KINETICS OF A RIGID BODY. [239. 



II. Moments of Inertia and Principal Axes. 



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I. INTRODUCTION. 



239. As will be shown in Sections III. and IV., the rotation 

 of a rigid body about any axis depends not only on the forces 

 acting on the body, but also on the way in which the mass 

 is distributed throughout the body. This distribution of mass 

 is characterized by the position of the centroid and by that of 

 certain lines in the body called principal axes. 



It has been shown in Part II., Art. 13, that the centroid of a 

 mass is found by determining the moments, or more precisely, 

 the moments of the first order, ^mx, ^my, *Zmz, of the mass with 

 respect to the co-ordinate planes, i.e. the sums of all mass- 

 particles m each multiplied by its distance from the co-ordinate 

 plane. 



The principal axes of a mass or body can be found by deter- 

 mining .the moments of the second order, ^mx 2 , ^my 2 , ^mx*, 

 ^myz, ^mzx, ^mxy of the mass with respect to the same 

 planes. We proceed, therefore, to study the theory of such 

 moments. 



240. If in a rigid body the mass m of each particle be multi- 

 plied by the square of its distance r from a given point, plane, 

 or line, the sum 



2 mr* = m^rf + m^rf -f - , 



extended over the whole body, is called the quadratic moment, 

 or, more commonly, the moment of inertia of the body for that 

 point, plane, or line. 



If the body is not composed of discrete particles, but forms 

 a continuous mass of one, two, or three dimensions, this mass 

 can be resolved into elements of mass dm, and the sum 

 becomes a single, double, or triple integral (r*dm. 



