FLOATING BODIES

FLOATING BODIES are such as swim on the surface of a fluid, of which the most important are ships and all kinds of vessels employed in war and in commerce. Every seaman knows of how much consequence it is to determine the stability of such vessels, and the positions which they assume when they float freely and at rest on the water. To accomplish this, it is necessary to state the principles on which that stability and these positions depend; and this has been done with so much ingenuity and science by GEORGE ATWOOD, Esq; F. R. S. in the Philosophical Transactions for the year 1796, that we are persuaded a large class of our readers will thank us for inserting an abstract of his memoir in this place.

A floating body is pressed downwards by its own weight in a vertical line that passes through its centre of gravity; and it is sustained by the upward pressure of a fluid, acting in a vertical line that passes through the centre of gravity of the immersed part; and unless these two lines be coincident, so that the two centres of gravity may be in the same vertical line, the solid will revolve on an axis, till it gains a position in which the equilibrium of floating will be permanent. Hence it appears that it is necessary, in the first place, to ascertain the proportion of the part immersed to the whole; for which purpose the specific gravity of the floating body must be known; and then it must be determined, by geometrical or analytical methods, in what positions the solid can be placed on the surface of the fluid, so that the two centres of gravity already mentioned may be in the same vertical line when a given part of the solid is immersed under the surface of the fluid. When these preliminaries are settled, something still remains to be done. Positions may be assumed in which the circumstances just recited concur, and yet the solid will assume some other position in which it will permanently float. If a cylinder, e. g. having its specific gravity to that of the fluid on which it floats as 3 to 4, and its axis to the diameter of the base as 2 to 1, be placed on the fluid with its axis vertical, it will sink to a depth equal to a diameter and a half of the base; and while its axis is preserved in a vertical position by external force, the centres of gravity of the whole solid and of the immersed part will remain in the same vertical line: but when the external force that sustains it is removed, it will decline from its upright position, and will permanently float with its axis horizontal. If the axis be supposed to be half of the dia-

meter of the base, and be placed vertically, the solid will sink to the depth of three-eighths of its diameter; and in that position it will float permanently. If the axis be made to incline to the vertical line, the solid will change its position until it settles permanently with the axis perpendicular to the horizon.

Whether, therefore, a solid floats permanently, or oversets when placed on the surface of a fluid, so that the centre of gravity of the solid and that of the part immersed shall be in the same vertical line, it is said to be in a position of equilibrium; and of this equilibrium there are three species, viz. the equilibrium of stability, in which the solid floats permanently in a given position; the equilibrium of instability, in which the solid, though the two centres of gravity already mentioned are in the same vertical line, spontaneously oversets, unless supported by external force; and the equilibrium of indifference, or the insensible equilibrium, in which the solid rests on the fluid indifferent to motion, without tendency to right itself when inclined, or to incline itself farther.

If a solid body floats permanently on the surface of a fluid, and external force be applied to incline it from its position, the resistance opposed to this inclination is termed the stability of floating. Among various floating bodies, some lose their quiescent position, and some gain it, after it has been interrupted, with greater facility and force than others.

Some ships at sea (e. g.) yield to a given impulse of the wind, and suffer a greater inclination from the perpendicular than others. As this resistance to heeling or pitching, duly regulated, has been deemed of importance in the construction of vessels, several eminent mathematicians have investigated rules for determining the stability of ships from their known dimensions and weight, without recurring to actual trial. To this class we may refer Bouguer, Euler, Fred. Chapman, and others; who have laid down theorems for this purpose, founded on a supposition that the inclinations of ships from their quiescent positions are evanescent, or, in a practical sense, very small.

"But ships at sea (says our ingenious author) are known to heel through angles of $10^\{\backslash circ\}$, $20^\{\backslash circ\}$, or even $30^\{\backslash circ\}$, and therefore a doubt may arise how far the rules, demonstrated on the express condition that the angles of inclination are of evanescent magnitude, should be admitted as practically applicable in cases where the inclinations are so great." "If we admit that the theory of statics can be applied with any effect to the practice of naval architecture, it seems to be necessary that the rules, investigated for determining the stability of vessels, should be extended to those cases in which the angles of inclination are of any magnitude likely to occur in the practice of navigation."

A solid body placed on the surface of a lighter fluid, at the depth corresponding to the relative gravities, cannot change its position by the combined actions of its weight and the pressure of the fluid, except by revolving on some horizontal axis which passes through the centre of gravity: but as many axes may be drawn through this point of the floating body in a direction parallel to the horizon, and the motion of the solid respects one axis only, this axis must be determined by the figure of the body and the particular nature of the case. When this axis of motion, as it is called, is de-

Floating. terminated, and the specific gravity of the solid is known, the positions of permanent floating will be obtained, first by finding the several positions of equilibrium through which the solid may be conceived to pass, while it revolves round the axis of motion; and secondly, by determining in which of those positions the equilibrium is permanent, and in which of them it is momentary and unstable."

Such as we have now briefly stated are the general principles, on which are founded Mr Atwood's investigations for determining the positions assumed by homogeneous bodies, floating on a fluid surface; and also for determining the stability of ships and of other floating bodies. We cannot farther accompany him in his elucidation of them, in the problems to the solution of which they lead, and in the important practical purposes of naval architecture to which they are referred. The whole paper, comprehending no less than 85 pages, is curious and valuable; it abounds with analytical and geometrical disquisitions of the most elaborate kind; and it serves to enlarge our acquaintance with a subject that is not only highly interesting to the speculative mathematician, but extremely useful in its practical application.

With this latter view, the author seems to have directed his attention to the various objects of inquiry which this article comprehends. They are such as intimately relate to the theory of naval architecture, so far as it depends on the pure laws of mechanics, and they contribute to extend and improve this theory. The union of those principles that are deduced from the laws of motion, with the knowledge which is derived from observation and experience, cannot fail to establish the art of constructing vessels on its true basis, and gradually to lead to farther improvements of the greatest importance and utility. To this purpose, the author observes, that

"If the proportions and dimensions adopted in the construction of individual vessels are obtained by exact geometrical mensurations, and calculations founded on them, and observations are made on the performance of these vessels at sea; experiments of this kind, sufficiently diversified and extended, seem to be the proper grounds on which theory may be effectually applied in developing and reducing to system those intricate, subtle, and hitherto unperceived causes, which contribute to impart the greatest degree of excellence to vessels of every species and description. Since naval architecture is reckoned amongst the practical branches of science, every voyage may be considered as an experiment, or rather as a series of experiments, from which useful truths are to be inferred towards perfecting the art of constructing vessels: but inferences of this kind, consistently with the preceding remark, cannot well be obtained, except by acquiring a perfect knowledge of all the proportions and dimensions of each part of the ship; and secondly, by making and recording sufficiently numerous observations on the qualities of the vessel, in all the varieties of situation to which a ship is usually liable in the practice of navigation."

In the valuable miscellany entitled the Philosophical Magazine, there is a paper on this subject by Mr John George English, teacher of mathematics and mechanical philosophy; which, as it is not long, and is easily understood, we shall take the liberty to transcribe.

"However operose and difficult the calculations necessary to determine the stability of nautical vessels may, in some cases, be, yet they all depend, says this author, upon the four following simple and obvious theorems, accompanied with other well known stereometrical and statical principles.

"Theorem 1. Every floating body displaces a quantity of the fluid in which it floats, equal to its own weight: and consequently, the specific gravity of the fluid will be to that of the floating body, as the magnitude of the whole is to that of the part immersed.

"Theorem 2. Every floating body is impelled downward by its own essential power, acting in the direction of a vertical line passing through the centre of gravity of the whole; and is impelled upward by the reaction of the fluid which supports it, acting in the direction of a vertical line passing through the centre of gravity of the part immersed: therefore, unless these two lines are coincident, the floating body thus impelled must revolve round an axis, either in motion or at rest, until the equilibrium is restored.

"Theorem 3. If by any power whatever a vessel be deflected from an upright position, the perpendicular distance between two vertical lines passing through the centres of gravity of the whole, and of the part immersed respectively, will be as the stability of the vessel, and which will be positive, nothing, or negative, according as the metacentre is above, coincident with, or below, the centre of gravity of the vessel.

"Theorem 4. The common centre of gravity of any system of bodies being given in position, if any one of these bodies be moved from one part of the system to another, the corresponding motion of the common centre of gravity, estimated in any given direction, will be to that of the aforesaid body, estimated in the same direction, as the weight of the body moved is to that of the whole system.

"From whence it is evident, that in order to ascertain the stability of any vessel, the position of the centres of gravity of the whole, and of the part immersed, must be determined; with which, and the dimensions of the vessel, the line of flotation, and angle of deflection, the stability or power either to right itself or overturn, may be found.

"In ships of war and merchandise, the calculations necessary for this purpose become unavoidably very operose and troublesome; but they may be much facilitated by the experimental method pointed out in the New Transactions of the Swedish Academy of Sciences, first quarter of the year 1787, page 48.

"In river and canal boats, the regularity and simplicity of the form of the vessel itself, together with the compact disposition and homogeneous quality of the burden, render that method for them unnecessary, and make the requisite calculations become very easy. Vessels of this kind are generally of the same transverse section throughout their whole length, except a small part in prow and stern, formed by segments of circles or other simple curves; therefore a length may easily be assigned such, that any of the transverse sections being multiplied thereby, the product will be equal to the whole solidity of the vessel. The form of the section ABCD is for the most part either rectangular, as in fig. 1. trapezoidal as in fig. 2. or mixtilinear

Fig. 3. in all which MM represents the line of floatation when upright, and EF that when inclined at any angle MXE; also G represents the centre of gravity of the whole vessel, and R that of the part immersed.

If the vessel be loaded quite up to the line AB, and the specific gravity of the boat and burden be the same, then the point G is simply the centre of gravity of the section ABCD; but if not, the centres of gravity of the boat and burden must be found separately, and reduced to one by the common method, namely, by dividing the sum of the momenta by the sum of weights, or areas, which in this case are as the weights. The point R is always the centre of gravity of the section MMCD, which, if consisting of different figures, must also be found by dividing the sum of the momenta by the sum of the weights as common. These two points being found, the next thing necessary is to determine the area of the two equal triangles MXE, MXF, their centres of gravity $o,\; o$, and the perpendicular projected distance $nn$ of these points on the water line EF. This being done, through R and parallel to EF draw RT—a fourth proportional to the whole area MMCD, either triangle MXE or MXF, and the distance $nn$; through T, and at right angles to RT or EF, draw TS meeting the vertical axis of the vessel in S the metacentre; also through the points G, B, and parallel to ST, draw NGW and BV; moreover through S, and parallel to EF, draw WSV, meeting the two former in V and W; then SW is as the stability of the vessel, which will be positive, nothing, or negative, according as the point S is above, coincident with, or below, the point G. If now we suppose W to represent the weight of the whole vessel and burden (which will be equal to the section MMCD multiplied by the length of the vessel), and P to represent the required weight applied at the gunwale B to sustain the vessel at the given angle of inclination; we shall always have this proportion: as VS : SW :: W : P; which proportion is general, whether SW be positive or negative; it must only in the latter case be supposed to act upward to prevent an overturn.

In the rectangular vessel, of given weight and dimensions, the whole process is so evident, that any further explanation would be unnecessary. In the trapezoidal vessel, after having found the points G and R, let AD, BC be produced until they meet in K. Then, since the two sections MMCD, EFDC are equal, the two triangles MMK, EFK are also equal; and therefore the rectangle $EK\; \backslash times\; KF\; =\; KM\; \backslash times\; KM\; =\; KM^2$; and since the angle of inclination is supposed to be known, the angles at E and F are given. Consequently, if a mean proportional be found between the fines of the angles at E and F, we shall have the following proportions:

As the mean proportional thus found: fine $\backslash angle\; E$ : $KM$ : $KF$, and as the said mean proportional: fine $\backslash angle\; F$ : $KM$ : $KE$; therefore ME, MF become known: from whence the area of either triangle MXE or MXF, the distance $nn$, and all the other requisites, may be found.

In the mixtilinear section, let AB = 9 feet = 108 inches, the whole depth = 6 feet = 72 inches, and the altitude of MM the line of floatation 4 feet or 48 inches; also let the two curvilinear parts be circular

quadrants of two feet, or 24 inches radius each. Then the area of the two quadrants = 9.47808 square inches, and the distance of their centres of gravity from the bottom = 13.8177 inches very nearly; also the area of the included rectangle $abie$ = 1440 square inches, and the altitude of its centre of gravity 12 inches; in like manner, the area of the rectangle $ABed$ will be found = 5184 square inches, and the altitude of its centre of gravity 48 inches: therefore we shall have

Now the sum of the momenta, divided by the sum of the areas, will give $\backslash frac\{27861.98966016\}\{7528.7808\}\; =\; 37.006$ inches, the altitude of G, the centre of gravity of the section ABCD above the bottom. In like manner, the altitude of R, the centre of gravity of the section MMCD, will be found to be equal $\backslash frac\{123093.98966016\}\{4936.7808\}\; =\; 24.934$ inches; and consequently their difference, or the value of GR = 12.072 inches, will be found.

Suppose the vessel to heel $15^\backslash circ$, and we shall have the following proportion; namely, As radius : tangent of $15^\backslash circ$ :: MX = 54 inches : 14.469 inches = ME or MF; and consequently the area of either triangle MXE or MXF = 390.663 square inches. Therefore, by theorem 4th, as 4936.7808 : 390.663 :: 72 = $nn$ = $\backslash frac\{1\}\{2\}$ AB : 5.6975 inches = RT; and, again, as radius : fine of $15^\backslash circ$ :: 12.072 = GR : 3.1245 inches = RN; consequently RT—RN = 5.6975—3.1245 = 2.573 inches = SW, the stability required.

Moreover, as the fine of $15^\backslash circ$ : radius :: 5.6975 = RT : 22.013 = RS, to which, if we add 2.4934, the altitude of the point R, we shall have 46.947 for the height of the metacentre, which taken from 72, the whole altitude, there remains 25.053; from which, and the half width = 54 inches, the distance BS is found = 59.529 inches very nearly, and the angle SBV = $80^\backslash circ\; -\; 06\text{'}\; -\; 42\text{'}\text{'}$; from whence SV = 58.645 inches.

Again: Let us suppose the mean length of the vessel to be 40 feet, or 480 inches, and we shall have the weight of the whole vessel equal to the area of the section MMCD = 4936.7808 multiplied by 480 = 2369654.784 cubic inches of water, which weighs exactly 85708 pounds avoirdupoise, allowing the cubic foot to weigh 62.5 pounds.

And, finally, as SV : SW (i.e.) as 58.645 : 2.573 :: 85708 : 3760 +, the weight on the gunwale which will sustain the vessel at the given inclination. Therefore a vessel of the above dimensions, and weighing 38 tons, 5 cwts. 28 lbs. will require a weight of 1 ton, 13 cwts. 64 lbs. to make her incline $15^\backslash circ$.

In this example, the deflecting power has been supposed to act perpendicularly on the gunwale at B; but if the vessel is navigated by sails, the centre velique must be found; with which, and the angle of deflection, the projected distance thereof on the line SV may be obtained; and then the power, calculated as above, necessary

necessary to be applied at the projected point, will be that part of the wind's force which causes the vessel to heel. And conversely, if the weight and dimensions of the vessel, the area and altitude of the sails, the direction and velocity of the wind be given, the angle of deflection may be found."