CENTER OF GRAVITY AND TIOMENT OF INERTIA.
It wilt be shown later that the strength of a beam depends partly on the form of its cross-section. The following discussion relates principally to cross-sections of beams, and the results reached (like shear and bending moment) will be made use of later in the subject of strength of beams.
47. Center of Gravity of an Area. The student probably knows what is meant by, and how to find, the center of gravity of any flat disk, as a piece of tin. Probably his way is to balance the piece of tin on a pencil point, the point of the tin at which it so balances being the center of gravity. (Really it is midway between the surfaces of the tin and over the balancing point.) The center of gravity of the piece of tin, is also that point of it through which the resultant force of gravity on the tin (that is, the weight of the piece) acts.
By "center of gravity" of a plane area of any shape we mean that point of it which ,corresponds to the center of gravity of a piece of tin when the latter is cut out in the shape of the area. The center of gravity of a quite irregular area can be found most readily by balancing a piece of tin or stiff paper cut in the shape of the area. But when an area is simple in shape, or consists of parts which are simple, the center of gravity of the whole can be found readily by computation, and such a method will now be described.
48. Principle of floments Applied to Areas. Let Fig. 21 represent a piece of tin which has been divided off into any num ber of parts in any way, the weight of the whole being W, and that of the parts W,, etc. Let c„ etc., be the centers of gravity of the parts, C that of the whole, and c„ c„ etc., and c the distances from those centers of gravity respectively to some line (L L) in the plane of the sheet of tin. When the tin is lying in a horizontal posi tion, the moment of the weight of the entire piece about L L is We, and the moments of the parts are WA, WA, etc. Since
the weight of the whole is the resultant of the weights of the parts, the moment of the weight of the whole equals the sum of the moments of the weights of the parts; that is, Ne=WA etc ....
Now let A„ etc. denote the areas of the parts of the pieces of tin, and A the area of the whole; then since the weights are proportional to the areas, we can replace the W's in the preceding equation by corresponding A's, thus: (4) If we call the product of an area and the distance of its center of gravity from some line in its plane, the "moment" of the area with respect to that line, then the preceding equation may be stated in words thus: The moment of an area with respect to any line equals the algebraic sum of the moments of the parts of the area.
If all the centers of gravity are on one side of the line with respect to which moments are taken, then all the moments should be given the plus sign; but if some centers of gravity are on one side and some on the other side of the line, then the moments of the areas whose centers of gravity are on one side should be given the same sign, and the moments of the others the opposite sign. The foregoing is the principle of moments for areas, and it is the basis of all rules for finding the center of gravity-of an area.
To find the center of gravity of an area which can be divided np into simple parts, we write the principle in forms of equations for two different lines as "axes of moments," and then solve the equations for the unknown distances of the center of gravity of the whole from the two lines. We explain. further by means of specific examples.
Examples. 1. It is required to find the center of gravity of Fig. 22, a, the width being uniformly one inch.
The area can be divided into two rectangles. Let C, and 02 be the centers of gravity of two such parts, and C the center of gravity of the whole. Also let a and b denote the distances of C from the two lines OL' and OL" respectively.