Hydrostatic is the study of fluid at rest. A substance that can flow from one point to another is called fluid. The density of a substance is defined as mass per unit volume.
$$ \text {Density,} \rho = \frac mV $$
The relative density of a substance is the ratio of its density of water at 4^{o}C. It is also called the specific gravity.
$$ \text {Relative density,} \rho _r = \frac {\rho }{\rho _w} $$
Where ρ is the density of the substance and ρ_{w} is the density of water at 4^{0}C.
Thrust
A force acting perpendicularly to a surface is called the thrust. For example, our weight on the ground, the weight of the liquid in a beaker at the bottom etc. are examples of thrust.
Force per unit area acting normally on the surface is called the pressure. Mathematically,
$$ P =\frac FA $$
So, pressure is also defined as the thrust per unit area of a surface. The direction of the force resulting from the pressure is determined by the orientation of the surface, and therefore, fluid pressure acts as a scalar quantity.
The unit of pressure is N/m^{2} in SI-units which is also called Pascal, Pa. the dimension of pressure is
[MLT^{-2}]/[L^{2}] = [ML^{-1}T^{-2}]
Derivation of Pressure
Liquids exert pressure at the container due to their weight.
Suppose a liquid in a container such as water in a beaker as shown in the figure. To find the pressure at a point inside it, let us consider a horizontal surface X of area A at that point at a depth h from the free surface of the liquid.
The force acting normally on X is weight of the liquid column directly above it a height h and cross-sectional area A. Since the volume of this liquid column, V = A × h and its mass, m = ρ × V, where ρ is its density, so weight of the liquid is
\begin{align*} W &= mg = \rho V g\\ \text {As the weight acts normally on X, the pressure at the surface is} \\ P &= \frac FA = \frac WA &= \frac {\rho A hg}{A} \rho &= h\rho g \end{align*}
This is an expression for the pressure exerted by a liquid at a depth h. It can be shown that the pressure exerted by a liquid at the sides of the container is the same as the same depth downward.
Laws of Liquid Pressure
From the expression, P = hρg, the liquid pressure follows the following laws:
Pascal’s law of pressure is the law of transmission of liquid pressure. It states that when a pressure is applied to an enclosed liquid, the pressure is equally transmitted to every portion of it.
Suppose a vessel containing water with three opening X,Y and Z of different cross-sectional area A, 2A and A/2 respectively as shown in the figure. These opening areas are closed with three tight pistons to keep water in the vessel. When a force F is applied to X inward, the forces needed to keep the pistons at the same position in Y and Y are F/2 and 2F respectively.
That is, the pressure at each opening,
$$ P = \frac {2F}{2A} = \frac FA = \frac {F/2}{A/2} = \frac FA $$
So, pressure is equally transmitted in all parts of the vessel.
Applications of Pascal’s Law
Hence, a larger force is produced at the larger cylinder. The hydraulic press is a force multiplying device with multiplication factor equal to the ratio of the area of pistons.
The upward force exerted by a fluid on an object which is completely or partially immersed in the fluid is called the upthrust or buoyancy. Because of the upthrust, we can easily lift up a heavy object in water and so, the object has lesser weight in water. When a body is completely immersed in water as shown in the figure, the pressure at its bottom B is greater than at its top T. So, a net upward force acts on a body due to the pressure difference and upthrust or buoyancy is produced in liquid.
Let W_{a}, be the weight of the object in air and W_{w}, the weight in water. Then, upthrust of the liquid,
\begin{align*} U &= \text {weight of the body in air} – \text {weight of the body in water} \\ &= W_a-W_w \end{align*}
So, upthrust in the loss in weight of the object in the fluid.
From the expression, P = hρg, the liquid pressure follows the following laws:
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