474 X- STATICS OF DEFOEMABLE BODIES. 



If we take for the axes of x and y the principal axes of the area 



of notation the integral / / xydxdy vanishes. Accordingly a rota- 



tion about a principal axis through the center of mass of the plane 

 of notation developes a couple about that axis of magnitude ggdoSnl 

 tending to right the body. We have accordingly for the whole 

 moment of the righting couple 



50) L = gdco (px| mV). 



On account of the change in the immersed part the center of 

 buoyancy has moved from B to B'. If we draw a vertical in the 

 new position through B\ the point M in which it cuts the line BG 

 is called the metacenter and the distance MG = h, the metacentric 

 height. Since the couple acting on the body is composed of the two 

 forces, mg acting downward at G and upward at B', it is evident 

 that if the equilibrium is stable or the righting couple is positive M 

 must be above G. The arm of the couple being the horizontal 

 projection of MG is equal to h x -d(o 9 and L = mgh x d(D. We 

 accordingly have, inserting this value of L in equation 50) for the 

 metacentric height 



51) mh x 



dividing by m and writing = V, the volume of the displaced liquid, 



52) k_j?_ 6 . 



The equilibrium is stable or unstable according as this is positive or 

 negative. 



For the displacement about the F-axis we have in like manner 

 a couple proportional to the displacement, with a new metacentric 



53) ft*=y-&, 



where K y is the radius of gyration of the plane of notation about 

 the F-axis. It is evident that the metacentric height is greater for 

 the displacement about the shorter principal axis of the section. Thus 

 it is easier to roll a ship than to tip it endwise. 



Since the rotation about either axis is resisted by a couple 

 proportional to the angular displacement, the body will perform 

 small harmonic oscillations about the principal axes with the periodic 



times 



and ! 



where K x and K y are respectively the radii of gyration of the solid 

 about the principal axes in the plane of notation. 



