17. If the components parallel to the axes of the forces acting on an element of fluid at (x, y, z) be proportional to y3 +2λyz+z2, z2+2μzx+x2, x2+2vxy+y2, shew that if equilibrium be possible we must have 2λ = 2μ = 2v = 1. 18. A closed cylinder, with its axis vertical, is just filled with liquid which rotates uniformly about a generating line; find the whole pressures on the base, the upper end, and the curved surface. 19. A sphere of 5 feet diameter revolves about a vertical axis 3 feet distant from its centre with angular velocity 2√g; if it be just filled with homogeneous fluid, prove that the resultant pressure and the whole pressure on the sphere are respectively and 323 times the weight of the fluid. 20. If the force at any point is given by a potential, and if a tube of small but variable circular section be imagined in the liquid, the whole pressure upon which is P, prove that dᏢ døds + 2πpг = 0 where r is the radius of the section, and s is measured along the axis of the tube. 21. The density of a liquid, contained in a cylindrical vessel, varies as the depth; it is transferred to another vessel, in which the density varies as the square of the depth; find the shape of the new vessel. 22. A circular cone, of vertical angle, is just filled with water, and has a generating line rigidly attached to a horizontal plane. The plane is caused to revolve with uniform angular velocity about a vertical axis through the apex of the cone find the greatest velocity which will allow of the pressure being zero at the highest point; and in this case find the whole pressure on the base. 23. A straight rod, every particle of which attracts with a force varying inversely as the square of the distance, is surrounded by a mass of homogeneous incompressible fluid; find the form of the surfaces of equal pressure. 24. A quantity of heavy liquid is attracted to a fixed centre, by a constant force the intensity of which is equal to the force of gravity, and is supported by a horizontal plane. Find the form of the surfaces of equal pressure; and also the pressure on the plane, proving that when the plane passes through the centre of force it is equal to four-thirds of the weight of the liquid. Find also expressions for the pressure on the plane when it is either above or below the centre of force. 25. The interior of a homogeneous shell, bounded by two non-concentric spherical surfaces, and attracting according to the law of nature, is partially filled with homogeneous liquid which revolves uniformly with it round the line passing through the centres of the spheres; prove that the free surface is a paraboloid of revolution. 26. A rigid spherical shell is filled with homogeneous inelastic fluid, every particle of which attracts every other with a force varying inversely as the square of the distance; shew that the difference between the pressures at the surface and at any point within the fluid varies as the area of the least section of the sphere through the point. 27. At the vertex of a solid cone (vertical angle 2a) there is a centre of force the attraction to which varies as the distance; and a given quantity of liquid is in equilibrium under the action of this force alone. Determine the form of α 2' 4 its free surface. If the volume of the liquid beπа3 cos2 3 prove that the whole pressure on the surface of the cone upasin a, where p is the density of the liquid and μ the absolute force. = μ 28. An open vessel containing liquid is made to revolve about a vertical axis with uniform angular velocity. Find the form of the vessel and its dimensions in order that it may be just emptied. 29. A quantity of liquid (gravity being supposed not to act) just fills a hollow sphere, and is repelled from a point in = the surface of the sphere by a force μ x distance: if the liquid revolve round the diameter passing through the centre of force with uniform angular velocity w, find the whole pressure on the surface of the sphere. If, by diminishing the angular velocity one half, the pressure is also diminished one half, shew that w2 = 6μ. 30. A rectangular plate of thin metal of given size is bent and held so that two opposite edges are parallel and in the same horizontal plane, and the vertical ends are then closed by flat plates; if this vessel be filled with water, find its form when the whole pressure upon its curved surface is a maximum. 31. An infinite mass of homogeneous fluid surrounds a closed surface and is attracted to a point (0) within the surface with a force which varies inversely as the cube of the distance. If the pressure on any element of the surface about a point P be resolved along PO, prove that the whole radial pressure, thus estimated, is constant, whatever be the shape and size of the surface, it being given that the pressure of the fluid vanishes at an infinite distance from the point 0. 32. A right cone, whose weight may be neglected, is suspended from a point in its rim; it contains as much fluid as it can prove that the whole pressure upon its surface is (cos (0 + a)}3 cos2 a (cos (-a) 1 πpghs 3 sin a cos where h, 2a, are the height and vertical angle of the cone, and is determined from 3 sin 20 4 sin 2 (0-a). = 33. A vessel formed by the revolution of a cardioid r = a(1 cos ) about its axis which is vertical (vertex upwards) is just filled with water and rotates about that axis with uniform angular velocity. Find this velocity, when π the line of no pressure is given by =. Find also the 0= pressure at any other point, and the points of maximum pressure. 34. A closed vessel full of liquid is made to revolve with uniform angular velocity about a vertical axis through its highest point; shew that the total pressure of the liquid on the surface is increased by Akpo2; A being the area of the surface, and k the radius of gyration of the surface about the vertical axis. 35. All space being supposed filled with an elastic fluid the particles of which are attracted to a given point by a force varying as the distance, and the whole mass of the fluid being given, find the pressure on a circular disc placed with its centre at the centre of force. 36. A closed vessel in the form of the surface formed by the revolution of the curve, r = a (1 — cos 9), is just filled with water, and held with the cusp upwards, and the axis vertical; calculate the whole pressure and the resultant vertical pressure on its surface, and prove that these quantities are in the ratio 263 210. 37. Circles are drawn having their centres on the axis of z and touching at the origin the plane xy, and the position of a point P is defined by r, 0, 4, where r is the radius of the circle through P, centre C, is the angle OCP, and the inclination of the plane OCP to a fixed plane through the axis of z; prove that dp P = · R (1 − cos 0) dr + T sin ✪ dr + Trdė + Nr sin 0 dø, where mR, mT, mN are the forces, on an element m of liquid at P, along CP, along the tangent to the circle at P, and perpendicular to the plane of the circle. 38. A mass m of elastic fluid is rotating about an axis with uniform angular velocity w, and is acted on by an attraction towards a point in that axis equal to μ times the distance, being greater than w2; prove that the equation of a surface of equal density p is μ (x2 + y2 + z2) − w2 (x2 + y2) = k log 39. A quantity of liquid, the density of which varies as the depth, fills an inverted paraboloid, of latus rectum c, to a height h; prove that, if it be poured into a vessel of the form generated by the revolution round the axis of x of the curve, a*y2 = 2ch2x (a — x) (2a — x), where a is any constant, its density will vary as the square of its depth. 40. A mass of self-attracting liquid, of density p, is in equilibrium, the law of attraction being that of the inverse square: prove that the mean pressure throughout any sphere of the liquid, of radius r, is less by pr than the pressure at its centre. 2 5 41. A quantity of liquid, the density of which varies as the depth, is transferred from a cylinder to a hemispherical bowl, which it just fills; find the new law of density, and prove that the whole pressures on the curved surface of the cylinder and on the surface of the bowl are in the ratio of 80: 243. 42. A closed hollow right circular cone, standing on its flat base on a horizontal plane, is just filled with a liquid, the density of which varies as the depth. The vessel is then inverted and held with its axis vertical and its vertex on the horizontal plane. Prove that the resultant pressure on the curved surface is unchanged in magnitude, and that the potential energy of the liquid is changed in the ratio 2 {T (3)}2 : 3г (3), it being assumed that the potential energy is zero when the liquid is let out on the plane. |