MECHANICS 



263 



The center of gravity of any irregular plane figure can be 

 determined by applying the following principle: The static 

 moment of any plane figure with regard to a line in its plane 

 that is, the product of its area A by the distance D of its center 

 of gravity from that line is equal to the algebraic sum of the 

 static moments of the separate parts into which the figure 

 may be divided, with regard to the same axis, or 



AD = aidi-}-asd2, etc., 



in which, 01 at, etc., denote the areas of the subdivided parts of 

 the figure, and di, dz, etc. are the distances of their respective 

 centers of gravity from the reference line. Solving this equa- 

 tion for the value of D, 



D-- 



The figure whose center of gravity is required is divided into 

 separate parts whose centers of gravity are easily ascertained, 

 usually into rectangles 

 or triangles. A suita- 

 ble axis is then assumed 

 with reference to which 

 the expressions aidi, 

 a%dz, etc. are found, and 

 their sum is divided by 

 A = ai + az + etc., the 

 quotient giving D. The 

 center of gravity of the 

 whole figure lies, there- 

 fore, on a line parallel 

 to the assumed axis and 

 distant D from it. In a 

 similar manner, another 

 line containing the cen- 

 ter of gravity is ob- 

 tained, the intersection 

 of the two lines giving 

 its exact position. 



EXAMPLE 1. Find the center of gravity of the cross-section 

 of the dam shown in Fig. 5. 



