Test: Hydrostatic Force on Plane Area - Civil Engineering (CE) MCQ

 Hydrostatic force per unit width on a vertical side of a beaker = ¹ ⁄ ₂ x  ρgh²,
where ρ = density of the liquid and
h= height of liquid column.
The hydrostatic force when the beaker is completely filled = ¹ ⁄ ₂ ρg(2h)² = 2ρgh².
Thus, percentage increase in hydrostatic force = 
= 300%

Hydrostatic force per unit width on a vertical side of a beaker = ¹ ⁄ ₂ x ρgh²,
where ρ = density of the liquid and
h= height of liquid column.
Thus if ρ₁ > ρ₂, F₁ > F₂ and F₁ ≠ F₂, when the h is constant.

The depth of the centroid and the centre of pressure y_CP are related by:

where I = the moment of inertia and A = area. If = h ⁄ 2; I = bh³/12 ;A = bh, then

The centroid of the lamina will be located at it’s centre. ( ^d ⁄ ₂). Thus, the depth of the centre of pressure will be h + ^d ⁄ ₂.

Total liquid pressure on the lamina = F = ,
where γ = specific weight of the liquid, 
 = depth of centroid of the lamina from the free surface,
A= area of the centroid.
Now, γ = 9:81 x 10³ N / m³;  
= 0.5 + ¹ ⁄ ₂  x  2m = 1.5 m,
A = 2 * 2 m² = 4 m².
Hence, F = 58.86 kN.

Hydrostatic force per unit width on a vertical side of a beaker = ¹ ⁄ ₂  x  ρgh²,
where ρ = density of the liquid and
h= height of liquid column.
Thus, if the liquid column is halved, the hydrostatic force on the vertical face will become one-fourth.

The depth of the centroid and the centre of pressure y_CP are related by:

where I = the moment of inertia and A= area. None of the quantities I, A and  can be negative. Thus, Y_CP > . For horizontal planes, I = 0, hence Y_CP = 

Hydrostatic force per unit width on a vertical side of a beaker F_v = ¹ ⁄ ₂  x  ρgh²,
where ρ = density of the liquid and
h= height of the liquid column.
Hydrostatic force per unit width on the base of the beaker = F_b = ρgh  x  h = ρgh².
Thus, F_b : F_v = 2 : 1.

Concept:

• When a stationary fluid comes in contact with a solid surface either plane or curved, a force is exerted by the fluid on the surface. This force is called total pressure or pressure force or hydrostatic force.
• Since for a liquid at rest, no tangential force exists, the hydrostatic force acts in the direction normal to the surface.
• The point of application of total pressure on the surface is called the center of pressure (CP).
• The direction of total Hydrostatic force (F) on the Horizontal Plane Surface is normal to the surface as it acts in the vertically downward direction. 

Centre of pressure:
The point of intersection of the line of action of the resultant hydrostatic force and the corresponding surface is called the center of pressure. The Centre of pressure is also defined as the point of application of the total pressure on the corresponding surface.

• According to Pascal’s Law, the pressure or intensity of pressure at a point in a static fluid is equal in all directions.
• For a plane surface of arbitrary shape immersed in a liquid in such a way that the plane of the surface makes an angle θ with the free surface of the liquid:

A = Total area of an inclined surface, h̅ = Depth of centre of gravity of inclined area from a free surface, h* = Distance of centre of pressure from the free surface of a liquid


For vertical plane surface: θ = 90°

The following facts can be concluded from the above equations:

• The Center of pressure lies below the centroid because for any plane surface, the factor I_G/A_h is always positive
• The deeper the surface is lowered into the liquid (i.e. greater is the value of h̅), the closer comes the center of pressure to the centroid of the area
• The depth of the center of pressure is independent of the specific weight of the liquid and is consequently the same for all liquids

Total Pressure:

The total pressure on a surface immersed in a liquid depends on its inclination to the liquid surface.