FLOATING BODIES.] MECHANICAL PHILOSOPHY. HYDROSTATICS. 



759 



the upward force at the meta- 

 centre, above this centre, actin 

 with a leverage, and unopposed, 

 must restore to the axis its ver 

 ticality. But if the metacentre 

 be below the centre of gravity of 

 the body, then the upward pres- 

 sure has a contrary effuct, and 

 the oblique axis, on which it 

 acts, is turned still more out of 

 the vertical direction. 



For instance, G (Fig. 174) 

 being the centre of gravity of 

 the floating body, if G A' be the 

 axis disturbed from its original 

 vertical position G A, by any force which no longer acts, 

 and if m be the metaceutre, or point where the line of 

 support gm, passing through the centre of gravity g of 

 the displaced fluid, meets G A', it is plain that the pro- 

 gress of the disturbance is ar- 

 rested, and G A' restored to its 

 vertical position. If, on the 

 other hand, the metacentre be 

 situated, in reference to the 

 centre of gravity G of the body, 

 as in Fig. 175, that is, below G, 

 then the upward pressure upon it 

 assists the deviation from the 

 vertical ; and thus G A' goes on 

 inclining more and more. 



It is thus obvious, that when 

 the metacentre is above the centre 

 of gravity of the disturbed body, 

 the disturbance will be rectified, and the equilibrium 

 which has been disturbed is stable. 



But when the metacentre is below the centre of gravity, 

 the disturbance becomes aggravated ; and the equilibrium 

 thus disturbed is unstable. It is possible that the line of 

 support gm may pass through G, the centre of gravity of 

 the body : in this case, the body remains at rest in its 

 new position, for the conditions of equilibrium are then 

 as rigorously fulfilled as at first : the equilibrium in such 

 circumstances is said to be indifferent ; it is also some- 

 times called neutral equilibrium. This kind of equi- 

 librium evidently has a place in a floating sphere and 

 cylinder, and also in a body, of whatever shape, floating 

 in a fluid of equal specific gravity as itself that is, if it 

 be allowed to call the equilibrium in this case floating, 

 no part of the body being above the surface of the fluid. 

 From what has now been said, it ia obvious how im- 

 portant it is, in the construction and loading of ships, to 

 make the centre of gravity of the whole body so low 

 that the metacentre may always be above that point, 

 even for a considerable disturbance. The centre of 

 gravity is sometimes elevated above its original position, 

 in a gale of wind, by the shifting of the cargo : if the 

 elevation be sufficient to bring it above the metacentre, 

 the vessel must capsize. 



Example. A rectangular beam, the transverse vertical 

 section of which is a square, and the specific gravity of 

 which is one-half that of the water on which it floats, 

 rests with one of its faces parallel to the surface of the 

 water : it is required to determine whether the equi- 

 librium is stable or unstable in reference to a slight 

 transverse disturbance. 



As the transverse vertical sections are all equal squares, 

 it will be sufficient to consider merely the middle section 

 A H C D (Fig. 176). 



The centre of gravity of this is at G, the middle of the 

 square ; that is, G H $ A B ; and if g H= JA B, then g 

 will be the centre of gravity of the displaced fluid ; for 

 the body floats half in and half out of the water, as 

 by hypothesis it is only half the weight of its bulk of 

 water. E F is the plane of flotation, and H K the ver- 

 tical axis through the centre of gravity G. 



Let now the body be turned round its centre of 



gravity, through a small angle (Fig. 177), FG/= 6, 



: the line of notation to be e/, and the former 



vertical axis to become the oblique line H K ; we have 



to find a 1 , the centre of gravity of the displaced fluid 



e f C B in this new position of the body, and thence the 



Fig. 178. 



point O where the vertical g' cuts the axis H K : this 

 point O is the metacentre. 



Fig. 177. 



gm 



Let m, n be the centres of gravity of the portions 

 E G e, F G/, and let h be the centre of gravity of the 

 portion eGFCB. 



Then by STATICS, hg : mg : : EGe : eGFCB and 

 hg 1 : n/ :: FG/: eGFCB. 



But E G e = F G/; consequently, hg : mg : : hg' : ntf 

 therefore gg' is parallel to inn 



e^JFCB' by the &ni P r P ortion > 



EGe 



" EFCB KGe 



Now, as the angle of disturbance 6 is considered as 

 very small, the portion EGe must be insignificant in 

 comparison with EFCB: hence 



EGe > 



Moreover, on account of the smallness of 0, sin. and 

 may be regarded as equal ; and we shall then have for 

 the three quantities m n, E G e, E F C B, the values 

 mn = 2Gm=4GE, GE being the distance of the 

 vertex G horn the base, or from the middle of the base, 

 the base being small in consequence of the minuteness 

 of 0. 



EGe = 4GE 2 x 0, EFCB = 2GE 2 



' O), 99' = i GE X i = 4 GE x = *GE 



X angle gOg' 



But 



gg 



GE = Og, that is 



Oy X angle gOg' 



Og = i GH 

 .'. OH = ^ + i) GH = j. GH .'. GO = J GH 



Hence, the equilibrium is unstable, and therefore the 

 beam will roll partly over, though displaced through only 

 a very small angle. When such a position is reached 

 that the diagonal A C becomes vertical, the body will 



