7-1 



MECHANICAL PHILOSOPHY. HYDROSTATICS. [FLCID PEEBSTOB. 



t..- 



>sity to the weight of the fluid displaced by it, and 

 d vertically upwards through tho centre of 

 | gravity of the fluid displaced. 



\\ 'hi-ii the solid floats at rest, the weight of tho fluid 

 displaced is equal to the weight of the solid ; and tho 

 centra of gravity of the solid and that of the displaced 

 fluid are in the same vertical line. 



he pressure of the fluid on the portion of surface 

 immersed, and the weight of the solid, are the only forces 

 acting : hence, as the body is at rest, the resultant of 

 the pressures on it must be equal and opposite to the 

 downward pressure, or weight, of the body, and act in 

 the same vertical straight line. But since, as just shown, 

 the resultant of the pressures (equal to the weight of the 

 displaced fluid) is directed upwards through the centre 

 ravity of the fluid displaced, and the weight of the 

 * >lid is directed downwards through the centre of gravity 

 r.f the solid, it follows that the resultant of the pressures, 

 that is, the weight of the fluid displaced, is equal to the 

 weight of the solid, and that the centres of gravity of 

 Ix.th an- in tho same vertical line. If, instead of floating 

 on the surface, the body be kept from sinking by a force, 

 acting along a string, just sufficient to keep the body at 

 rest, the conditions of equilibrium may be found as 

 f ollows : 



Let G be the centre of gravity of the body suspended 

 in the fluid by the string 

 FA: let E C, G B be ver 

 ticals through the centre of 

 gravity of the displaced 

 fluid and through the centre 

 of gravity of the body. 

 I/et also 



W = weight of the body, 

 and W' = weight of 

 fluid displaced ; 

 T = tension of the string ; 

 V = volume of the fluid 



displaced ; 



and D= density of the fluid. 

 The body is kept at rest by the forces W = D Vg, 

 acting in the direction E C ; W, acting in the direction 

 G B ; and T, acting in the direction F A. These three 

 forces must therefore be all in one plane ; and T must 

 be = W W. Through G draw E G perpendicular to 

 E C, F A ; then as the body is at rest, the moments of 

 the forces W, W to turn the body about K in opposite 

 directions, are equal (STATICS, p. 716) . 



.'. W.GK=W'.EK=DVj.EK. 



Hence the weight acting over the pulley A upon the 

 body at K, just sufficient to keep the body from sinking, 

 i- \V W; and in order that it may be kept from turn- 

 there must be the condition \V.GK = W'.EK. 

 ild tho body be lighter than the fluid, and tend to 

 float instead of to sink, then the force on F to prevent 

 its rising will of course bo W' W ; the other condition 

 to prevent turning remaining the same. 



As noticed at page 749, if the vessel be full before 



Slinging the body into the fluid, the quantity of the 

 ii'1 which the immersion of the body causes to run 

 over will occasion no diminution of the weight of the 

 vessel and contents, nor yet any modification of the 

 pressures on the bottom and sides : for the body merely 

 fills the place of the bulk of fluid which its immersion 

 drive* out of the vessel The circumstances as to the 

 weight and pressures are the same as if the fluid, that 

 originally occupied the space now filled by the body, 

 bad become solidified while at rest in the vessel The 

 weight P in the above diagram, and which measures the 

 tension of the string, measures the excess of weight in 

 the body above the weight of the water it displaces. 



<TRE OF PRESSURE. The pressures of 

 a flui.l against the different points of a plane surface may 

 be regarded as a system of parallel forces, acting perpen- 

 dicular to the plane : the resultant of these forces is 

 therefore perpend icuUr to tho plane ; and the magnitude 

 or intensity of the resultant has already been shown 

 (page 761) to be equal to a column of the fluid whose 



Tig. KS. 



base is tho surface pressed, and whose altitude is equal 

 to the depth of the centre of gravity of the surface below 

 the level of the fluid. Tho centre of pressure is that 

 point of the surface to which, if a single force equal and 

 opposite to the resultant of the pressures were applied, 

 the plane would be kept at rest. 



PROBLEM. To find the centre of pressure of a fluM <>n 

 a triangle whose base is horizontal and at tho surface of 

 the fluid. 



Let A B C (Fig 168) be the triangle, and draw C M to 

 bisect the base AB ; also let D E, F G be two lines 

 parallel to A B, and drawn 

 so that the distances M m, 

 n C may be equal These 

 lines being horizontal are 

 uniformly pressed through- 

 out, so that the centre of 

 pressure on each is at its 

 middle point; and, as already 

 proved at page 751, the 

 pressure on one, is the same 

 as the pressure on the other. 

 Consequently we may regard 

 the extremities m, n of the 

 line m n as pressed by equal 

 forces : the resultant of these 

 is therefore equal to the sum of both applied to the 

 middle point P, which point is evidently the middle 

 point of C M. 



Whatever two lines be taken equidistant from M and 

 C, the point of application P of the resultant remains the 

 same ; and as the whole pressure on the triangle may be 

 regarded as made up of all these linear pressures, it 

 follows that the resultant pressure must pass through 

 P, which is therefore the centre of pressure on the 

 triangle. 



PROBLEM. To find the centre of pressure of a fluid 

 on a parallelogram, one of whose sides coincides with the 

 surface of the fluid. 



Let AC (Fig. 169) be the parallelogram, and draw 

 E F bisecting the opposite sides D C, A B. The centre 

 of pressure is necessarily in E F, as the pressures on 

 each side of it are equal. Draw E A, E B, as also 

 horizontal lines H J, K L, &c. Then the pressure on 

 Fig. 169. one of these lines, as H J, 



I is to the pressure on A B 



"* as E o to E F ; that is, as 

 m n to A B. Hence, re- 

 presenting the pressure on 

 A B by tho line A B, the 

 lines m, n, p, q, &c., will 

 correctly represent the 

 pressures on H J, K L, etc. 

 Consequently if the fluid 

 were all removed, and a pressure equal to that originally 

 on A B be applied to that line, and also a pressure to 

 every line pq, mn, <fec., in the triangle, the pressures 

 being always proportional to the lengths of these paral- 

 lels, the parallelogram, in which the pressed triangle is 

 inscribed, will still bo at rest. But the resultant of all 

 the pressures, thus uniformly diffused over the triangle, 

 must pass through the centre of gravity of the triangle. 

 Hence the centre of gravity of the triangle is tho centre 

 of pressure of the fluid on the parallelogram : and con- 

 sequently the centre of pressure on the parallelogram, 

 one of whose sides is at the surface of the fluid, is on the 

 bisecting line E F, and at a depth equal to two-thirds 

 the depth of the opposite or lowest side of the paral- 

 lelogram. Again : suppose the upper side of the paral- 

 lelogram to be below the surface of the fluid, but parallel 

 to it ; let H J, for instance, be the upper side of the paral- 

 lelogram, and 1 > C the surface of the fluid. Then, as 

 shown alxive, the pressure on the parallelogram H C may 

 be replaced by a pressure uniformly diffused over the 

 triangle E m n ; and the pressure on the parallelogram 

 A C, by the extension of the pressure on E m n uniformly 

 over the triangle E A B : hence the pressure on the 

 parallelogram A J may be replaced by a pressure 

 uniformly spread over the trapezium A m n B. Con- 



