Fluid Statics

 Introduction

The branch of physics which deals with study of fluid at rest is called Hydrostatics.$\\$“Hydro”, meaning water/water like fluid$\\$“Statics”, meaning at rest$\\$

A fluid is a substance that flows from one point to another such as liquid and gas.

Density

The ratio of mass of a body to its volume is called density.$\\$Mathematically, Density(ρ)=Mass(m)/Volume(v) $\to$ $ρ = \frac{m}{v}$

When two liquids of equal volume are mixed whose densities are $d_1$ and $d_2$ :$\\$Density of mixture =$\frac{d_1+d_2}{2}$

When two liquids of equal mass are mixed whose densities are $d_1$ and $d_2$ :$\\$Density of mixture =$(\frac{2.d_1.d_2}{d1+d2})$

Relative Density/Specific Gravity

Relative density is different from density. It’s the comparison of a substance’s density with density of pure water at 4 degree celsius.$\\$Ways to calculate relative density/specific gravity:$\\$

Specific Gravity$\\$= $\frac{weight- of- the- body}{weight- of- same- volume -of- water- at- 4 \degree C.} \\$=$\frac{density- of- the- body}{density -of- water- at -4 \degree C}$$\\$$\textbf {NOTE:}$ density of water at 4 degree celsius is 1gm/cc(CGS unit)$\\$Specific Gravity =density of body(only when CGS unit is chosen)$\\$

Some other deviations of Specific Gravity:$\\$

Sp.Gr=$(\frac{wt. -of -object -in -air}{loss -of -wt. -in -water})$ x sp. gr of water at room temperature$\\$Sp.Gr(for a given liquid)=$\frac{wt.- of- given -volume- of- liquid}{wt.- of- equal- volume- of- water}$x sp. gr of water at room temperature.

PRESSURE: 

Laws of liquid pressure

The pressure of a liquid is directly proportional to its density.$\\$$P \propto d$

The pressure of a liquid is directly proportional to its depth from the free surface.$\\$$P \propto h$

Pressure at a point in a liquid is the same in all directions.

Pressure of a liquid is independent of the shape of the vessel.

ARCHIMEDES’ PRINCIPLE

The principle states that , ‘When a body is partially or completely immersed in liquid then loss of weight of body is equal to weight of the displaced liquid.$\\$Mathematically, Loss of weight = weight of liquid displaced

$\textbf{Upthrust:} \\$An upward force acting on a body when it is partially or completely immersed.

According to Archimedes’ principle,$\\$" Upthrust = wt. of liquid displaced"$\\$"Upthrust = wt. of body in air - wt. of body in water"

Mathematically, Upthrust(U)$\\$= d.g.v$\\$(d= density of the liquid in which the body is)$\\$(v= volume of body inside the liquid)

Pascal’s Law

This is the law of transmission of liquid pressure.

The law states that pressure applied to an enclosed liquid(i.e liquid within a container like system), pressure is equally transmitted to every portion of the vessel/container.$\\$We get the relation$\\$$\frac{F_1}{A_1} = \frac{F_2}{A_2}$ (where $F_1$ and $F_2$ are forces and $A_1$ and $A_2$ are area of each opening)

Application of Pascal’s Law

$\text{Hydraulic Press:}$$\\$Smaller force is magnified to a larger force. Force applied at the smaller opening magnifies to a larger magnitude at bigger opening, thus helping to lift heavier loads easily.

As pressure on both ends are equal(From Pascal’s law)$\\$$F_2=\frac{A_2}{A_1} \times F_1$( as $A_1$ is smaller and $A_2$ is larger $F_2$ is a bigger quantity )

$\textbf{Hydraulic Brakes:}$

$\textbf{Hydraulic Cranes:}$

Principle of Floatation

Considering a body of density $d$ is placed in a liquid of density $L$ then,$\\$

$\to$ $d>L$ i.e wt. of body> upthrust , the body sinks$\\$$\to$ $d=L$ i.e wt. of body= upthrust , the body just floats on the surface$\\$$\to$ $d<L$ i.e wt. of body< upthrust , the body floats with certain part of it inside the liquid and certain part of it outside,$\\$fraction outside $= 1-\frac{d}{l}$

Equilibrium in Floating Bodies

For a body such as a ship to keep floating, the Metacenter(MC) of the body should be above its Center of Gravity(CG), else it sinks.$\\$

$\textbf{Metacenter:}$ Point of intersection of the line from CG and the line from Center of Buoyancy(CB).$\\$$\textbf{Center of Buoyancy:}$ The geometric center of the submerged portion of the body.$\\$

Some Important points:

$\to$ Liquid obeys volumetric conservation at all times under normal conditions.$\\$

$\to$ If a body just floating in a liquid is pressed down and released it will sink.$\\$

$\to$ Force of buoyancy depends on the mass of liquid displaced.$\\$

$\to$ Apparent weight of a floating body is 0.$\\$

$\to$ When an ice block floating in water melts the water level remains constant.$\\$

$\to$ If ice containing metal floating in water melts, water level falls.$\\$

$\to$ Even though g(acceleration due to gravity) varies, the barometric height remains constant.$\\$

$\to$ If P is atmospheric pressure total pressure at depth h is$\\$

        P’= P+ d.g.h