186 EXERCISES ON STABILITY (12) Find the dynamical stability in foot-tons at 30° of a rectangular pontoon 100 ft x 20 ft x 10 ft draft, having a GM. of 2 ft. (13) The curve of statical stability of a vessel is a segment of a circle of radius twice the ordinate of maximum statical stability, which is 2500 tonsfeet; estimate the total dynamical stability of the vessel, the angle of vanishing stability being 85°. (14) The curve of stability of a vessel is a common parabola, the angle of vanishing stability is 70°, and the maximum moment of stability 4000 ft-tons. Prove that the statical and dynamical stabilities at 30° are 3918 and 1283 ft-tons. (15) A cylindrical vessel with a flat bottom is free to turn about a horizontal axis through its C.G. Prove that if a little molten metal be poured in, the vertical position is unstable; that it does not become stable until the depth of the metal exceeds c-c2-a2, where a is the radius of the cylinder, and c the height of the centre of gravity above the base; and that it is again unstable when the depth exceeds c+√c2—a2. Determine the weight which must be fixed to the bottom of the vessel so as to make the equilibrium stable at first. (16) A canister containing water floats in a liquid, with its axis vertical. Prove that its stability for angular displacements will be unaffected if a certain weight of water is removed and a spherical ball of equal weight is placed in the cylinder so as to float in the water partially immersed, even though the ball touch the cylinder. OF FLOATING BODIES. 187 (17) A vessel is constructed to carry petroleum in tanks formed by the sides of the vessel and a middle line and transverse bulkheads. If the water plane of the vessel be rectangular, and the tanks extend over 3rds the length, and are not full, investigate the stability at a small angle of heel, having given -Length of vessel 240 ft, breadth 36 ft, c.G. of laden vessel 14 ft from top of keel, C.B. of laden vessel 8 ft from top of keel, displacement 2500 tons, and S.G. of petroleum 08. (18) A vessel is of box form, 300 ft x 50 ft, and draws 20 ft of water when intact. A bunker, 10 ft wide, 10 ft deep (6 ft below, 4 feet above the water line), containing coal, extends a length of 100 ft amidships at each side of the vessel. The C.G. of the vessel is 18 ft above the keel, find the GM.— (1) In the intact condition. (2) With both bunkers riddled, the inner and end bulkheads remaining intact. (19) A vessel in the form of a cube of side 12a containing liquid is placed so as to rest on the top of a fixed sphere of radius 5a. Neglecting the weight of the vessel prove that there will be stability provided the depth of the liquid is between 4a and 6a. (20) Prove that a cylindrical kettle of radius a and height h will be in stable equilibrium on the top of a spherical surface of radius c, when the water inside occupies a height intermediate to the roots of the equation x2-hx+}a2+n(h2 — 2ch) = 0 ; the weight of the kettle being n times the weight of water which fills it. 188 EXERCISES ON STABILITY. (21) A cup whose outside surface is a paraboloid of revolution of latus-rectum 7, and whose thickness measured horizontally is the same at every point and very small compared with 7, has a circular rim at a height h above the vertex, and rests on the highest point of a sphere of radius r. If water be now poured in until its surface cuts the axis of the cup at a distance from the vertex, and if the weight of water be four times that of the cup, the equilibrium will be stable, if (22) Prove that if a thin conical vessel of vertical angle 2a and weight W, whose c.G. is at a distance h from the vertex, is resting upright in a horizontal circular hole of radius c, it will become unstable when a weight P of liquid is poured into it to a depth x, so as to make Px−2(P+W)c cot a+ Wh cos2a positive. (23) A cylindrical vessel, floating upright in neutral equilibrium, will really be stable if the radius of curvature at the water line of the vertical cross section is greater than the normal cut off by the medial plane. (24) Prove that the metacentric height given by (P/W)b cosec ($ 94) can be made correct to the second order for the ship, when P is removed, by adding to it (§ 112) (P/W)(c—r1), where 1 CHAPTER V. EQUILIBRIUM OF FLOATING BODIES OF REGULAR FORM AND OF BODIES PARTLY SUPPORTED. OSCILLATIONS OF FLOATING BODIES. 126. The Equilibrium of a floating Cylinder, Cone, Paraboloid, Ellipsoid, Hyperboloid, etc. When the body has the shape of one of these regular mathematical forms, the curves of flotation F and of its evolute C, of buoyancy B, and of the prometacentres M, or the metacentric evolute, can be determined by various theorems introducing interesting geometrical applications of the properties of these curves and surfaces. For a prismatic or cylindrical body like a log, floating horizontally in water, the various surfaces are cylindrical and we need only consider their curves of cross section. If the section is an ellipse, these curves of flotation and of buoyancy are also ellipses; and the determination of the position of equilibrium will depend on the problem of drawing normals from the c.G. of the body to the ellipse of buoyancy, or tangents to its metacentric evolute; and two or four normals or tangents can be drawn according as the C.G lies outside or inside this evolute. If the sides of the log in the neighbourhood of the water line are parallel planes, the curve of flotation reduces to a point, and the curve of buoyancy becomes a parabola (§ 103). 190 CURVES OF BUOYANCY If the submerged portion of the log is triangular, or more generally if the log is polygonal or if the sides in the neighbourhood of the water line are intersecting planes, the curves of flotation and of buoyancy are similar hyperbolas of which the cross section of these planes are the asymptotes. When the cross section of the log is rectangular or triangular, the curves of flotation and of buoyancy are composed of parabolic and hyperbolic arcs, interesting figures of which, by Messrs. White and John, will be found in the Trans. Inst. Naval Architects, March, 1871; also by M. Daymard, I.N.A., 1884. If the outside shape of the body is an ellipsoid or other quadric surface, then according to well-known theorems the surfaces of flotation and buoyancy are similar coaxial surfaces; just as in the sphere, from which the ellipsoid may be produced by homogeneous strain. If the surface of the body is a quadric cone, the surfaces of flotation and of buoyancy will be portions of hyperboloids of two sheets, asymptotic to the cone. |