28 



HYDRAULICS AND ITS APPLICATIONS 



Here is the angle between the tangent plane to the surface at the 

 point (x ?/) and the plane X Y. It follows that 2 (6 A cos 0} is the 

 projection of the surface on this plane. Call this A f , and let x',y', be 

 the co-ordinates of the centroid of this projection on the reference 

 plane. 



Then (8) becomes 



W A' x' X = PF2 (8 A cos x 2 ) 



Similarly 



v _ 2 (B A cos B x*} 



A' x' 

 S (B A cos xy) 



A' x' ~ 



(5) 



(6) 



It follows that the centre of pressure of any curved surface has the 



same co-ordinates as that of its pro- 

 jection on the plane of reference. 



Example. The centre of pressure of the 

 curved surface of a hemisphere having its 

 diametrical plane vertical, and immersed with 

 its upper edge in the surface of the water, is 

 at the same depth as that of the vertical 

 diametrical plane. 



Since 2 (8 A cos 6 x*) is the 

 moment of inertia of the projection 

 of the surface, about the axis Y, 

 expression (5) may be written 



A ' If' 2 l.f 2 



*r a. K K 



FlG - 12 - where k' is the radius of gyration 



about the axis O Y of the projection 

 of the surface on the plane X Y. 



In the case of the hemispherical surface just considered radius r 



x' = r A' = 77 r 2 



A' k'* = 



77 r' 



+ TT r 2 . r 2 = TT r 2 - 



Where the surface is plane, the axes X, O Y may be taken in tho 

 plane itself, when = 0, and the above expressions reduce to 



. 



A x 



v _ 



