The center of masses – the most important geometrical and technical characteristic of a body. Without calculation of its coordinates it is impossible to present designing in mechanical engineering, the solution of tasks of construction and architecture. Exact determination of coordinates of the center of weight is made by means of integral calculus.

## Instruction

1. It is necessary to begin always from simple, gradually passing to more difficult situations. You recognize that the center of mass of a continuous flat figure of D which density ρ is constant and evenly distributed in its limits is subject to definition. The argument x changes from and to b, y from with to d. Break a figure a grid vertical (x=x(i-1), x=xi (i=1.2, …, n)) and horizontal straight lines (y=y(j-1), y=xj (j=1.2, …, m)) into elementary rectangles with the bases khi=xi-x (i-1) and heights ∆yj=yj-y (j-1) (see fig. 1). At the same time the middle of an elementary piece of xi find as ξi= (1/2) [xi+x(i-1)], and ∆yj height as ηj= (1/2) [yj+y(j-1)]. As density is distributed evenly, the center of mass of an elementary rectangle will coincide with its geometrical center. That is Htsi=ξi, Yцi=ηj.

2. The mass of M of a flat figure (if it is unknown), calculate as the work of density on the square. Replace the elementary square at ds= ∆ with xiyj=dxdy. Present ∆mij in the form of dM=ρdS=ρdxdy and receive its weight on the formula given on the drawing. 2a. At small increments consider that the mass of ∆mij, is concentrated in a material point with coordinates of Htsi=ξi, Yцi=ηj. From problems of mechanics it is known that each coordinate of the center of mass of a system of material points is equal to fraction which numerator contains the sum of the static moments of mass of mν of rather corresponding axis, and the denominator is equal to the sum of this masses. The static moment of mass of mν, concerning an axis of 0th is equal уν*mν, and relatively 0u хν*mν.

3. Apply this rule to the considered situation and receive approximate values of the static moments _kh and _u in a look _u ≈ {∑ ξνρ ∆ xν ∆ yν }, _kh ≈ {∑ ηνρ ∆ xν ∆ yν } (summation was made on ν from 1 to N). The expressions of the sum entering the last are integrated. Pass to limits from them at ∆хν → 0 ∆yν → 0 and write down final formulas (see fig. 2b). You find coordinates of the center of masses division of the corresponding statistical moment into M figure lump.

4. The methodology of obtaining coordinates of the center of mass of a spatial figure G differs only in the fact that there are threefold integrals, and the static moments are considered rather coordinate planes. It is worth to remember and that density is not necessarily constant, that is ρ (x, y, z≠const. Therefore final and samya the general answer has an appearance (see fig. 3).