CALCULATION OF RADII OF GYKATION 



293 



The integration is over the lamina, and therefore between the limits 

 x a to x = a and y = b to ?/ 



y = b. On integrating, we find 







On taking 6 = 0, the lamina 

 becomes a thin rod, and the re- FlG - 144 



suit agrees with that obtained in the last section. 



240. Homogeneous solid ellipsoid. Let the semi-axes of the 

 ellipsoid be a, b, c, and let us find the radius of gyration about the 

 marjor axis. Taking the principal axes of the ellipsoid as axes of 

 coordinates, and denoting the density of the ellipsoid by p, we have 



ra 



fj J 



where the integration is over the whole volume of the ellipsoid. 

 On performing the integrations, we obtain 



EXAMPLES 



1. Find the radius of gyration of a rod 12 inches lo"ng ab6ut a point distant 

 4 inches from one end. 

 is 2. Find the radius of gyration of a circular disk, 



(a) about an axis through its center perpendicular to its plane ; 



(6) about a diameter. 



3. Show that the radius of gyration of a sphere, radius a, about any diameter 

 is f a 2 , and about any tangent line is | a 2 . 

 V 4. What is the radius of gyration of a cube about an edge ? 



5. What is the radius of gyration of a square lamina about a diagonal ? 



6. Find the radius of gyration of a solid circular cylinder, 

 (a) about an axis ; 



(6) about a generator ; 



(c) about a diameter of one of its ends. 



7. Prove that the radius of gyration of a solid conical spindle about its axis 

 is V^~ a, where a is the radius of its base. 



