Hydrostatics

Hydrostatics Points : hydrostatics chapter, total pressure, centre of pressure resultant pressure, equilibrium of floating bodies, archimedes principle Hydrostatics We have discussed in the at section that a liquid, at rest, exerts some pressure on all sides of the container. We have been solving numerical problems to know the intensity of such pressures. In this chapters we snail discuss the total presser on a surface and its position (i.e., the point, where the total pressure acts). The term hydrostatics means the study of pressure exerted by a liquid at rest it has been observed that the direction of such pressure a always at right angles to the surface, on which it acts. In this chapter only the study of water will be dealt with, unless specified, otherwise, Total Pressure We total pressure on an immersed surface, may be defined as the total pressure extorted by the liquid on it. Mathematically total pressure.

P = P₁a₁ + P₂a₃ + P _aa₃
Where P₁a₁ intensities of pressure on different
strips of the surface and
a₁a₂a₃ areas of the corresponding strips Centre of Pressure Resultant Pressure The intensity of pressure on an immered surface is not uniform), but increase with depth. As the pressure is greater over the lower portion of the figure, therefore the resultant pressure, on an immersed surface, will act at some point below the centre of gravity of the Immersed surface and towards the lower edge of the figure. The point through which this resultant pressure acts, is known as centre of pressure and is always expressed In terms of depth from the liquid surface, In the following pages we shall discus the centre of pressure of vertically as well as Inclined immersed surface, Total pressure on a Horizontally Immersed surface.
Consider a plane horizontal swfeco immersed iii a liquid as shown.
Let
w = specific weight of the liquid,
A = Area of the immersed surface in m²,
x = Depth of the horizontal surface from the liquid level in metres

We know that the total pressure on the surface,
P = Weight of the liquid above the immersed surface = Sp. wt. of liquid x Volume of liquid = Sp. wt. of liquid x Area of Surface x Depth of liquid = w A x kN
Where w is the specific weight of the liquid in kN/m3.

Note: If the given liquid is water, then its specific weight taken as 9.81 kN/m3.

Example. A rectangular tank 4 metres long 2 metres wide contains water up to a depth of 2.5 metres. Calculate the pressure on the base of the tank.

Solution

Centre of pressure of a vertically Immersed surface:
Consider a plane surface immersed vertically, in a liquid as shown.

First of all, let us divide the whole immersed surface into a number of small parallel strips as shown in the figure.
Let
w = Specific weight of the liquid,
A = Area of the immersed surface, and
x = Depth of centre of gravity of the immersed surface from the liquid surface.

Let us consider a strip of thickness dx, width b and at a depth of x from the free surface of the liquid as shown in Fig.

We know that intensity of pressure on the strip wx and area of the strip = bdx
Pressure on the strip, p - Intensity of pessure x Area = wx . bdx
Moment of this pressure about the liquid surface,
=(wx . bdx) x = 2² d.x Now the ‘sum of Equilibrium of Floating Bodies We see that whenever a body is placed over a liquid, either it sinks down or floats on the liquid. If we analyse the phenomenon of floatation, we find that the body, placed over a liquid, is subjected to the following two forces.

1. Gravitational force
2. Upthrust of the liquid.

Since the two forces act opposite to each other, therefore we have to study the comparative effect of these forces, A little consideration will show, that if the gravitational force is more than the upthrust of the liquid, the body will sink down, But if the gravitational force is less than the upthrust of the liquid, the body will float. This may be best understood by the Archimedes principle as discussed below. Archimedes Principle The Archimedes’ principle states, “Whenever a body is immersed wholly or partially in a fluid, it is buoyed up (i.e., lifted up) by a force equal to the weight of fluid displaced by the body.” Or in other words, whenever a body is immersed wholly or partially in a Quid, the result** force acting on it, is equal to the difference between the upward pressure of the fluid on its bottom the downward. Q.1.Define the total pressure The total pressure on an immersed surface, may be defined as the total pressure exerted by the liquid on it. Mathematically total pressure

P = P₁ x a₁ + P₂ x a₂ + P₃ x a₃ + ........ Q.2. Write down the different type acting total pressure on the plate Ans: (1) Horizontal immersed surface
(2) Vertical immersed surface
(3) Inclined immersed surface Q.3. Define the center of pressure? Ans. The intensity of pressure on an immersed surface is not uniform, but increase with depth. As the pressure is greater over the lower portion therefore the resultant pressure on an immersed surface will act at some points blow the center of gravity. The point through which this resultant pressure acts is known as center of pressure .