251.] MOMENTS OF INERTIA. 137 



If, in particular, this plane contains the centroid G of the 

 mass M t we have 2w/ = o, so that the formula reduces to 



^mq^ = 2mg 2 + Md*. (3) 



Introducing the radii of inertia ^ ', ^ , this can be written 



(3') 



249. Similar considerations hold for the moments of inertia 

 ^mp with respect to two parallel planes TT, TT' at the 

 distance d from each other. We have, in this case, /'=/ d\ 



, (4) 



and if the plane TT contain the centroid G, 



(5) 



250. Of special importance is the case in which one of the 

 lines (or planes), say / (TT}, contains the centroid. The formulae 

 (3)> (3 ; )> an d (5) hold in this case ; and if we agree to designate 

 any line (plane) passing through the centroid as a centroidal 

 line (plane), our proposition can be expressed as follows : The 

 moment of inertia for any line (plane] is found from the moment 

 of inertia for the parallel centroidal line (plane] by adding to 

 the latter the product Md 2 of the whole mass into the square of 

 the distance of the lines (planes]. 



It will be noticed that of all parallel lines (planes) the 

 centroidal line (plane) has the least moment of inertia. 



251. Exercises. 



Determine the radius of inertia of the following homogeneous masses : 



(1) Rectangular plate of length /and width h, for a centroidal line 

 perpendicular to its plane. 



(2) Area of equilateral triangle of side a : (a) for a centroidal line 

 parallel to the base ; (b) for an altitude ; (c) for a centroidal line per- 

 pendicular to its plane. 



(3) Circular disc of radius a: (a) for a tangent; (d) for a line 

 through the centre perpendicular to the plane of the disc ; (c) for a 

 perpendicular to its plane through a point in the circumference. 



