TOTAL PRESSURE AND CENTRE OF PRESSURE

 

TOTAL PRESSURE AND CENTRE OF PRESSURE

Total Pressure: - It is the force exerted by fluid in rest on a surface.

Centre of Pressure: - It is the point, of a surface, where total pressure is act.

 

Total pressure and centre of pressure on Immersed surfaces: -

1.   Hydrostatic Force on a Horizontally immersed surface:

F = ρgℏA        [ℏ = Depth of the C.G. or centroid from free surface]

2.   Hydrostatic Force on a Vertically immersed surface:

F = ρgℏA      [ℏ = Depth of the C.G. or centroid from free surface]

 

3.   Hydrostatic Force on an inclined plane:

 

Fig.1 | Hydrostatic force on Inclined Plane

A = Area of the Plane

ρ = Density of fluid

θ = Inclination of plane with free surface

y = Distance of element from ‘o’

ȳ = Distance between centroid of plane and ‘o’.

ℏ = Depth of the centroid from free surface.

Let’s assume an elementary stripe of area dA at a distance of y from ‘o’ and at depth h.

Here, h/y= sin θ = ℏ/ ȳ

Force on elementary stripe, dF = P.dA

= ρgh.dA

= ρgysinθ.dA

Therefore Total Pressure on the plane –

F= ∫dF = ∫ ρgysinθ.dA

      = ρgsinθ ∫y.dA

      = ρgsinθȳA    [∵ ∫y.dA = ȳA]

       = ρgℏA           [∵ ℏ =ȳsinθ]

By applying Varignon’s theorem, -

Moment of Total force (F) about ‘o’ = Moments of all forces acting on plane about ‘o’.

F.y* = ∫dF.y     [y*= Assuming a poit where Total pr. Will acts]

      = ∫ (ρgysinθ.dA)y

Or, ρgℏA. y*= ρgsinθ ∫ y².dA

Or, ℏA. y*= I_osinθ         [∵ I_o = ∫ y².dA = MI of the plane about ‘o’]

Or, y*= I_osinθ / ℏA

h* = Depth of the point, where total pr. Act or centre of pr.

h*/sinθ = [(I_g+A ȳ²) sinθ] / ℏA  [∵ Applying parallel axes theorem]

h* = [I_g(sinθ)² / ℏA] + ℏ

 

∵  h* > ℏ

∴ Centre of pressure of  a plane is below the centroid of the plane.

Case – 1 Surface is vertical (i.e. θ =90⁰)

∴ Sinθ = 1 Then, h* = [I_g / ℏA] + ℏ

Case – 2 Surface is Horizotal (i.e. θ =0⁰)

∴ Sinθ = 0 Then, h* = ℏ

 

4.   Hydrostatic Force on a curved plane:

Fig.2 | Hydrostatic force on Curved Surface

                                                                         

Total force on the curved surface (F) = √ (F_x²+F_y²)

Where, F_x =∫dF_x = ∫ρgh.dA.sinθ = Force due to pressure on projected area on vertical Plane.

F_y =∫dF_y = ∫ρgh.dA.cosθ = Weight of liquid supported by the Plane.

 

Direction of force θ = tan^-1(F_y/ F_x)

 

[NOTE:

1.   If w is the sp. Wt. of liquid and h is the depth of any point from the surface, then pr. intensity at the point will be wh

2.   A vertical wall is subjected to a pr. due to one kind of liquid, on one of its sides. The total pr. on the wall per unit length is - wh.(h/2) =wh²/2.

3.   Water pr. per meter length on a vertical masonry wall of dam is - wh²/2 ]