EQUILIBRIUM OF FLOATING BODIES 



33 



displaced position shown. Through H' draw a vertical H' M to meet 

 H G in M. 



The weight of the body now acts vertically downwards through G, and 

 the equal force of buoyancy vertically upwards through H' , each of these 

 forces being equal to W, the weight of the body, and together forming a 

 couple whose arm is G N, the perpendicular from G on to H' M. 



Obviously if M is above G this couple tends to restore the body to 

 its equilibrium position, so that the equilibrium is stable, unstable, or 

 neutral, according as M is above, below, or coincides with G. 



If the angle of displacement = 0, the magnitude of the righting 

 moment = W . G N = W . G M sin 9. 



As 6 is increased the position of the intersection M of the verticals 

 through H and H 1 will in general 

 move and will approach or recede 

 from G. The point M, for an 

 infinitely small angle of displace- 

 ment, is called the Metacentre of 

 the body, and the distance G M 

 is called the Metacentric Height. 



Evidently in ship design it is 

 of the highest importance that 

 the metacentre should be above 

 G, under all conditions of loading 

 and under any circumstances of 

 rolling. 



The height G M may be 

 determined experimentally by 

 placing two equal weights P at equal distances x from the centre line of 

 the vessel, when floating on an even keel and in its equilibrium position. 



Let PFbe the weight of the vessel, including the weights P. 



Then if both weights be moved through a distance 6 x to the right, 

 the C. G- of the vessel will move through a distance G G' (Fig. 17) 

 where 



Also G G' will be parallel to the direction in which P is moved, i.e., will 

 be perpendicular to G M, since if any portion of a body be moved in a 

 given direction, the C. G. of the whole moves in the same direction. 

 If 6 be the angle of heel produced by this shift of the weights 

 G G' = G M tan 0. 



H.A. 9 



FIG. 17. 



