243-] MOMENTS OF INERTIA. 



Expressions of the form ^mr^r^ or r^dm, where r v r% are 

 the distances of m or of dm from two planes (usually at right 

 angles), are called moments of deviation or products of inertia. 



241. The determination of the moment of inertia of a con- 

 tinuous mass is a mere problem of integration ; the methods 

 are similar to those for finding the moments of mass of the first 

 order required for determining centroids (see Part II., Chapter 

 III.), the only difference being that each element of mass must 

 be multiplied by the square, instead of the first power, of the 

 distance. 



A moment of inertia is not a directed quantity ; it is not 

 a vector, but a scalar ; indeed, it is a positive quantity, provided 

 the masses are all positive, as we shall here assume. 



The moment of inertia of any number of bodies or masses 

 for any given point, plane, or line is obviously the sum of the 

 moments of inertia of the separate bodies or masses for the 

 same point, plane, or line. 



242. The moment of inertia ^mr* of any body whose mass 

 is M= 2<m can always be expressed in the form 



where r is a length called the radius of inertia, arm of inertia, 

 or radius of gyration. This length r Q is evidently a kind of 

 average value of the distances r, its value being intermediate 

 between the greatest r 1 and least r" of these distances r. For 

 we have 2,mr' 2 >2mr 2 >2mr" 2 , or, since 



243. As an example, let us determine the moment of inertia 

 of a homogeneous rectilinear segment (straight rod or wire of 

 constant cross-section and density) for its middle point (or, 

 what amounts to the same thing, for a line or plane through 

 this point at right angles to the segment). 



