The vessel being filled with water, compare the tension of the string with the weight of the water.

35. A hollow cone open at the top is filled with water; find the resultant pressure on the portion of its surface cut off, on one side, by two planes through its axis inclined at a given angle to each other; also determine the line of action. of the resultant pressure, and shew that, if the vertical angle be a right angle, it will pass through the centre of the top of

the cone.

36. A vessel in the form of an elliptic paraboloid, whose axis is vertical, and equation + =

h

is divided into four

equal compartments by its principal planes. Into one of these water is poured to the depth h, prove that, if the resultant pressure on the curved portion be reduced to two forces, one vertical and the other horizontal, the line of action of the latter will pass through the point (a, b, &h).

37. A bowl in the form of a hemisphere is filled with water; find the direction and magnitude of the resultant pressure on the upper portion of the bowl cut off by a plane through its centre inclined at a given angle to the horizon.

38. An open conical shell, the weight of which may be neglected, is filled with water, and is then suspended from a point in the rim, and allowed gradually to take its position of 2 -1 equilibrium; prove that, if the vertical angle be cos ̄1 the surface of the water will divide the generating line through the point of suspension in the ratio 2: 1.

39. A regular polygon wholly immersed in a liquid is moveable about its centre of gravity; prove that the locus of the centre of pressure is a sphere.

40. A hemispherical bowl is filled with water, and two vertical planes are drawn through its central radius, cutting off a semi-lune of the surface; if 2a be the angle between the

planes, prove that the angle which the resultant pressure on the surface makes with the vertical

41. A volume

Απα
3

of fluid of density p surrounds a fixed

sphere of radius b and is attracted to a point at a distance c(<b) from its centre by a force ur per unit mass; supposing the external pressure zero, find the resultant pressure on the fixed sphere.

42. A vessel in the form of a surface of revolution has the following property; if it be placed with its axis vertical, and any quantity of water be poured into it, the ratio of the total normal pressure to the resultant vertical pressure varies as the depth of the water poured in. Shew that the equation to the generating curve is

cs = xy.

43. Find the equation of a curve symmetrical about a vertical axis, such that, when it is immersed with its highest point at half the depth of its lowest, the centre of pressure may bisect the axis.

44. Find the surfaces of floatation and of buoyancy in the case of a right circular cylinder floating with one end immersed.

45. The vertices A, B, C of a triangular lamina are sunk in a homogeneous liquid to depths h1, h2, h ̧ respectively prove that if P1, P2, P, be the respective perpendiculars from A, B, C on BC, CA, AB, then the trilinear coordinates of the centre of pressure are

:

h2

4 h1+ h2+h ̧

46. A triangular lamina is totally immersed in a homogeneous liquid, the depths of the angular points being P, q, r; prove that if the centre of pressure of the triangle coincide with the mean centre of its angular points for multiples l, m, n, then

p: qr: 31-(m + n) : 3m − (n + 1) : 3n − (l +m).

B. H.

5

CHAPTER IV.

THE EQUILIBRIUM OF FLOATING BODIES.

48. To find the conditions of equilibrium of a floating body.

We shall suppose that the fluid is at rest under the action of gravity only, and that the body, under the action of the same force, is floating freely in the fluid. The only forces then which act on the body are its weight, and the pressure of the surrounding fluid, and in order that equilibrium may exist, the resultant fluid pressure must be equal to the weight of the body, and must act in a vertical direction.

Now we have shewn, that the resultant pressure of a heavy fluid on the surface of a solid, either wholly or partially immersed, is equal to the weight of the fluid displaced, and acts in a vertical line through its centre of gravity.

Hence it follows that the weight of the body must be equal to the weight of the fluid displaced, and that the centres of gravity of the body, and of the fluid displaced, must lie in the same vertical line.

These conditions are necessary and sufficient conditions of equilibrium, whatever be the nature of the fluid in which the body is floating. If it be heterogeneous, the displaced fluid must be looked upon as following the same law of density as the surrounding fluid; in other words, it must consist of strata of the same kind as, and continuous with, the horizontal strata of uniform density, in which the particles of the surrounding fluid are necessarily arranged.

If for instance a solid body float in water, partially immersed, its weight will be equal to the weight of the water displaced, together with the weight of the air displaced; and if the air be removed, or its pressure diminished by a diminution of its density or temperature, the solid will sink in the water through a space depending upon its own weight, and upon the densities of air and water. This may be further explained by observing that the pressure of the air on the water is greater than at any point above it, and that this surface pressure of the air is transmitted by the water to the immersed portion of the floating body, and consequently the upward pressure of the air upon it is greater than the downward pressure.

49. We now proceed to illustrate the application of the above conditions, by the discussion of some particular cases.

Ex. 1. A portion of a solid paraboloid, of given height, floats with its axis vertical and vertex downwards in a homogeneous liquid, required to find its position of equilibrium.

Taking 4a as the latus rectum of the generating parabola, h its height, and the depth of its vertex, the volumes of the whole solid and of the portion immersed are respectively 2πаh2 and 2παx2; and if p, σ, be the densities of the solid and liquid, one condition of equilibrium is

which determines the portion immersed, the other condition being obviously satisfied.

Ex. 2. It is required to find the positions of equilibrium of a square lamina floating with its plane vertical, in a liquid of double its own density.

The conditions of equilibrium are clearly satisfied if the lamina float half immersed either with a diagonal vertical, or with two sides vertical.

To examine whether there is any other position of equilibrium, let the lamina be held with the line DGC in the surface, in which case the first condition is satisfied.

But, if the angle CGA=0, and if 2a be the side of the square, the moment about G of the fluid pressure, which is the same as the difference between the moments of the rectangle AK, and of twice the triangle GBD,

Hence there is no other position of equilibrium.

Ex. 3. A triangular prism floats with its edges horizontal, to find its positions of equilibrium.

Let the figure be a section of the prism by a vertical plane through its centre of gravity.

PQ is the line of floatation and H the centre of gravity of the liquid displaced. When there is equilibrium the area APQ is to ABC in the ratio of the density of the prism to the density of the liquid, and therefore for all possible positions of PQ the area APQ is constant; hence PQ always touches, at its middle point, an hyperbola of which AB, AC, are the asymptotes.