Chat*. XVII.
1155
ported, it will fall and float horizontally: a cylinder C inches in length, and of the samediameter, will sink 4‘ inches, and in that position will float permanently: there are,consequently, different kinds of equilibrium, as that of stability, instability, and of in-difference.
The stability of bodies floatingon a fluid at any angle of in-clination from a given position ofequilibrium may be thus deter-mined : —suppose E DII F a ver-tical section through the centre ofgravity G of a homogeneous solid,whose figure is symmetrical withregard to the axis of motion, andthat it floats on the surface H A B Lof the fluid, O being the centre ofgravity of the part immersed : theline GOC will be perpendicularto A B: if, by an external force,the solid is inclined through anangle KGS, it will take theposition SRLMN, and the partimmersed will be W R M N P :hence, as the part X W I is raised out of the water, and the corresponding and equal partX N P immersed, the centre of gravity, which would otherwise have been at E, will betransferred to the point Q: by drawing QS parallel to G O and E Y, and ZGz perpen-dicular to SQ, it is obvious that the upward pressure of the fluid will be exerted in theline QS, with a force equal to the weight of the body, or that of the fluid displaced, andthis force will have the same tendency to turn the body round its axis of motion, as if itwere applied at the point Z: to determine, then, the position which bodies assume on afluid surface, and the stability with which they float, it is only required to find the perpen-dicular distances G, Z, between the two vertical lines which pass through the centre ofgravity of the solid and the part immersed: as the weight of the body continues the same,the portion I X W elevated from the fluid, in consequence of the inclination, must be equalto P X N, the portion immersed : supposing a to represent the centre of gravity I X W, and/ that of N X P, the centre of gravity Q will be at a distance from E corresponding tothe change produced by removing the fluid IWX to the position NXP. To determineby a geometrical construction the line GZ, let fall the perpendicular ab,fc, and in theline E Y, drawn parallel to A B, take ET, so that ET: 5c = volume IWK : volumeWRMP; through T draw FTS parallel to GO, the required centre of gravity will thenbe somewhere in FS: and because ER : EG=sin. KGS : rad. ; the line GO = EG,being supposed given, the line E R will be determined, and being taken from E T, alreadyfound, will leave ET or GZ, the perpendicular distance required.
"When the body of a system is removed from its place, the corresponding motion of thecommon centre of gravity, estimated in any given direction, is to the motion of the bodymoved, and estimated in the same direction, as the weight of that body is to the weight ofthe whole system : considering, then, I It M N as a system of bodies, whose common centreof gravity is E, and that the body I \V X, whose centre of gravity is a , has moved, or beentransferred to NXP, whose centre of gravity is /, we shall have volume WRMP, orA DII P : volume I W X, or WXP = 6c : E T, the motion of one body, and the centreof gravity of the system.
Suppose, then, V= the volume of the immersed portion of the floating body,
A = the volume NXP, or the part immersed in consequence of theinclination,
h = G O, the distance between the centre of gravity of the wholesolid, and that of the part immersed,s = to the sine of the angle of inclination KGS ,b = be, the space through which A has been transferred.
Then, 6: E T = V : A, andET=—
but, E R : EG, or GO = i: 1, we have ER=As;
6 A
consequently, RT = ET-ER, or GE = -y— hs.
When a solid body floats upon a fluid of greater specific gravity than itself, and has at-tained a state of equilibrium, the magnitude of the body is to that of the part immersedbelow the plane of floatation, as the simple specific gravity of the fluid is to that ofthe floating body : therefore, to find the magnitude of any solid so floating, subtract the
4 e 2
s
.L
Fig.1754