CENTRE

(Lai. centrum.) This term has numerous applications in Geometry and Mechanics: thus, centre of a circle, or of an ellipse, is the middle point of any diameter; centre of a curve is the point where two diameters intersect each other; and in mechanics we speak of centres of attraction, conversion, equilibrium, gravity, percu ion, oscillation, . Centre of Attraction, also called Centre of Gravitation,is the point to which bodies tend in consequence of the action ot gravity. Centre of Conversion is the point in a body about which it turns when a force is applied to any part of it, or unequal forces to its different parts. For example, suppose a rod laid on a table to be struck near one extremity in a direction perpendicular to its length; the rod will turn round, but there will be one point in it which remains at rest, or about which, as a centre, the other points turn. This point is the centre of conversion. Centre of Equilibrium of a system of bodies is a point such that if the system were suspended from it, the whole would remain in equilibrium. Thus, the fulcrum or point of support of a lever is its centre of equilibrium. Centre of Gravity. A term employed in Mechanics to denote a certain point in the interior of a body, or system of bodies connected with each other in an invariable manner, so situated that any plane whatever which pa es through it divides the body into two segments of which the weights are exactly equal. Hence, if the centre of gravity of any body or system of bodies be sustained, the whole will remain at rest; for the weights on both sides of a vertical plane pa ing through the point of support being equal, the body can have no tendency to angular motion. Let there be a system of bodies A, B, C, D, E, placed horizontally in the same straight line, S A B O C D E and connected with each other in an invariable manner; and suppose O to be the centre of gravity of the system: by the definition of the centre of gravity, if the system is supported at O, the weights on both sides of O, or the effort which the bodies make to tum about that point on opposite sides of O, will exactly counterbalance each other, and the whole will remain at rest. But the effect of A in taming the system about O depends on the quantity of matter in A, and its distance from O, or the length of the arm of the lever at the extremity of which it acts. The whole effect of the body A is therefore proportional to the ma of A multiplied into the length of the line O A. The same reasoning obviously applies to each of the other bodies,B,C, D, and E; consequently, taking these letters to represent the ma es of the bodies respectively, we have, by the definition, A O A + B O B = C O C + DO D + E O E. Now suppose it were required to determine from this property the position of the centre of gravity of the united bo. dies, or its distance from a given point S in the same straight line: on making SAeo, S B = 6, S C = c, 8D = d, S E z=e, and the unknown distance SO = x, we shall haveOA=i-a,0 B = r-J,OC=c-i,OD = (i-*, and OE = ( — x. Therefore by substitution, K(.x — a)+B(x — 6) = C(c — *) + D(d— i) + E(ex); and by transposition, (A+B+C + D + E)x=aA-f6B + cC + dD + eE; sA+JB+cC+dD+eE. whence * = A+B-t-C + D + E > that is to say, the distance of the centre of gravity of the given system from a given point is equal to the sum of the products of all the ma es into their distances from the point divided by the sum of the ma es. It is obvious, from the equation now given, that the effect of all the bodies to produce motion about the point S, is the same as if they were all united in their common centre of gravity O. It is also obvious that the form of the equation will be exactly the same, whatever may be the number of bodies belonging to the system; and as every body may be regarded as composed of elementary particles, the reasoning208which applies to a system of particles, con- [s. 222]