Moment of Inertia. 235 



m -or m r* v -_ 



w x r ' or -jw- 



will be the moment of this resistance ; therefore the sum of the 

 moments of these resistances which the particles of L would op- 

 pose to the motion of rotation, produced by p, upon these par- 



ticles, is n J, T V y or.^rj Tmr 2 , for the two expressions are 



r Jvl f JVL / 



the same, since v and FM do not change, whatever be the par- 

 ticle ra, which we consider. 



We hence perceive, that, other things being the same, the re- 

 sistance which the particles of a body oppose to the motion of 

 rotation, communicated to them, is so much the greater as fm r 2 

 is greater. 



The quantity -^ - f mr 2 is called the moment of inertia of 



a body, and fmr 2 the exponent of the moment of inertia. 



360. We shall see soon how the exponent of the moment of 

 inertia in any body may be determined ; but when this expo- 

 nent has been determined w r ith respect to any axis whatever, it 

 is very easy thence to infer, what it must be with respect to any 

 other axis parallel to the former. 



Let AB be any axis, and A'B' another axis parallel to it, 

 passing through the centre of gravity G of the body. Let m be 

 any particle of this body ; and through m suppose a plane mFF', 

 perpendicular to the two axes AB, A'B' ; mF, m F', being drawn, 

 and the perpendicular mP being let fall upon FF', the lines 

 m F, m F', will be perpendicular respectively to AB, A'B 1 . 3^3 m< 



This being supposed, we shall have, 



m F*= m F 1 -f FF' -f 2 FF 1 x PP ; 



Geom. 

 hence, 192. 



j*m- mF^fm- mF' -f f m ' FF' -f/m . 2 FF' X PP. 



Now, since the distance FF' is always the same, whatever be the 



a 2 



particle m under consideration, Jm FF / is simply FF 1 Jm, or 



