Inclined Plane Surface Submerged in Liquid

Inclined Plane Surface Submerged in Liquid :-

Center of Pressure:- It is defined as the point of  application of the total pressure on the surfaces.

There are four case of submerged surfaces on which the total pressure force and center of pressure is to be determined:-

(1) vertical plane surface

(2) horizontal plane surface

(3) Inclined plane surface

(4) Curved surface

Inclined plane surface:

consider a plane surface of arbitrary shape immersed in a liquid in such a way that plane of surface makes an angle Ө  with free surface.

Here, 

= Depth of  C.G. of inclined area from free surface.
A= Total area of inclined surface.
h*= Diameter of center of pressure from free surface of liquid.
Ө= Angle made by the plane of the surface with free liquid surfaces.

Assume that plane of surface, if produced meet the free liquid surface at O. then O-O is the axis perpendicular to plane of the surface.
here, ȳ= Distance of C.G. of the inclined surface from O-O.
         y*=Distance of the center of pressure from O-O.
consider small strip of area dA
pressure intensity on the strip     p=९gh
pressure force dF,           
dF=p x Area of strip= ९gh x dA
Total pressure force on whole area,
F= ∫ dF=∫ ९ghdA
h/y =/ ȳ=h*/y*= sinӨ
F=९g sinӨ∫ y dA
∫y dA=A ȳ
F=९g sinӨ ȳ x A
F =९gAħ

Center of Pressure (h*):-
Moment of force , dF, about axis O-O
                                    =dF x y=९g y sinӨdA x y=९g sinӨdA
Sum of all moment of all forces about O-O= ∫dA
                                                                     = ९g sinӨIo

                                                                     =F x y*
F x y* = ९g sinӨI0      
y*=h*/ sinӨ           
F= ९gA 

By theorem of parallel axis Io=Ig + A2
 and by substituting all the values above we get,
h*
If Ө=90°, it is also applicable for vertically plane submerged surfaces.

for more understanding refer this video:

            

Problems:-

PROBLEMS:
(1)A rectangular plate surface 3 m wide and 5 m deep lies in fluid of specific gravity .9 such that its plane makes an angle of 45⁰ with water surface, upper edge 3 m below free water surface. Determine the total pressure?

Solution:
Total pressure , F=W x A x Ӯ
                            =w x A x ( y + d sinθ )
                             9.81 x 900 x 15 x (3 + 5 sin 45)
                             =86.5 N/c

(2)A rectangular plane surface 3 m wide and 4 m deep lies in water in such a way that its plane makes an angle of 30° with the free surface of water determine the total pressure force and position of center of pressure and the upper edge is 2 m below the free surface.
solution:
b=3 m   d=4 m  θ=30°
Distance of upper edge from free surface of water=2 m
total pressure force,   f=ৎgA
where, ৎ=1000 kg/ 
 A=b x d =3 x 4=12 
= Depth of C.G. of plane from free surface
  = 2 + BCSinθ
  = 2 + 2Sin30°
  = 3 m
F= 1000 x 9.81 x 12 x 3
   = 352.167 k N

Center of pressure(h*)

h*

            =16 

h*= 3.111 m