2692. A rough sphere of radius a rolls in a spherical bowl of radius a + b in a state of steady motion, the normal making an angle a with the vertical: prove that the time of small oscillations about this position is

HYDROSTATICS.

[In the questions under this head, a fluid is supposed to be uniform, heavy, and incompressible, unless otherwise stated: and all cones, cylinders, paraboloids, &c. are supposed to be surfaces of revolution and their bases circles.]

2693. 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: prove that the mean pressures on the corresponding portions of the curve surfaces are in the ratios

1* : 2 ; 3 :...: n

2694. A hollow cylinder containing a weight W of fluid is held so that its axis makes an angle a with the horizon: prove that the resultant pressure on its curve surface is W cos a in a direction making an angle a with the vertical.

2695. Equal volumes of three fluids are mixed and the mixture separated into three parts; to each of these parts is then added its own volume of one of the original fluids, and the densities of the mixtures so formed are in the ratios 3: 45: prove that the densities of the fluids are as 1: 2: 3.

2696. A thin tube in the form of an equilateral triangle is filled with equal volumes of three fluids which do not mix and held with its plane vertical: prove that the straight lines joining the common surfaces of the fluids form an equilateral triangle whose sides are in fixed directions; and that, if the densities be in A. P., the straight line joining the surfaces of the fluid of mean density will be always vertical.

2697. A thin tube in the form of a square is filled with equal volumes of four fluids which do not mix, whose densities are p1, Pa Pa P1 and held with its plane vertical; straight lines are drawn joining adjacent points where two fluids meet so as to form another square: prove that, if p, +P1 = P2+P2, the diagonals of this square will be vertical and horizontal respectively; but, if p1 = P3 and P=P, every position of the fluids will be one of equilibrium.

2698. A fine tube in the form of a regular polygon of n sides is filled with equal volumes of n different fluids which do not mix and held with its plane vertical: prove that the sides of the polygon formed by joining adjacent points where two fluids meet will have its sides in fixed directions; and, if the densities of the fluids satisfy two certain conditions, every position will be one of equilibrium.

2699. A circular tube of fine uniform bore is half filled with equal volumes of four fluids which do not mix and whose densities are as 1487, and held with its plane vertical: prove that the diameter joining the free surfaces will make an angle tan 2 with the vertical.

2700. A triangular lamina ABC, right-angled at C, is attached to a string at A and rests with the side AC vertical and half its length immersed in fluid: prove that the density of the fluid is to that of the lamina as 8: 7.

2701. A lamina in the form of an equilateral triangle, suspended freely from an angular point, rests with one side vertical and another side bisected by the surface of a fluid: prove that the density of the lamina is to that of the fluid as 15: 16.

2702. A hollow cone, filled with fluid, is suspended freely from a point in the rim of the base: prove that the total pressures on the curve surface and on the base in the position of rest are in the ratio 1 + 11 sin2 a : 12 sin3 a,

where 2a is the vertical angle of the cone.

2703. A tube of small bore, in the form of an ellipse, is half filled with equal volumes of two given fluids which do not mix: find the inclination of its axes to the vertical in order that the free surfaces of the fluids may be at the ends of the minor axis.

2704. A hemisphere is filled with fluid and the surface is divided by horizontal planes into n portions, on each of which the whole pressure is the same: prove that the depth of the 7th of these planes is to the radius as √r: √n.

2705. A hemisphere is just filled with fluid and the surface is divided by horizontal planes into n portions, the whole pressures on which are in a geometrical progression of ratio : prove that the depth of the 7th plane is to the radius as

J-1 1.

2706. A lamina ABCD in the form of a trapezium with parallel sides AB, CD is immersed in fluid with the parallel sides horizontal: prove that the depth of the centre of pressure below E, the point of intersection of AB, CD is

where h is the depth of E, c the distance between AB, CD, and m the ratio CD: AB; and that, when the centre of pressure is at E, the depths of AB, CD will be as

3m2-1: 3-m3.

2707. The co-ordinates of the centre of pressure of a triangular lamina immersed in fluid are

where (x1, 1), (x, y), and (x, y) are the co-ordinates of the middle points of the sides of the lamina, the axis of y being the intersection of the plane of the lamina with the surface of the fluid and the axis of x any other straight line in the plane of the lamina.

2708. The co-ordinates of the centre of pressure of any lamina immersed in fluid are

where (~1, y1), (x ̧, Y1), (x ̧, Y.) are the co-ordinates of the corners of a maximum triangle inscribed in an ellipse whose equation referred to the principal axes of the lamina at its c. G. is

JA, B being the principal radii of gyration. The axes to which the centre of pressure is referred are as in the previous question.

2709. Prove the following construction for finding the centre of pressure of a lamina always totally immersed in fluid which is capable of motion in its own plane about its c. G.: find A, B the highest and lowest positions of the centre of pressure, through A draw a straight line parallel to that straight line of the lamina which is horizontal when A is the centre of pressure, and another straight line similarly determined through B; their point of intersection is the centre of

pressure.

2710. Prove that, when the c. G. is fixed below the surface of a fluid and the lamina move about the c. G. in its own plane, the centre of pressure describes a circle in space and in the lamina the ellipse whose equation referred to the principal axes is

where A, B are the principal radii of gyration, and c the depth

W. P.

30

of the C.G. measured in the plane of the lamina below the surface of the fluid: also that, of the four points in which the circle and ellipse intersect, the centre of pressure is the lowest and the other three are corners of a triangle whose sides touch a fixed circle with its centre at the C.G.

2711. A lamina totally immersed in fluid moves in its own plane so that the centre of pressure is a point fixed in space: prove that the path of the C.G. is the curve whose equation is, referred to the centre of pressure as origin,

x2 {(x − a)2 + y2} +x (x − a) (A + B) + AB = 0;

where A, B are the principal radii of gyration at the C. G., and a the depth (measured in the plane of the lamina) of the centre of pressure below the surface of the fluid.

2712. A rectangular lamina ABCD is immersed in fluid with the side AB in the surface of the fluid; a point P is taken in CD and the lamina divided into two parts by the straight line AP: determine for what position of P the distance between the centres of pressure of the two parts is a maximum.

[If the sides AB, BC be denoted by a, b, and DP by x, the distance will be a maximum when 27ax = 4 (9a3 – 2b2), and since x must be positive and less than a, there will be no maximum unless b: a lie between 3:2√2 and 3 : √2.]

2713. A lamina in the form of the sector of a circle is immersed in fluid with the centre of the circle in the surface: prove that the coordinates of its centre of pressure are

where the axis of y is in the surface of the fluid, and angles which the bounding radii make with the axis of y.

a, 0+ a are the

2714. A lamina, bounded by the epicycloid generated by a circle of radius a rolling on a circle of radius 2a, is placed in fluid with the cusp line in the surface: prove that the co-ordinates of the centre of pressure of half the part immersed are 1312ma, jia; and those of the centre of

1575
2048

73

pressure of the part lying outside the fixed circle are

axis of y lying in the surface.

132 πα,

5 1455 Ia; 4

a; the

2715. An isosceles triangle is immersed with its axis vertical and its base in the surface of a fluid: prove that the resultant pressure on the area intercepted between any two horizontal planes acts through the C.G. of that portion of the volume of a sphere, described with the axis for diameter, which is intercepted between the planes.

2716. A conical shell is placed with its vertex upwards on a hori zontal table and fluid is poured in through a small hole in the vertex; the cone begins to rise when the weight of the fluid poured in is equal to its own weight: prove that this weight bears to the weight of fluid which would fill the cone the ratio 9 – 3/3 : 4.