456 



H Y D R O D Y NA M I C S. 







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 Klosting 

 Bodies. 



CHAP. IV. 



Definition. 



ON THE EQUILIBRIUM AND STABILITY OF FLOATING 

 BODIES. 



We have already seen, (Chap. 1 1. Prop. I. p. 429,) that 

 when a body is in equilibrium in a fluid, its weight is 

 always equal to that of the fluid displaced ; and that 

 the centre of gravity of the floating body when ho- 

 mogeneous, must be. situated in the same vertical line 

 with the centre of gravity of the part submersed, or 

 of the fluid displaced, Prop. II. p. 430. From the 

 equality between the weight of the body and that of 

 the displaced fluid, the upward pressure of the fluid is 

 exactly capable of balancing . the downward tendency 

 of the body ; but unless these two forces are directly 

 opposed to each other by passing through the same 

 point, the solid body will have a rotatory motion, in- 

 stead of a position of perfect equilibrium. In order, 

 therefore, to determine the positions in which a body 

 will float permanently on the surface of a fluid, we have 

 only, after the specific gravity of the body has been as- 

 certained, to discover in what positions the solid can be 

 placed, in order that the centre of gravity of the solid 

 and of the part immersed may be in the same vertical 

 line. The solid, however, will not float permanently 

 in every case, when these centres of gravity are situated 

 in the same vertical line ; for there are examples, in 

 which the body cannot remain in this position of equi- 

 librium, but will actually, assume another, in which 

 it will continue to float permanently. Mr Atwood 

 has illustrated this by the case of a cylinder, whose 

 specific gravity is to that of the fluid on which it floats 

 as 3 to 4-, and whose axis is to the diameter of its base 

 as 2 to 1. When the .cylinder which we suppose to 

 be 2 feet long, and its base 1 foot in diameter, is held 

 in the fluid, with its axis in a vertical line, it will 

 sink to a depth of 1^ feet; but as soon as it ceases to 

 be supported, it instantly oversets, and remains float- 

 ing with its axis horizontal. If the cylinder, instead of 

 being 2 feet long is only 6 inches, or one-half the dia- 

 meter of its base, it will sink to the depth of |ths of its 

 diameter, or 4^ inches, and will float permanently in 

 that position. In this last case, if the axis of the cylin- 

 der is not exactly in a vertical line, but a little inclined 

 to it, the cylinder will still settle permanently with its 

 axis in a vertical line. 



Hence it is obvious, that there are different kinds of 

 equilibrium. 



1st, The equilibrium of stability, or that which is ex- 

 hibited in the short cylinder 6' inches long, which floats 

 permanently in a given position. 



2d, The equilibrium of instability, or that which is ex- 

 hibited in the cylinder 2 feet long, which -oversets, al- 

 though the centre of gravity of the solid, and that of 

 the part immersed, are in the same vertical line. In 

 this case, the equilibrium is as perfect as in the first 

 case ; for while the centres of gravity are in the same 

 vertical line, the solid must continue erect ; but the 

 slightest deviation of the centres of gravity from that 

 line creates a rotatory motion, from which the solid ne- 

 cessarily oversets. 



3d, The equilibrium of indifference, or the insensible 

 equilibrium in which the solid floats indifferent to mo- 

 tion, and without any tendency to recover its position 

 when inclined from it, or to incline itself farther. The 

 equilibrium of indifference takes place, when the pro- 

 portion between the axis of the cylinder and the dia- 

 meter of its base is greater than 1 to 2, and less than 2 

 to 1. This kind of equilibrium is exhibited in a ho- 



Equilibti- 

 "n and 



mogeneous sphere, or in a homogeneous cylinder, float- 

 ing with its axis horizontal.- 



If a solid floats permanently on a fluid surface, and 

 if it is moved from its position of equilibrium by 

 any external force, the resistance which the solid op- 

 poses to this inclination is called the stability offloating ; 

 and the horizontal line round which it moves, is call- 

 ed the axis of motion. 



It would be impossible in a work like this, to enter 

 at great length into a subject so difficult and profound 

 as the present. We shall, therefore, content ourselves 

 with stating the general principles relative to the stabi- 

 lity of floating bodies, and with investigating the differ- 

 ent positions of stability and instability which they as- 

 sume ; and in doing this shall freely avail ourselves of 

 the labours of Mr Atwood, whose papers on the stability 

 of floating bodies are remarkable for their perspicuity. 

 In arranging, abridging, and sometimes simplifying his 

 demonstrations, we trust we shall .do an important ser- 

 vice to the reader. 



PROP. I. 



To determine the stability of bodies floating on a 

 fluid at any angle of inclination from a given position 

 of equilibrium. 



Let EDHF be a vertical section through the centre PLATE 

 of gravity G, of a homogeneous solid, whose figure is CCCXV. 

 symmetrical with regard to the axis of motion, and let it F'g' 1< 

 float on the surface HABL of the fluid, O being the 

 centre of gravity of the part immersed. The line 

 GOC will therefore be perpendicular to AB. If by an 

 external force the solid is inclined through an angle 

 KGS, the solid will take the position IRLMN, and the 

 part immersed will now be WRMNP. Hence, as the 

 part XWI is raised out of the water, and the corre- 

 sponding and equal part XNP immersed, the centre of 

 gravity which would otherwise have been at E (taken 

 so that GO = GE) will now be transferred to some 

 other point Q. Having drawn QS parallel to GO, and 

 EYand ZGz perpendicular to SQ, it is obvious that the 

 upward pressure of the fluid will be exerted in the line 

 QS, with a force equal to the weight of the body, or 

 that of the fluid displaced, and this force will have the 

 same tendency to turn the body round its axis of mo- 

 tion, as if it were applied at the point Z. In determin- 

 ing, therefore, the position which bodies assume on a 

 fluid surface, and the stability with which they float, 

 it is necessary only to find the perpendicular distances 

 GZ between the two vertical lines which pass through 

 the centre of gravity of the solid and the part immersed. 



Since the weight of the body continues the same, 

 the portion IXW, elevated from the fluid in consequence 

 of the inclination, must always be equal to the portion 

 PXN which is immersed. Hence, supposing a to be the 

 centre of gravity of IXW, and/that of NXP, then the 

 centre of gravity Q will be at a distance from E, corre- 

 sponding to the change produced, by removing the fluid 

 1WX to the position NXP. In order to determine, 

 by a geometrical construction, the line GZ, let fall 

 the perpendiculars a b, fc, and in the line EY drawn 

 parallel to AB, take ET, so that ET : b c = volume 

 IWX : volume WRMP. Through T draw FTS paral- 

 lel to GO, then the centre of gravity required will be 

 somewhere in FS, and because ER: EG = sin. KGS 

 : rad. the line GO = EG being supposed given, the 

 line ER will be determined, and being taken from ET 

 already found, will leave RT or GZ the perpendicular 

 distance required. 



S 



