CHAPTER XVI 



120. The moment of inertia of a body about an axis is the sum 

 of the products of the elementary masses which make up the whole 

 body, and the squares of the perpendicular distances of these 

 elementary masses from the given axis. 



The moment of inertia of a lamina could be found in the same 

 way by considering the elementary areas which make up the 

 whole area and the squares of the perpendicular distances of these 

 elementary areas from the given axis. 



Thus, if there is a system of elementary masses m lt m z , m 3 , . . . 

 whose perpendicular distances from a given axis are x lt x z , x 3 . . . 

 respectively, the moment of inertia of that system about the 

 given axis is m^x^ + m 2 # 2 2 + ra 3 # 3 2 + ... or 2 mx z . Or if there 

 is an area which is made up of a number of elementary areas a lt 

 a z , 3 . . . whose perpendicular distances from a given axis are 

 #!, # 2 , i 3 . . .. respectively, the moment of inertia of that area 

 about the given axis is a^x-^ + a 2 x<f + a 3 x 3 2 + ... or 2o# 2 . 



The two following examples will illustrate how the expression 

 for the moment of inertia is introduced in actual problems : 



(1) A body is rotating about a fixed axis with a uniform angular 

 velocity of w radians per second. Taking a small elementary 

 mass m^ situated at a perpendicular distance x 1 feet from the 

 axis ; in 1 second this mass turns through an angle of w radians, 

 and therefore describes a circumferential distance of tax^ feet. 



The circumferential velocity of the mass m 1 = wx^ ft. per sec. 



Kinetic energy of rotation = 5 w 1 (tJi 1 ) 2 ft. pdls. 



u 



The kinetic energy of rotation of the body will be the sum of 

 the kinetic energies of all the elementary masses m^ ra 2 , m 3 . . . 

 situated at distances x^, x z , x 3 . . . respectively from the axis. 



Hence the total kinetic energy 



= -m^ 2 *?! 2 + -w 2 mr 2 2 + ^m 3 wx 3 2 + . . . 



ft. pdls. 



222 



