39 CHAPTER V. THE CENTER OF GRAVITY. 39. DEF. THE CENTER OF GRAVITY of a body, or of a system of any number of bodies connected together, is that point upon which the body (or the system) will balance when placed in any position whatever; the point itself being maintained at rest, and the body (or the system) being acted on by no other force than the pressure arising from the weight of the matter composing it. [The Center of Gravity is (as it were) that fulcrum, round which the body, when placed in any position, has no kind of tendency whatever to turn, although the body be capable of being moved in any plane, and in any direction, about that fulcrum.] 40. PROP. XVII. If a body balance itself upon a line in all positions, the center of gravity of the body is in that line.* That is to say; If there be a line round which a body can be made to revolve, such that if when the line is put into any position, the body (after being made to revolve round the line into any position) remains at rest without having any tendency to move, the center of gravity of the body lies in that line. B H If possible, let G, the center of gravity of the body AB, not lie in the line CD C on which the body balances in all positions; and let the body and line be so placed, that G and CD may be in the same horizontal plane. From G draw A GH perpendicular to CD, and GF vertical. Now since (by Def. Art. 39,) if G be supported the body remains at rest when placed in any position, the vertical pressure which arises from the weight of the matter composing the body must always pass through G, or else that pressure could not be rendered inoperative (as it is supposed to be) by G being maintained at rest. This pressure therefore, in the position of the body represented in the figure, will act perpendicularly at an arm GH, and will have a tendency to turn the body round the point H, which, since it is a point in a line CD fixed in position, may be considered to be a fixed fulcrum. But, by hypothesis, when CD is supported the body balances in all positions upon it. Wherefore G does not lie out of the line CD, i. e. it lies in it. Q. E. D. 42. PROP. XVIII. To find the center of gravity of two heavy points*, and to shew that the pressure at the center of gravity is equal to the sum of the weights in all positions. *The "heavy points" spoken of in this Proposition and the next, are exceedingly small material bodies, and not geometrical points, which (being defined by Euclid to be "without parts") are without length, or breadth, or thickness, and therefore can be of no weight, since they can contain no Matter. Let P and Q be the weights of two heavy points A and B. Join AB, and take AC AB: Q : P+Q; : M N Let ACB be in any position. Through C draw MCN horizontal; and through A and B draw the vertical lines PAM and QNB,-these last are the lines in which the weights P and Q act. Then the angles at M and N being right angles, and the angle ACM being equal to the opposite angle BCN, the triangles ACM and BCN are equiangular. Now supposing P to act on the horizontal lever MCN at M at right angles to CN, (Art. 15), and Q at N at right angles to CN, since PQ CB: CA : :: CN CM, by equiangular triangles, therefore P and Q will balance, and the pressure on C is PQ, acting vertically. Props. II. and VII. And this, being true for any position, is true for all positions of A and B. Wherefore, by definition of Center of Gravity, C is the Center of Gravity of A and B, and the pressure at C is P + Q, acting vertically downwards. 43. PROP. XIX. To find the center of gravity of any number of heavy points; and to shew that the pressure on the center of gravity is equal to the sum of the weights in all positions. Let A, B, C, three heavy points whose weights are P, Q, R, be connected together and placed in any position. Join AB, and take D a point a in AB such that E : AD AB Q: P+Q; P+Q VR .. AD: AB-AD, or DB, :: Q: {P+Q}-Q, or P; therefore D is the center of gravity of A and B, and the pressure produced by P and Q in all positions of the system, is a pressure P+Q acting vertically at D. Prop. xvIII. Join DC, and take in DC a point E such that .. DE: EC :: R : P+Q; therefore E is the center of gravity of the weights P+Q acting at D and R acting at C; and if E be supported, those weights are supported in any position of the system. Since therefore the system will balance itself in all positions on E, that point is its Center of Gravity;—and the Pressure on E is P+Q+ R. The construction here applied to a system of three bodies is applicable to a system of any number of bodies. Wherefore, "The center of gravity of any number of heavy points may always be found, and the pressure on the center of gravity is equal to the sum of the weights." Q. E. D. 44. [By the definition given in Art. 39, of "the Center of Gravity" of a body, it will be understood that to have a center of gravity a body must have Weight. Now in the next two Propositions we are required to find the centers of gravity of a line, and of a plane; the former of which is defined by Euclid to have length merely, without either breadth or thickness, and the latter, though possessing length and breadth, is defined to be without thickness. A geometrical line, or plane, therefore, can have no weight, since there can be no weight where matter does not exist, and matter existing under any form is of three dimensions, or has length, and breadth, and thickness. The line therefore and the plane, of which we have to find the centers of gravity, are not the line and plane of geometry. But the line of which the center of gravity is determined in the next Proposition, is supposed to be formed of very small equal heavy bodies placed at the same distance one from the other along the whole length of the line. And the plane triangle referred to in the next Proposition but one, is supposed to be made up of such lines arranged parallel to any one of the sides of the triangle, and at equal distances one from the other.] 45. PROP. XX. To find the center of gravity of a straight line. Let AB be a straight line composed of small equal heavy bodies ranged at equal distances one from the other from end to end of it. Bisect AB in C, and let P and Q be two of the small heavy bodies which are a P equally distant from C. Then 9 B |