Every element of the liquid moves in a straight line with a constant acceleration g sin a, and since the forces on any element are the resultant fluid pressure upon it and its weight, it follows that the resultant of these forces is mg sin a, parallel to the plane, m being the mass of the element. It is hence easy to see that the resultant pressure is perpendicular to the plane and is equal to mg cos a. Whatever be the shape of the element, the resultant fluid pressure upon it in the direction parallel to the plane is zero, and therefore it follows that the surfaces of equal pressure are planes parallel to the inclined plane, and that the surface of the liquid is the plane through its highest point parallel to the inclined plane. If there be no air within the vessel the pressure at the surface is zero, it being given that the vessel is only just filled, or, which is the same thing, just not filled. Taking z as the depth of a point in the liquid below the surface thus defined, and drawing a thin cylinder or prism from this point to the surface, the pressure on the base will be the resolved part of the weight of the prism perpendicular to the plane, and, as before, As in the previous article the whole pressure and resultant pressure may be obtained, employing g cos a for g. The reasoning employed in this and the preceding example is applicable to any analogous case, that is, to any case in which the fluid, while bodily in motion, is within its own mass in a state of relative equilibrium. MISCELLANEOUS PROBLEMS. 1. A triangle ABC is immersed in a fluid, its plane being vertical, and the side AB in the surface. If O be the centre of the circumscribing circle, prove that pressure on triangle OCA: pressure on triangle OCB :: sin 2B: sin 24. 2. Water is gently poured into a vessel of any form; prove that when so much water has been poured in that the centre of gravity of the vessel and water is in the lowest possible position, it will be in the surface of the water. 3. A closed hollow cone is just filled with liquid, and is placed with its vertex upwards; divide its curved surface by a horizontal plane into two parts on which the whole pressures are equal. Also do the same when the vertex is downwards. 4. If the cone be placed on its side on a horizontal table, compare the whole pressures on the curved surface and the base. 5. A triangle ABC has its plane vertical and the side AB in the surface of a liquid; divide it by straight lines drawn from A into n triangles on each of which the pressure shall be the same. 6. A solid displaces 1 1 and 1/ of its volume respectively when it floats in 3 different fluids; find the volume it displaces when it floats in a mixture formed, 1st, of equal volumes of the fluids, 2nd, of equal weights of the fluids. 7. A float is made by attaching to a hemisphere (radius r) a cone of the same base, and axis of length 2r. If this will float in a fluid A with the cone just immersed, and in a fluid B with the hemisphere just immersed, compare the densities of A and B. 8. Compare the whole pressures on the curved surface and plane base of a solid hemisphere, radius r, immersed in water with its base horizontal and at a depth (r). Note. The centre of gravity of the portion of the surface of a sphere contained between two parallel planes which intersect or touch the surface is equidistant from the planes. 9. A parabolic lamina floats in a liquid with its axis vertical and vertex downwards; having given the densities, σ, p, and the height (h) of the parabola, find the depth to which its vertex is immersed. 10. A heavy sphere, weight W, is placed in a vertical cylinder, filled with atmospheric air, which it exactly fits. Find the density of the air in the cylinder when the sphere is in a position of permanent rest, r being the radius and h the height of the cylinder. 11. If half a second be the unit of time, and the acceleration of a falling body that of acceleration, determine the ratio of the unit of density to the density of distilled water, in order that the formula, p=gpz, may give the pressure in pounds. 12. A cone, of given weight and volume, floats in a given fluid with its vertex downwards; shew that the surface of the cone in contact with the fluid is least, when the vertical angle of the cone is 2 tan-1 1 √2* 13. A hollow sphere is filled with fluid and a plane drawn through the centre divides the surface into two parts, the total normal pressures upon which are as m: 1; find the position of the plane and the greatest and least values of m. 14. A uniform tube is bent into the form of a parabola, and placed with its vertex downwards and axis vertical: supposing any quantities of two fluids of densities p, p' to be poured into it, and ", r' to be the distances of the two free surfaces respectively from the focus, then the distance of the common surface from the focus will be 15. If water be the standard substance, 4 feet the unit of length, and 2 seconds the unit of time, find the unit of weight in the equation W=gpV, assuming 32 as the value of g when a foot and a second are units. 16. If there be n fluids arranged in strata of equal thickness, and the density of the uppermost be p, of the next 2p, and so on, that of the last being np; find the pressure at the lowest point of the nth stratum, and thence prove that the pressure at any point within a fluid whose density varies as the depth is proportional to the square of the depth. 17. A fine tube, bent into the form of an equilateral triangle with its vertex upwards and base horizontal, contains equal quantities of two liquids, each liquid filling a length of the tube equal to a side of the triangle. Prove that the height of the surface of the lighter fluid above that of the heavier the altitude of the triangle :: p'-p : p'+p, p and p' being the densities. 18. A cylinder is filled with equal volumes of n different fluids which do not mix; the density of the uppermost is p, of the next 2p, and so on, that of the lowest being np: shew that the whole pressures on the different portions of the curved surface of the cylinder are in the ratio 12: 22: 32 :...: n2. 19. Equal volumes of n fluids are disposed in layers in a vertical cylinder, the densities of the layers, commencing with the highest, being as 1 : 2 ......: n; find the whole pressure on the cylinder, and deduce the corresponding expression for the case of a fluid in which the increase of density varies as the depth. Also, if the n fluids be all mixed together, shew that the pressure on the curved surface of the cylinder will be increased in the ratio 3n: 2n+1. 20. A hollow cone floats with its vertex downwards in a cylindrical vessel containing water. In the position of equilibrium the area of the circle in which the cone is intersected by the surface of the fluid bears to the base of the cylinder the ratio of 6: 19. Prove that, if a volume of water equal to 19 ths 8 of the volume originally displaced by the cone be poured into the cone, and an equal volume into the cylinder, the position in space of the cone will remain unaltered. 21. A body is wholly immersed in a liquid and is capable of motion about a horizontal axis. It is found that the total pressure of the fluid on the surface is increased by A when the body is turned through one right angle, and further increased by B when it is turned through another right angle. Prove that the difference between the greatest and least pressures on the surface is √2(42+ B2). 22. A frustum of a right cone, formed by a plane parallel to the base and bisecting the axis, is closed and filled with fluid by means of a thin vertical pipe, which is also filled. If the top of this pipe be on a level with the vertex of the cone, find the whole pressure on the curved surface, and if this bear to the pressure on the base the ratio of 7 to 6, find the vertical angle of the cone. 23. If in the last example the base be removed, and the vessel then placed on a horizontal plane, and filled to the top of the pipe, find the least weight of the vessel which will prevent its being lifted. 24. An open cylindrical vessel, axis vertical, contains water, and a cone the radius of which is equal to that of the cylinder is placed in the water vertex downwards. Prove that, in the position of equilibrium, if the density of the cone be one-eighth of the density of water, the surface of the water will be raised above its original level through a height equal to one-twentyfourth the height of the cone. 25. A solid cone of wood (density σ) rests with its base on the plane base of a large vessel, and water (density p) is then poured in to a given height; B a piece of the same wood is then attached by a string to the vertex of the cone so as to be wholly immersed; find what the size of the piece must be in order that it may just raise the cone. 26. An elliptic lamina floats with its plane vertical in a liquid of twice the density of the lamina, 1st, with its axis vertical, 2ndly, with its axis horizontal; determine in each case whether the equilibrium is stable or unstable, the lamina being displaced in its own plane. 27. A regular tetrahedron has one of its faces removed and is filled with fluid; the other faces, which are capable of moving round the lowest point, are kept together by means of strings which join the middle points of the horizontal edges of the vessel; shew that the tension of the strings is to the weight of the fluid as√3 to 4√2. N |