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Hydrostatics

HYDROSTATICS Hydrostatics is a science which grew originally out of a number of isolated practical problems; but it satisfies the requirement of perfect accuracy in its application to phenomena, the largest and smallest, of the behaviour of a fluid. At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.

1. The Different States of a Substance or Matter. All substance in nature falls into one of the two classes, solid and fluid; a solid substance, the land, for instance, as contrasted with a fluid, like water, being a substance which does not flow of itself.

A fluid, as the name implies, is a substance which flows, or is capable of flowing; water and air are the two fluids distributed most universally over the surface of the earth.

Fluids again are divided into two classes, termed a liquid and a gas, of which water and air are the chief examples.

A liquid is a fluid which is incompressible or practically so, i.e. it does not change in volume sensibly with change of pressure. . A gas is a compressible fluid, and the change in volume is considerable with moderate variation of pressure.

Liquids, again, can be poured from one open vessel into another, and can be kept in an uncovered vessel, but a gas tends to diffuse itself indefinitely and must be preserved in a closed reservoir.

The distinguishing characteristics of the three kinds of substance or states of matter, the solid, liquid and gas, are summarized thus in O. Lodge's Mechanics:

A solid has both size and shape. A liquid has size but not shape. A gas has neither size nor shape.

2. The Change of State of Matter. By a change of temperature and pressure combined, a substance can in general be made to pass from one state into another; thus by gradually increasing the temperature a solid piece of ice can be melted into the liquid state of water, and the water again can be boiled off into the gaseous state as steam. Again, by raising the temperature, a metal in the solid state can be melted and liquefied, and poured into a mould to assume any form desired, which is retained when the metal cools and solidifies again; the gaseous state of a metal is revealed by the spectroscope. Conversely, a combination of increased pressure and lowering of temperature will, if carried far enough, reduce a gas to a liquid, and afterwards to the solid state; and nearly every gaseous substance has now undergone this operation.

A certain critical temperature is observed in a gas, above which the liquefaction is impossible; so that the gaseous state has two subdivisions into (i.)a true gas, which cannot be liquefied, because its temperature is above the critical temperature, (ii.) a vapour, where the temperature is below the critical, and which can ultimately be liquefied by further lowering of temperature or increase of pressure.

3. Plasticity and Viscosity. Every solid substance is found to be plastic more or less, as exemplified by punching, shearing and cutting; but the plastic solid is distinguished from the viscous fluid in that a plastic solid requires a certain magnitude of stress to be exceeded to make it flow, whereas the viscous liquid will yield to the slightest stress, but requires a certain length of time for the effect to be appreciable.

According to Maxwell (Theory of Heat) " When a continuous alteration of form is produced only by a stress exceeding a certain value, the substance is called a solid, however soft and plastic it may be. But when the smallest stress, if only continued long enough, will cause a perceptible and increasing change of form, the substance must be regarded as a viscous fluid, however hard it may be." Maxwell illustrates the difference between a soft solid and a hard liquid by a jelly and a block of pitch; also by the experiment of supporting a candle and a stick of sealingwax; after a considerable time the sealing-wax will be found bent and so is a fluid, but the candle remains straight as a solid.

4. Definition of a Fluid. A fluid is a substance which yields continually to the slightest tangential stress in its interior; that is, it can be divided very easily along any plane (given plenty of time if the fluid is viscous). It follows that when the fluid has come to rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane. This mechanical axiom of the normality oj fluid pressure is the foundation of the mathematical theory of hydrostatics.

The theorems of hydrostatics are thus true for all stationary fluids, however, viscous they may be; it is only when we come to hydrodynamics, the science of the motion of a fluid, that viscosity will make itself felt and modify the theory; unless we begin by postulating the perfect fluid, devoid of viscosity, so that the principle of the normality of fluid pressure is taken to hold when the fluid is in movement.

5. The Measurement of Fluid Pressure. The pressure at any point cf a plane in the interior of a fluid is the intensity of the normal thrust estimated per unit area of the plane.

Thus, if a thrust of P Ib is distributed uniformly over a plane area of A sq. ft., as on the horizontal bottom of the sea or any reservoir, the pressure at any point of the plane is P/A ft per sq. ft., or P/I44A ft per sq. in. (ft/ft. 2 and ft/in. 2 , in the Hospitalier notation, to be employed in the sequel). If the distribution of the thrust is not uniform, as, for instance, on a vertical or inclined face or wall of a reservoir, then P/A represents the average pressure over the area ; and the actual pressure at any point is the average pressure over a small area enclosing the point. Thus, if a thrust AP ft acts on a small plane area AA ft. 2 enclosing a point B, the pressure p at B is the limit of AP/AA ; and =lt(AP/AA)=<fP/<ZA, (i)

in the notation of the differential calculus.

6. The Equality of Fluid Pressure in all Directions. This fundamental principle of hydrostatics follows at once from the principle of the normality of fluid pressure implied in the definition of a fluid in 4. Take any two arbitrary directions in the plane of the paper, and draw a small isosceles triangle abc, whose sides are perpendicular to the two directions, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section. Let P, Q denote the normal thrust across the sides be, ca, and R the normal thrust across the base ab. Then, since these three forces maintain equilibrium, and R makes equal angles with P and Q, therefore P and Q must be equal. But the faces be, ca, over which P and Q act, are also equal, so that the pressure on each face is equal. A scalene triangle abc might also be employed, or a W tetrahedron.

It follows that the pressure of a fluid requires to be calculated in one direction only, chosen as the simplest direction for convenience.

7. The Transmissibility of Fluid Pressure. Any additional pressure applied to the fluid will be 5- =r transmitted equally to every point in the case of I2-^E-.J| a liquid; this principle of the transmissibility of ~ T pressure was enunciated by Pascal, 1653, and FIG. la. applied by him to the invention of the hydraulic press.

This machine consists essentially of two communicating cylinders (fig. I a), filled with liquid and closed by pistons. If a thrust P ft is applied to one piston of area A ft. 2 , it will be balanced by a thrust W ft applied to the other piston of area B ft. 2 , where p = p/A=\V/B, (i)

the pressure p of the liquid being supposed uniform; and, by making the ratio B/A sufficiently large, the mechanical advantage can be increased to any desired amount, and in the simplest manner possible, without the intervention of levers and machinery.

Fig. ib shows also a modern form of the hydraulic press, applied to the operation of covering an electric cable with a lead coating.

8. Theorem. In a fluid at rest under gravity the pressure is the same at any two points in the same horizontal plane; in other words, a surface of equal pressure is a horizontal plane.

This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB. As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B ; and so the pressure is the same at all points in the same horizontal plane. If the fluid is a liquid, it can have a free surface without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure, or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane.

Hence the theorem. The free surface of a liquid at rest under gravity is a horizontal plane. This is the characteristic distinguishing between a solid and a liquid; as, for instance, between land and water. The land has hills and valleys, but the surface of water at rest is a horizontal plane; and if disturbed the surface moves in waves.

9. Theorem. In a homogeneous liquid at rest under gravity the pressure increases uniformly with the depth.

This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically. In this case the thrust at the lower end B must exceed the thrust at A, the upper end, by the weight of the prism of liquid; so that, denoting the cross FIG. ib.

section of the prism by a ft. 2 , the pressure at A and By by pa and p ft/ft. 2 , and by w the density of the liquid estimated in ft/ft. 3 , pa. poa=wa. AB, (i)

p=w.AB+A>- (2)

Thus in water, where w=62-4lb/ft. 3 , the pressure increases 62-4 ft/ft. 2 , or 62-4 -5-144=0-433 ft/in. 2 for every additional foot of depth.

10. Theorem. If two liquids of different density are resting in vessels in communication, the height of the free surface of such liquid above the surface of separation is inversely as the density.

For if the liquid of density a rises to the height h and of density p to the height k, and po denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be ah-\-ps, and pk-{-po, and these being equal we have ah = pk. (i)

The principle is illustrated in the article BAROMETER, where a column of mercury of density a and height h, rising in the tube to the Torricellian vacuum, is balanced by a column of air of density p, which may be supposed to rise as a homogeneous fluid to a height k, called the height of the homogeneous atmosphere. Thus water being about 800 times denser than air and mercury 13-6 times denser than water, (2)

and with an average barometer height of 30 in. this makes k 27,200 ft., about 8300 metres.

1 1 . The Head of Water or a Liquid. The pressure a h at a depth h ft. in liquid of density a is called the pressure due to a head of h ft. of the liquid. The atmospheric pressure is thus due to an average head of 30 in. of mercury, or 30X13-6-^12=34 ft. of water, or 27,200 ft. of air. The pressure of the air is a convenient unit to employ in practical work, where it is called an " atmosphere "; it is made the equivalent of a pressure of one kg/cm 2 ; and one ton/inch 2 , employed as the unit with high pressure as in artillery, may be taken as 150 atmospheres.

12. Theorem. A body immersed in a fluid is buoyed up by a force equal to the weight of the liquid displaced, acting vertically upward through the centre of gravity of the displaced liquid.

For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.

When the body is floating freely like a ship, the equilibrium of this liquid thrust with the weight of the ship requires that the weight of water displaced is equal to the weight of the ship and the two centres of gravity are in the same vertical line. So also a balloon begins to rise when the weight of air displaced is greater than the weight of the balloon, and it is in equilibrium when the weights are equal. This theorem is called generally the principle of Archimedes.

It is used to determine the density of a body experimentally; for if W is the weight of a body weighed in a balance in air (strictly in vacua), and if W is the weight required to balance when the body is suspended in water, then the upward thrust of the liquid or weight of liquid displaced is W-W, so that the specific gravity (S.G.), defined as the ratio of the weight of a body to the weight of an equal volume of water, is W/(W-W')- As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its own volume of water; and he utilized it in the problem of the determination of the adulteration of the crown of Hiero. He weighed out a lump of gold and of silver of the same weight as the crown; and, immersing the three in succession in water, he found they spilt over measures of water in the ratio ft ' A : ft r 33 : 24 : 44 ; thence it follows that the gold : silver alloy of the crown was as 1 1 : 9 by weight.

13. Theorem. The resultant vertical thrust on any portion of a curved surface exposed to the pressure of a fluid at rest under gravity is the weight of fluid cut out by vertical lines drawn round the boundary of the curved surface.

Theorem. The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane curve; and then investigating the thrust on this plane area, which will be the same as on the curved surface.

The proof of these theorems proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship. In casting a thin hollow object like a bell, it will be seen that the resultant upward thrust on the mould may be many times greater than the weight of metal; many a curious experiment has been devised to illustrate this property and classed as a hydrostatic paradox (Boyle, Hydrostatical Paradoxes, 1666).

Consider, for instance, the operation of casting a hemispherical bell, in fig. 2. As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR', or fay*, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside hemisphere. Afterwards, when the metal has risen above B, to the level KK', the additional thrust is the weight of the cylinder of diameter KK' and height BH. The upward thrust is the same, however thin the metal may be in the interspace between the outer mould and the core inside ; and this was formerly considered paradoxical.

Analytical Equations of Equilibrium of a Fluid at rest under any System of Force.

14. Referred to three fixed coordinate axes, a fluid, in which the pressure is p, the density p, and X, Y, Z the components of impressed force per unit mass, requires for the equilibrium of the part filling a fixed surface S, on resolving parallel to Ox, JjlpdS=jjfpXdxdydz, (i)

where /, m, n denote the direction cosines of the normal drawn outward of the surface S.

But by Green's transformation ///pdS=///gd*<fy<fc, (2)

thus leading to the differential relation at every point The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

Hence the space variation of the pressure in any direction, or the pressure-gradient, is the resolved force per unit volume in that direction. The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure.

Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force ; i dp i dp i dp Y v 7 , , - -f-. - ~f~ t -f-, or A, Y, L (4)

p dx' pdy' p dz' are the partial differential coefficients of some function P, =fdp/p, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force m any direction is the downward gradient of V ; and then = o, or P + V = constant, (5)

in which P may be called the hydrostatic head and V the head of potential.

With variation of temperature, the surfaces of equal pressure and density need not coincide; but, taking the pressure, density and temperature as connected by some relation, such as the gas-equation, the surfaces of equal density and temperature must intersect in lines lying on a surface of equal pressure.

15. As an example of the general equations, take the simplest case of a uniform field of gravity, with Oz directed vertically downward ; employing the gravitation unit of force, I dp i dp i dp pd X =0 'pdy = <pdz =l ' O P=J<Zp/p=z+a constant. (2)

When the density p is uniform, this becomes, as before in (2) 9 p=pz+pt. (3)

Suppose the density p varies as some nth power of the depth below O, then p = t iz' 1 (4)

* M n + i n + i n + i supposing p and p to vanish together.

These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation p = pk = Rp8, (6)

where 8 denotes the absolute temperature ; and then dz~dz so that the temperature-gradient d6/dz is constant, as in convective equilibrium in (n).

From the gas-equation in general, in the atmosphere i dp_i dp__i dti_p__i dj>_i_ d9 .

pdz~p dz~8 dz~p~6dz~k~8dz' which is positive, and the density p diminishes with the ascent, provided the temperature-gradient dS/dz does not exceed 8/k.

With uniform temperature, taking k constant in the gas-equation, dp/dz = p=p/k, p=pee'"<, . (9)

so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures pi and pi at heights Zi and z 2 (zi-za)/* =log.(#j//>i) =2-3 logio(/>2//>i). (10)

In the convective equilibrium of the atmosphere, the air is supposed to change in density and pressure without exchange of heat by conduction ; and then p/po = (9/ffo)", Plpo = (0/ft>)" +1 , (n)

where y is the ratio of the specific heat at constant pressure and constant volume.

In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a, gravity varying inversely as the square of the distance r from the centre; so that, k = po/po, denoting the height of the homogeneous atmosphere at the surface, 8 is given by (n + i)ft(i-9/flo)=o(i-o/r), (13)

or if c denotes the distance where 8 = 0, i = -r-^ 04) ' When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as />-o = (p-po), or p/po =! + (- A, X = fcpo, (15) so that X is the pressure due to a head k of the liquid at density po under atmospheric pressure /v, and it is the gauge pressure required on this law to double the density. Then dp/dz = kdp/dz=p, p=p^' k , p-po = kpo(e""'-i); (16) and if the liquid was incompressible, the depth at pressure p would be (ppa)/pa, so that the lowering of the surface due to compression is ke*l k -k-z = \z l lk, when k is large. (17)

For sea water, X is about 25,000 atmospheres, and k is then 25,000 times the height of the water barometer, about 250,000 metres, so that in an ocean 10 kilometres deep the level is lowered about 200 metres by the compressibility of the water; and the density at the bottom is increased 4 %.

On another physical assumption of constant cubical elasticity X, (i 8)

pt p (19)

(3) (4) (5)

(6) (7) (8) (9)

and the lowering of the surface is i> Po p F i / z \ z* , * <- ^ Z = k log Z= k log I I j-1 ZKr (2O)

Po 6 Po V / 2 as before in 17).

16. Centre of Pressure. A plane area exposed to fluid pressure on one side experiences a single resultant thrust, the integrated pressure over the area, acting through a definite point called the centre of pressure (C.P.) of the area.

Thus if the plane is normal to Oz, the resultant thrust R=ffpdxdy, (i)

and the co-ordinates *, y of the C.P. are given by xR=ffxpdxdy, yR = ffypdxdy. (2)

The C.P. is thus the C.G. of a plane lamina bounded by the area, in which the surface density is p.

If p is uniform, the C.P. and C.G. of the area coincide. For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface, i.e. to the perpendicular distance from the line of intersection of the plane of the area with the free surface of the liquid.

If the equation of this line, referred to new coordinate axes in the plane area, is written x cos o +y sin a h = O, R=fjp(hxcosay sin a)dxdy, xR = ffpx(hxcos o y sin a)dxdy, yR = fjpy(hxcos a y sin a)dxdy. Placing the new origin at the C.G. of the area A, ffxdxdy = o, ffydxdy = o, R = pA, xhA= cos affxydA sin affxydA, yhA = cos affxydA sin aff) 3 dA.

Turning the axes to make them coincide with the principal axes of the area A, thus making ffxydA = O, xh = a* cos a, y&= V sin a, (10)

where a and 6 denoting the semi-axes of the momental ellipse of the area.

This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G. of the area.

Thus the C.P. of a rectangle or parallelogram with a side in the surface is at f of the depth of the lower side; of a triangle with a vertex in the surface and base horizontal is J of the depth of the base; but if the base is in the surface, the C.P. is at half the depth of the vertex; as on the faces of a tetrahedron, with one edge in the surface.

The core of an area is the name given to the limited area round its C.G. within which the C.P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area. Thus the core of a circle or an ellipse is a concentric circle or ellipse of one quarter the size.

The C.P. of water lines passing through a fixed point lies on a straight line, the antipolar of the point ; and thus the core of a triangle is a similar triangle of one quarter the size, and the core of a parallelogram is another parallelogram, the diagonals of which are the middle third of the median lines.

In the design of a structure such as a tall reservoir dam it is important that the line of thrust in the material should pass inside the core of a section, so that the material should not be in a state of tension anywhere and so liable to open and admit the water.

17. Equilibrium and Stability of a Ship or Floating Body. The Metacentre. The principle of Archimedes in 12 leads immediately to the conditions of equilibrium of a body supported freely in fluid, like a fish in water or a balloon in the air, or like a ship (fig. 3) floating partly immersed in water and the rest in air. The body is in equilibrium under two forces: (i.) its weight W acting vertically downward through G, the C.G. of the body, and (ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upward through E, the C.G. of the displaced fluid; for equilibrium these two forces must be equal and opposite in the same line.

The conditions of equilibrium of a body, floating like a ship on the surface of a liquid, are therefore:

(i.) the weight of the body must be less than the weight of the total volume of liquid it can displace; or else the body will sink to the bottom of the liquid; the difference of the weights is called the " reserve of buoyancy."

(ii.) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body ; and (iii.) the C.G., B, of the liquid displaced and G of the body, must lie in the same vertical line GB.

18. In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, so as to keep upright; if displaced slightly from this position, the forces called into play must be such as to restore the ship to the upright again. The stability of a ship is investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced.

Suppose P tons is moved c ft. across the deck of a ship of W tons displacement ; the C.G. will move from G to Gi the reduced distance GiG2 = c(P/W); and if B, called the centre of buoyancy, moves to BI, along the curve of buoyancy BBi, the normal of this curve at Bi will be the new vertical Bid, meeting the old vertical in a point M, the centre of curvature of BBi, called the metacentre.

If the ship heels through an angle 9 or a slope of I in m, GU=GGicot0=mc(PfW), (i)

and GM is called the metacentric height; and the ship must be ballasted, so that G lies below M. If G was above M, the tangent drawn from G to the evolute of B , and normal to the curve of buoyancy, would give the vertical in a new position of equilibrium. Thus in H.M.S. " Achilles " of 9000 tons displacement it was found that moving 20 tons across the deck, a distance of 42 ft., caused the bob of a pendulum 20 ft. long to move through 10 in., so that also cot 9=24, e =2" 24'.

(3)

In a diagram it is conducive to clearness to draw the ship in one position, and to incline the water-line; and the page can be turned if it is desired to bring the new water-line horizontal.

Suppose the ship turns about an axis through F in the water-line area, perpendicular to the plane of the paper; denoting by y the distance of an element dA if the water-line area from the axis of rotation, the change of displacement is ~LydA tan 9, so that there is no change of displacement if 2y<2A=o, that is, if the axis passes through the C.G. of the water-line area, which we denote by F and call the centre of notation.

The righting couple of the wedges of immersion and emersion will be ZwydA. tan O.y=w tan 02/dA = w tan e.Ak 1 f t. tons, (4)

w denoting the density of water in tons/ft. 3 , and W = a>V, for a displacement of V ft. 3 This couple, combined with the original buoyancy W through B, is equivalent to the new buoyancy through B, so that W.BBi=wA* 2 tan9, (5)

BM = BBi cot 9 = A 2 /V, (6)

giving the radius of curvature BM of the curve of buoyancy B, in terms of the displacement V, and Ak 1 the moment of inertia of the water-line area about an axis through F, perpendicular to the plane of displacement.

An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A. For if the ship turns through a small angle 6 about the line FF', then 61, 6 2 , the C.G. of the wedge of immersion and emersion, will be the C.P. with respect to FF' of the two parts of the water-line area, so that bibi will be conjugate to FF' with respect to the momental ellipse at F.

The naval architect distinguishes between the stability of form, represented by the righting couple W.BM, and testability of ballasting, represented by W.BG. Ballasted with G at B, the righting couple when the ship is heeled through is given by W.BM. tan0; but if weights inside the ship are raised to bring G above B, the righting couple is diminished by W.BG. tan 8, so that the resultant righting couple is W.GM. tan B. Provided the ship is designed to float upright at the smallest draft with no load on board, the stability at any other draft of water can be arranged by the stowage of the weight, high or low.

19. Proceeding as in 16 for the determination of the C.P. of an area, the same argument will show that an inclining couple due to the movement of a weight P through a distance c will cause the ship to heel through an angle 6 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a?-hV/A, 6 2 -/rV/A, (i)

h denoting the vertical height BG between C.G. and centre of buoyancy. The varying direction of the inclining couple PC may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK = ( 2 -fcV/A)/FQ sin QFF' (2)

through an angle or a slope of one in m, given by where k denotes the radius of gyration about FF' of the water-line area. Burning the coal on a voyage has the reverse effect on a steamer.

Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)

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