METACENTRE is a point in a floating body, the position of which, relative to that of the centre of gravity, determines the conditions for the stability or instability of the equilibrium of that body. The equi librium is stable, if, when the body receives a slight disturbance from its position, it tends, by the combined action of its own weight and the pressure of the fluid in which it is partially immersed, to re-adjust itself in that position after some oscillations ; and the equilibrium is instable if a slight disturbance will cause the body to overeat and acquire a different position, which will then necessarily be one of stable equilibrium.
The surface of a heavy fluid at rest is a horizontal plane ; this portion of this plane which we may imagine to be within the floating body is called the plane of floatation.
When a body floating on a fluid is in equilibrium, the weight of the body applied downwards at its centre of gravity must be equal and exactly opposite to the pressure of the fluid, or, which is the same, to a force equal to the weight of the displaced fluid, applied upwards at the centre of gravity of this portion of the fluid ; hence in this position the right line joining these two centres is vertical, and is called the line of support.
When the body ie slightly disturbed from this position, the plane of floatation evidently alters its position in the floating body ; the centre of gravity of the part immersed also changes, and the thrust of the fluid will in general no longer pass through the centre of gravity of the whole body. The magnitude of this force will however undergo but a very small change, and the body is now subjected to the action of two forces which are equal and contrary, but no longer directly opposite.
The figure and density of a body may however possibly be such that the thrust of the fluid may, after the disturbance, continue to pass through the centre of gravity of the body. The equilibrium is then said to be indifferent, inasmuch as the disturbance communicated only produces a new position of equilibrium. This happens when a body floats in a fluid of equal density with itself, and in other cases, as in a floating sphere. We may observe that if the disturbance of the equi librium consisted merely of an elevation or depression of the centre of gravity, small vertical oscillations would be the consequence : the disturbance considered here is supposed such as to tend to turn the body round its centre of gravity, or to make the original line of support deviate In a vertical plane through a very small angle ; this line is called the axis passing through the centre of gravity. •
When the position of the body is thus disturbed, if the line of thrust when produced upwards meets the above-named axis, the point of intersection ie called the metacentre. The consequent motion of the body will then be the same as if the centre of gravity were fixed, and the thrust applied vertically at the metacentre; hence if the owta centre be sasses the centre of gravity, the thrust tends to re-adjust the axis, and the equilibt ium is stable ; if lel tic that force tend,' to carry the sus farther from its original place, and the equilibrium is in. stable : If the two centres coincide, the equilibrium is indiirt're?ut We give an example :— gg- 1. Allen is a vertical aluare section passing through o, the centre of gravity of a rectangular beam floating on a fluid of twice its specific gravity, this section being at right angles to the fame of the beam ; therefore o a and if gum Is n, g is the centre of gravity of the Anil displaced, o g in the lino of support, and r the plane of firaatat ion, Las. S.
Fig. 2 represents the same body turned round its centre of gravity through a small angle r of or 8. Let o r = 1 ; we must find g', the centre of gravity of ef c u, and draw g' o vertical or perpendicular to a f, cutting the axis 0 ft in o the metamntre. Let m, n, be the centres of gravity of the portions E 0 e, F u and 1t that of the portion fares, then hy : yin :: solid E o e : solid cc r en; and : yn :: solid r of : solid eoren; but the solids E u e, of, are equal : hence Ay : gut :: : ?yr, therefore gg' is parallel to ma, or nearly solid E o e horizontal, and = nearly. Now mu = 2 c D4 = and solid E c c= 4 e x length, solid EFCB= 2 x length ; therefore gg' = 4 o if ux—x L gOg'= 8; therefore go or G 0 =—Tf hence the equilibrium is instable. If the equilibrium were stable, the times of the oscillations would be found by supposing the thrust applied at. o, the point a remaining fixed.