CENTER of gravity, in mechanics, that point about which all the parts of a body do; in any situation, exactly balance each other. Hence, 1.1f a body be suspended by this point as the center of motion, it will remain at rest in any position indiffe rently. 2. If a body be suspended in any other point, it can rest only in two posi. tions, viz. when the said center of gravi ty is exactly above or below the point of suspension. 3. When the center of gra vity is supported, the whole body is kept from falling. 4. Because this point has a constant endeavour to descend to the center of the earth ; therefore, 5. When the point is at liberty to descend, the whole body must also descend, ei ther by sliding, rolling, or tumbling down. 6. The center of gravity in regu lar uniform and homogeneal bodies, as squares, circles, &c. is the middle point in a line connecting any. two opposite points or angles ; wherefore, if such a line be bisected, the point of section will be the center of gravity.
To find the center of gravity of a tri angle. Let B G (Plate III. Aliscell. fig. 1,) bisect the base A C of the AB C, it will also bisect every other line.
1) E drawn parallel to the base, conse. quently the center of gravity of the tri angle will be found somewhere in the line B G. The area of the triangle may be considered as consisting of an infinite number of indefinitely small parallelo grams, D, E, b, a, each of which is to be considered as a weight, and also as the fluxion of the area of the triangle, and so may be expressed by 2 y a, (putting B = x, and E = y) if this fluxionary weight be multiplied by its velocity x, we shall have 2 y x x for its momentum. Now put B -- a and A C = b, then B G (a) : A E (b) B F (x) : 1) E_ bx 2 y, therefore the fluxion of the a b x weight 2 y.b and the fluxion of" a b x x the momenta 2 y x x3 the fluent of the latter, viz. 3 — divided a l x by the fluent of the former, viz. will give 2 x for the distance of the point from B in the line B F, which has a velo city equal to the mean velocity of all the particles in the triangle D B E, and is therefore its center of gravity. Conse quently the centre of gravity of any tri angle A B C, is distant from the vertex B s 2 13 G, a right line drawn from the an gle B bisecting the base A C. And since the section of a superficial or hollow cone is a triangle, and circles have the same ratio as their diameters, it follows that the circle, whose plane passes through the center of gravity of the cone, is of the length of the side distant from the vertex of the said cone.
To find the center of gravity of a solid cone. As the cone consists of an infinite number of circular areas, which may be considered as so many weights, the ter of gravity may be found as before, by putting B E = x (fig. 2.) B G = a, the circular area D F B = y, and A G C ; and from the nature of the cone, a' : xs b xl b : y a' out y = flux a ion of the weights; and y b x3x = fluxion of the momenta, whence the b xa fluent of the latter, viz. — divided by ' 4a" b x3 the fluent of the will give x for the center of gravity of the part B E F, consequently the center of gra vity of the cone A B C G is distant from the vertex B of the side B C, in a circle parallel to the base.
To find the center of gravity in a paral lelogram and parallelopiped, draw the diagonal A D and B G (fig. 3,) likewise C B and H F; since each diagonal A 1) and C B divides the parallelogram ACDB into two equal parts, each passes through the center of gravity : consequently the point of intersection, I, must be the cen ter of gravity of the parallelogram. In like manner, since both the plane Cl1FH and A D G 1: divide the parallelopiped into two equal parts, each passes through its center of gravity, so that the common intersection I K is the diameter of gravi ty, the middle whereof is the center. After the same manner may the center of gravity be found in prisms and cylin ders, it being the middle point of the right line that joins the center of gravity of their opposite bases.
The center of gravity of a parabola is found as in the triangle and cone. Thus, let B F in the parabola A B C (fig. 4) be equal to x, D B = y, then will yi7 be the fluxionary weight, and yx.i: the fluxion of the momenta ; but from the nature of the curve we have y = x ; whence yx = a!, and y x3 = Aix, whose fluent 2 divided the the fluent of ski!: will give 5 x = B F for the distance of the center of 5 gravity from the vertex B in the part of D 13 E ; and so .; of B G is that center in the axis of the whole parabola A B C from the vertex B.
The center of gravity in the human body is situated in that part which is call ed the pelvis, or in the middle between the hips. For the center of gravity of segments, parabolics, conoids, spheroids, &c: we refer to Wolfius.