134 KINETICS OF A RIGID BODY. [244. 



Let 2 1 be the length of the rod (Fig. 28), O its middle point, 

 p its density (i.e. the mass of unit length), x the distance OP 



A O P B 



Fig. 28. 



of any element dm = pdx from the middle point. Then we 

 have, for the moment of inertia /, 



and for the radius of inertia r , since the whole mass is M= 2 pi, 



r*=*-=\l\ 



244. Exercises. 



Determine the radius of inertia in the following cases. When noth- 

 ing is said to the contrary, the masses are supposed to be homogeneous. 



(1) Segment of straight line of length /, for a perpendicular through 

 one end. 



(2) Rectangular area of length / and width h : (a) for the side h ; 

 () for the .side /; (c) for a line through the centroid parallel to the 

 side h ; (*/) for a line through the centroid parallel to the side /. 



(3) Triangular area of base b and height h, for a line through the 

 vertex parallel to the base. 



(4) Square of side a, for a diagonal. 



(5) Regular hexagon, for a diagonal. 



(6) Right cylinder or prism of height 2 h t for the plane bisecting 

 the height at right angles. 



(7) Segment of straight line of length /, for one end, when the density 

 is proportional to the nih power of the distance from this end. Deduce 

 from this: (a) the result of Ex. (i) ; (b) that of Ex. (3) ; (c) the 

 radius of inertia of a homogeneous pyramid or cone (right or oblique) 

 of height h, for a plane through the vertex parallel to the base. 



(8) Circular area (plate, disc, lamina) of radius a, for any diameter. 



(9) Circular line (wire) of radius a, for a diameter. 



