its centre of gravity, and the weight of the fluid displaced acting upwards in a vertical through the centre of gravity of the fluid displaced; hence the weight of the fluid displaced must be equal to the weight of the solid, and the line joining the centres of gravity of the solid and of the fluid displaced must be vertical. 23. To find the positions in which a solid can float in equilibrium. Let f (x, y, x)=0 be the equation to the surface of the solid, + + =1 the equation to the surface of the fluid, the α b C centre of gravity of the solid being the origin of the co-ordinates: V the volume of the fluid displaced by the solid; X, Y, Z the co-ordinates of the centre of gravity of the fluid displaced; p the density of the fluid; W the weight of the solid. Then gp V will be the weight of the fluid displaced, and = X Y Ꮓ the equations to the line joining the centres of gravity of the solid and of the fluid displaced. But when the solid is at rest, its weight is equal to the weight of the fluid displaced, and the line joining the centres of gravity of the solid and of the fluid displaced is perpendicular to the surface of the fluid, therefore V = £x£y£x 1, V.X= sx Sy fz ∞, V. Y=Sx Sy fz Y, V. Z= Sx fy Sz &• The limits of the integrations being determined by the equations And having found the different values of a, b, c, we know the equation to the surface of the fluid corresponding to each position of equilibrium of the solid. The section of a solid floating in equilibrium made by the surface of the fluid, is called the plane of floatation. 24. A solid floating in equilibrium is slightly elevated or depressed, and then left to itself; to determine its motion. Let ADB (fig. 14.) be the position of the solid at the end of the time t from the beginning of its motion, CP a vertical meeting the surface of the fluid a Cb in C and the plane of floatation APB in P, PC = x, a the space through which the solid was elevated or depressed, A the area of the plane of floatation, the volume of the fluid displaced by the solid when at rest, p the density of the fluid: then the moving force on the solid in the direction PC will be the difference between its weight and the weight of the fluid displaced = gp A. CP, and the mass of the solid = p V, therefore the accelerating force on the solid in the direction PC A A Hence it appears that the body will oscillate vertically, the time of an oscillation being 25. A To determine whether the equilibrium of a solid is stable or unstable. Let the equilibrium of the solid be slightly disturbed by making it revolve through a very small angle in a vertical plane, without altering the quantity of fluid displaced; then the equilibrium of the solid will be stable or unstable, according as the pressure of the fluid tends to make it return to, or recede farther from its original position, that is according as a force acting upwards in a vertical through the centre of gravity of the fluid displaced, tends to diminish or increase the angle through which the solid has revolved. 26. When the equilibrium of a solid is slightly disturbed as in the preceding Article; to find the vertical through the centre of gravity of the fluid displaced. Let G, H (fig. 15.) be the centres of gravity of the solid and of the fluid displaced by it, when floating in equilibrium. Let a plane through GH meet the plane of floatation in ACB, and the surface of the solid in ADB. Suppose the solid to revolve through a very small angle 0 in the plane ADB, so that the quantity of fluid displaced may be the same as before; and let ADB meet the surface of the fluid in a Cb. Draw MF ver tical through the centre of gravity of the fluid displaced by the solid in its new position, and mp, nq vertical through the centres of gravity of the wedges ACa, BCb. Then if the plane of floatation be symmetrical with respect to the plane ADB, mp, nq and consequently MF will be in the plane ADB. Draw HFE parallel to ab. Then from the two ways of making up the solid ADb, (vol. a Db). FE + (wedge AC a). C'm = (vol. ADB). HE - (wedge BCb). Cn. And if x, y be the co-ordinates of any point in the boundary of the plane of floatation, ACB being the axis of x, and CY (perpendicular to ACa) the axis of y, (wedge ACa). Cm=20 x2y, from C to A, (wedge BCb). Cn=20, x2y, from C to B. But 2fy from C to A+ 2y from C to B=2x2y from A to B =k2 A, k2 A being the moment of inertia of the plane of floatation round CY, .. (wedge ACa). Cm + (wedge BCb). Cn=0k2 A. And wedge AC a wedge BCb, therefore = 20fxy from C to A=20fxy from C to B, therefore C is the centre of gravity of the plane of floatation. Also if the volume of the fluid displaced = V, (vol. ADB). HE - (vol. a Db). FE=V.HF, HF=HM.0, .. V.HM= k2 A. The point M, in which FM ultimately cuts HG is called the metacentre. A force acting in the direction FM will tend to diminish or increase the angle HMF according as M is above or below G, therefore the equilibrium of the solid is stable or unstable according as M is above or below G. If the plane of floatation be not symmetrical with respect to ADB, let a Yb (fig. 16.) be the section of the solid made by the surface of the fluid; and let H, G, &c. be the projections of H, G, &c. in (fig. 15.) on the plane a Yb. Draw pr, q8, MN perpendicular to ab. It may be proved as before, that the centre of gravity of the plane of floatation lies in CY, and that V.HN= k* A0, Also V.MN+ (wedge Ya Y').pr = (wedge YbY'). q & ; (wedge Ya Y').pr=0. Sa fyxy, from C to a ; V.MN = ᎢᎾ . And the equilibrium will be stable or unstable according as H and G lie on the same or opposite sides of MN. 27. To determine the small oscillations of the solid DC (fig. 15.) when left to itself, after its equilibrium has been slightly disturbed, the solid being symmetrical with respect to the plane ADB Let the figure represent the position of the solid at the end of the time t from the beginning of the motion; and let p be the density of the fluid, K the radius of gyration of the solid revolving round G in the plane ADB, HG=c; then, retaining the notation of Art. 26., the moment of the pressure of the fluid tending to turn the solid round G in the direction FMG and the moment of inertia of the solid round G in the plane ADB=K2pV, SECTION III. ON THE EQUILIBRIUM OF ELASTIC FLUIDS ACTED ON BY GRAVITY. ART. 28. To measure the pressure of the atmosphere. Let a glass tube ABC (fig. 17.) closed at the end A, be bent at B, so that the branches AB, BC may be parallel and AB about thirty one inches longer than BC. Then if AB and part of BC be filled with mercury, and placed in a vertical position, the mercury will rise in BC, and sink in AB, (leaving a vacuum in the upper part of the tube,) till the pressure of the mercury at the common surface of the air and mercury in BC is equal to the pressure of the atmosphere. Let P, Q be points in the upper and lower surfaces of the mercury; through P, Q draw horizontal planes cutting a vertical HK in H and K. Let II be the pressure of the atmosphere, σ the density of the mercury; then (Art. 12. Cor. 1.) the pressure of the mercury at Q=go.HK; and this must be equal to the pressure of the atmosphere at Q when the mercury is at rest, .. Ilgo.HK. An instrument of this description furnished with a scale for measuring HK, is called a barometer. 29. The expansion of mercury between the temperatures 10 of melting snow and boiling water is of its volume at the 555 former temperature, and the increment of its volume is very nearly proportional to the increment of its temperature. Hence if σ, σ be the densities of mercury at 0°, t (Centigrade) |