FLUID PRESSURE.] 



MECHANICAL PHILOSOPHY. HYDROSTATICS. 



751 



Fig. 159. 





are the same abstract numbers ; but the former 

 refers to quantity of matter, and the latter to 

 weight. 



g = 1000 oz. avoirdupois : the weight or pressure 

 communicated by gravity to a cubic foot of dis- 

 tilled water, at a temperature of 60 Fahrenheit. 



All these symbols, except the pressure or weight g 

 produced by gravity, are abstract numbers : the symbol 

 W is of course the concrete quantity weight, as in common 

 language. And weight including under this term both 

 pressure and tension is the only concrete magnitude 

 with which we have to do in Statics. 



FLUID PRESSURES. 1. When a fluid is at rest, 

 its upper surface is horizontal. 



Let P Q (Fig. 159) be any two points in the upper sur- 

 face of a fluid at rest, and 

 the vertical lines P A, Q B 

 be drawn, A B being a 

 horizontal line of particles 

 of the fluid. Then as A B 

 is in equilibrio, the pres- 

 sures on the extremities 

 A, B in the horizontal di- 

 rection must be equal ; but the pressure on A, in the 

 direction A B, is the same as the vertical pressure of the 

 ;mn of particles PA ; and the pressure on B, in the 

 direction of B A, is the same as the vertical pressure of 

 olumn of particles Q B. But these pressures are 

 equal ; therefore the column of particles P A is equal to 

 the column of particles Q B ; that is, the points P, Q 

 are equally distant from the horizontal line A B, and are 

 therefore themselves in a horizontal line ; and in the 

 same way is it shown that any two points on tke upper 

 surface are in a horizontal line, and therefore that the 

 entire surface is a horizontal plane. It is evident, too, 

 that if A B be the upper surface of a fluid sustaining a 

 lighter fluid that does not mix with it, the separating 

 surface A B will be horizontal ; for, from what is shown 

 above, every point of A B is pressed alike. 



It must be observed, that what are called vertical 

 lines, are lines perpendicular to the surface of the earth ; 

 and the upper surface of a fluid, here shown to be per- 

 pendicular to these verticals, will therefore be a surface 

 parallel to that of the earth a spherical surface ; but it 

 is customary to call but a small portion of such a large 

 surface, a plane. 



It ia upon this property of fluids that the value of 

 rig lao> levelling instruments de- 



pends ; that is, instru- 

 ments which serve to 

 show whether or not any 

 two points are at the 

 same horizontal level. 

 The common level consists 

 of a bent tube (see Fig. 160), open at the ends, which 

 are turned up. The tube is nearly filled with a fluid, 

 generally mercury, which supports two floats bearing 

 sights, with a slender wire across, the wires being at equal 

 distances from the floats. When held in the hand, the 

 two surfaces bearing the floats are necessarily horizontal, 

 however the tube itself may be inclined ; and conse- 

 quently the two wires are always on the same horizontal 

 plane ; also, whatever other objects, seen through the 

 Bights, may be on the same level as the wires, must like- 

 wise be in the same horizontal plane, or on the same 

 leveL 



2. The pressure perpendicular to a surface immersed 

 in a fluid, is equal to the weight of a column of the fluid 

 whose base is the area A of the surface, and whose alti- 

 tude is the perpendicular depth of the centre of gravity 

 of the surface. 



For let the vertical length of any bnear column of 

 particles pressing on the surface be a,, and the point of 

 the surface pressed be P,. Regarding this point as a 

 small area, we have for the pressure or weight of the 

 column, S'yP^,, S being the specific gravity of the fluid. 

 In like manner, for another vertical column of length a 2 , 

 pressing on another point P 2) we have S^P^Oi ; and so 



on. Hence, the whole pressure perpendicular to the 

 surface is 



Sg (Pid! + 



+ P,a 3 -f . . . .) 



But if G be the depth of the centre of gravity of the 

 assemblage of points P 1} P 2 , P 3 , <fec., that is, of the pro- 

 posed surface A, then by Statics, 



P,a 1 +P 2 a 2 +P 3 a 3 - r -. . . .-(P x + P 2 + P 3 +- . . . ) G 

 .'. Pressure perp. to the surface = SAG . g 



where AG is the volume V of a column of fluid of base 

 equal to the area A and height G ; so that W=SV<jf. 

 Hence, if a given area A, immersed in a fluid, revolve 

 round its centre of gravity, the pressure perpendicular 

 to its surface must be the same in every position. Also 

 if the area be a rectangle, the pressure upon it, when it 

 forms the bottom of a vessel, will be double the pressure 

 upon it when it forms one of the vertical sides ; so that 

 the pressure upon the four sides of a cubical vessel filled 

 with liquid, is equal to twice the pressure on the base, 

 that is, to twice the weight of the fluid. 



If the sides of a vessel filled with fluid are all vertical, 

 the entire pressure on the sides is equal to the weight of 

 a column of the fluid whose base is the rectangle formed 

 by developing the sides into a plane, and whose height 

 is half that of the fluid. 



By means of the preceding proposition, it is easy to 

 find the amount of pressure sustained by a rectangular 

 dam, or by a pair of flood-gates. If we multiply the 

 area of the dam, or flood-gate, by half the depth of the 

 water, we shall have the volume of water the weight of 

 which will be the pressure. For example : let the water 

 be 8 feet deep, and the breadth of the flood-gate 6 feet ; 

 then the area of the surface pressed is 48 feet : hence, 

 48 X 4 = 192 cubic feet of water= 12000 pounds = 5 / T tons. 



Since the centre of gravity of a straight line is at its 

 middle point, if two straight lines a t , a 3 be placed 

 vertically in a fluid, the upper extremities of each 

 being on the surface, then pressure on o x : pressure on 



that is, the pressures are as the squares of the lengths. 

 But if the lines are inclined to the surface of the fluid 

 at the angles a,, a a respectively, the perpendicular 

 depths of the centre of gravity are 



(>! sin. t>i, and ia a sin. a s 

 The pressures are therefore as 



fli X i<i sin. ai : a 2 X ia sin. a,, 

 or as cti 2 sin. a^ ' &a sin. aa ; 



that is, as the squares of the lines into the sines of the 

 angles of inclination, or as the squares of the lines them- 

 selves if the inclinations are equal. 



If a triangle be immersed at any inclination in a fluid, 

 with its vertex downwards Fig. 161. 



and its base horizontal, and 

 at the surface of the fluid, 

 then the pressures on any 

 two lines DE, FG (Fig. 161), 

 the one as distant from the 

 base as the other is from 

 the vertex, will be equal. 



For draw C M bisecting 

 A B, and therefore bisecting 

 the parallels D E, F G in m 

 and n. 



Now the pressures on 

 D E, F F are as D E . Mm : 

 F G . M?i ; that is, as D E . Cn : F G . Cm. But Cn : 

 Cm : : F G : D E ; hence the pressures are as D E . F G : 

 F G . D E ; that is, they are equal. 



If the triangle be reversed, C being at the surface of 

 the fluid, the student may easily prove that the pressures 

 on any two parallels are as the squares of those parallels, 

 or as the squares of their depths. 



3. If one of the sides of a vessel filled with fluid be a 



