214-216] MOMENTS OF INERTIA. 229 



inertia about the original axis of the whole mass placed at the centre of 

 inertia. 



Let #, y, z be the coordinates of any particle of the system, m its mass, 

 x, y, z the coordinates of the centre of inertia, ^/, y , z those of the particle m 

 relative to the centre of inertia. 



Then x=x+x , y=y+y , z=z + 



Now 2m# 2 = 2m (x + # ) 2 = ^ 2 2^ + 2m.*/ 2 . 



So 2 wy 2 = j/ 2 2?ft -f 2my 2 . 



Hence 2m (^ 2 +y 2 ) = 2m (x 2 +y 2 ) + (^ 2 +y 2 ) 2m, 



which is the theorem stated. 



II. The moment of inertia of a plane disc of any form about any axis 

 perpendicular to its plane is the sum of those about any two rectangular axes 

 in the plane which meet on the first axis. 



For, taking the axes to be those of z, x, y, and taking the point where they 

 meet for origin, the moments of inertia about the axes of x and y are respectively 

 2m?/ 2 and 2m# 2 , and the moment of inertia about the axis of z is 2m(# 2 -fy 2 ). 



III. To compare the moments of inertia of a disc about different axes in 

 its plane. 



For parallel axes we can use Theorem I. and it will therefore be sufficient 

 to consider axes in different directions through the origin. Let 6 be the angle 

 which any line makes with the axis x. The distance of any point (x, y) from 

 this line is x sin 6+y costf, and thus the moment of inertia about the line 

 is 2m (y cos 6 - x sin fff 1 sin 2 6 2 (mx 2 } + cos 2 6 2 (my 2 ) - 2 sin 6 cos 6 2m^y. 



The expression for the moment of inertia about a perpendicular line 

 would be 



cos 2 & 2 (ma?) + sin 2 6 2my 2 + 2 sin 6 cos 02 (mxy\ 



The quantity 2 (mxy] is known as the product of inertia with respect to 

 the axes of x and y (in two dimensions). For new axes obtained by turning 

 through an angle 6 it has the value 



(cos 2 - sin 2 6} 2 (mxy) - 2 sin 6 cos 6 {2 (my 2 ) - 2 (m^ 2 )} . 



We can always choose the axes of (x, y} so that this quantity 2 (mxy) 

 vanishes. When this is done the axes of x and y are called Principal axes 

 of the disc. The directions of the principal axes vary with the point chosen 

 as origin. 



Now suppose the axes of x and y are principal axes of the disc at the 

 origin. Let A, =2 (my 2 ), be the moment of inertia with respect to the axis 

 of #, and Sj = 2 (mx 2 ), be the moment of inertia with respect to the axis y. 

 Then the moment of inertia about a line through the origin making an angle 

 6 with the axis x is A cos 2 B +B sin 2 0. 



If an ellipse whose equation is Ax*+By * = const, be supposed drawn on 

 the disc, then the moment of inertia about any diameter of it is inversely 



