressure on the bottom is measured by 

 m n x m #, which in Fig. 8 is much less than the weight 

 of water in the vessel, and in Fig. 9 much greater. 



Cor. 2. The truth of what is called the Hydrostatic 

 Paradox, is easily deduced from the preceding proposi- 

 tion. Let ABCDEFGH, Fig. 10, be a vessel filled with 

 water, then, by the proposition, the pressure upon GF= 

 GF X GI, however narrow be the column ABCD, that is, 

 the pressure exerted upon the bottoms of vessels filled with 

 Jluid does not depend upon the quantity ofthejiuid which 

 they contain, but solely upon its altitude. In like man- 

 ner, it is obvious from Prop. II. that the water will 

 .land at the same level ab AB, Fig. 1 1, in the two com- 

 municating vessels abed, ABCD, consequently, any par* 



firm of fluid abed, hotvever small, will balance any par* Pressure 

 lion of fluid ABCD, however great. * K 1 ui i'- 



Cor. 3. The pressure exerted upon the sides of a pj""j b 

 vessel, perpendicular to it* base, is equal to the weight ^_ _- 

 of a rectangular prism of the fluid, whose height ia 

 equal to that of the fluid, and whose base is a parallel- 

 ogram, one side of which is equal to the height of the 

 fluid, and the other to half the perimeter of the vessel. 



Cor. 4. The pressure against one side of a cubical vessel 

 is equal to half the pressure against the bottom; and the 

 pressure against the sides and bottom together, is equal 

 to three times the pressure against the bottom alone. 

 Hence, by Cor. 1. the pressure against both the sides 

 and bottom together, is equal to three times the weight 

 of fluid in the vessel. 



Cor. 5. The pressure exerted upon the surface of a 

 hemisphere full of fluid, is equal to the product of that 

 surface multiplied by its radius. 



Cor. 6. The pressure sustained by different parts of 

 the sides of a vessel, are as the squares of their depths 

 below the surface. Hence, these pressures will be re- 

 presented by the ordinates of a parabola, when the 

 depths are represented by its abscissae. 



DEFINITION. 



The centre of pressure is that point of a surface ex- Centre ef 

 posed to the action of a fluid, to which, if a force equal pressure. 

 to the whole pressure upon the surface were applied, 

 the effect would be the same as it is when the pressure 

 is distributed over the whole surface. 



PROP. V. 

 To find the centre of pressure. 



Let it be required to find the centre of pressure P, F ig. PLAT E 

 11, on the side of a cubical vessel ABCD. Let G be CCCXIIF. 

 the centre of gravity of the surface, then the pressure Fl B - llm 

 exerted against this surface will be BC X BC x GB, or 



, since in the case of a cube or rectangle, GB= , 



* 



and since the pressure must be equal to the sum of all 

 the elementary pressures upon the elementary portions 



F/ we have ^- X PB=/BC X F/X FB x FB, or 



BO 1 /* 



= /F/X FB 2 . But the sum of the elementary 



pressures F/X FD 2 compose a pyramid whose base is 

 =BC', and whose altitude is BC, consequently, by the 

 property of the centre of inertia (See MECHANICS) 



of pressure, is two-thirds of the depth of fluid in the 

 vessel. 



COR. The centre of pressure coincides with the cen- 

 tre of percussion, as the centre of percussion is also 

 two- thirds of the height of the body. 



.SECT. II. On the Pressure and Equilibrium of Fluids 

 of Variable Density. 



DEFINITION. 



THE absolute weights of different bodies that have 

 the same bulk are called their specific gravities or den- 

 sities, and any body that, under the same bulk, is hea- 

 vier than another, is said to be specifically heavier. 



