## Hydrostatic Forces

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# Hydrostatic Forces

## Total Pressure

• The force exerted by the static fluid on the surface in contact with the fluid is called as total pressure. It is denoted by ‘P’.

• It is always normal to the contact surface.

• The surface may be horizontal, vertical, inclined or curved.

P = γ ¯XA

• It is always normal to the contact surface.

• The surface may be horizontal, vertical, inclined or curved.

P = γ ¯XA

## Centre of Pressure

Te point at which the total pressure is supposed to act is called as centre of pressure. It is denoted by ¯h.

¯h = ¯x + I

¯h = ¯x + I

_{G}sin^{2}θ / A ¯X## Vertical Plate Immersed not he Liquid

Consider a plate AB vertically submerged in fluid.

Now consider small strip of area dA lying at a depth of x from ‘O’.

Pressure intensity on the strip,

dp = γ . x

Total pressure force on strip,

dp = Pressure x Area

= γ . x . dA

Total pressure force on the whole area.

Integrate Equation

∫ dp = ∫ γ . x . dA

= γ . ∫ x . dA

But ( ) xdA is the sum of moments of the area of the strip, which is equal to A ¯x.

P = γ . A ¯x

Centre of pressure can be calculated by using law of moment.

Law of moment states, “Sum of moment of all such pressure forces on strip about free surface is equal to the moment of whole body about free surface.

Total pressure force on strip,

dp = γ . x. dA

Moment of force on strip,

dM = dp. x

= (γ.x .dA) x

= γ . x

Now sum of moment of all such pressure forces on strip about free surface.

Integrate Equation

∫ dM = ∫ γ . x

= ∫ γ . x

But we know ∫ X

M = γ . I

But moment of whole body about free surface,

M = P ¯h

Equate the Equations and

P ¯h = γ . I

by parallel axis theorem I

γ (A. ¯x) ¯h = γ (I

¯h = I

¯h = I

The centre of pressure always lies below the centre of gravity by I

For greater depths, centre of pressure approaches centre of gravity as ¯x is larger and I

For more help in

Now consider small strip of area dA lying at a depth of x from ‘O’.

Pressure intensity on the strip,

dp = γ . x

Total pressure force on strip,

dp = Pressure x Area

= γ . x . dA

Total pressure force on the whole area.

Integrate Equation

∫ dp = ∫ γ . x . dA

= γ . ∫ x . dA

But ( ) xdA is the sum of moments of the area of the strip, which is equal to A ¯x.

P = γ . A ¯x

Centre of pressure can be calculated by using law of moment.

Law of moment states, “Sum of moment of all such pressure forces on strip about free surface is equal to the moment of whole body about free surface.

Total pressure force on strip,

dp = γ . x. dA

Moment of force on strip,

dM = dp. x

= (γ.x .dA) x

= γ . x

^{2}dANow sum of moment of all such pressure forces on strip about free surface.

Integrate Equation

∫ dM = ∫ γ . x

^{2}dA= ∫ γ . x

^{2}dABut we know ∫ X

^{2}dA = I_{O}= moment of inertia about free surfaceM = γ . I

_{O}But moment of whole body about free surface,

M = P ¯h

Equate the Equations and

P ¯h = γ . I

_{O}by parallel axis theorem I

_{O }= I_{G}_{ }+ A (¯x)^{2}γ (A. ¯x) ¯h = γ (I

_{G}+ A(¯x)^{2})¯h = I

_{G / }A¯x + A(¯x)^{2 }/ A¯x¯h = I

_{G}/ A¯x + ¯xThe centre of pressure always lies below the centre of gravity by I

_{G}/ A¯x . If I_{G}/ A¯x > 0.For greater depths, centre of pressure approaches centre of gravity as ¯x is larger and I

_{G }/ ¯x becomes smaller.For more help in

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