

FIRST AND SECOND MOMENTS 25 



centroid of the entire figure from the base is found to be 



x = Axad+Bx bd + C X ~cd 



X ~ A + B+C ' 



As another example, consider the circular disk with a circular 

 hole cut in it, shown in Fig. 19. Here also the centroid must lie 

 somewhere on the axis of symmetry C# C z . Therefore, denoting 



the radii of the circles by 

 R, r, as shown, and taking 

 moments about the tangent 

 perpendicular to the line of 

 centers, the distance X Q of 

 the centroid from this tan- 

 gent is found to be 



or, since x 2 R and x 1 = R e, where e denotes the eccentricity of 

 the hole, or distance between centers, 



20. Moment of inertia. In the analysis of beams, shafts, and 

 columns it will be found necessary to compute a factor, called the 

 moment of inertia, which depends only on the shape and size of the 

 cross section of the member. 

 This shape factor is usually 

 denoted by /, and is defined as 

 the sum of the products obtained 

 by multiplying each element of 

 area of the cross section by the 

 square of its distance from a 

 given line or point. Thus, in 



Fig. 20, if AA denotes an element of area and y its distance from any 

 given axis 00, then the moment of inertia of the figure with respect 

 to this axis is defined as 



FIG. 20 



