Hydrostatics | CEE Academy

📚

HYDROSTATICS

▢ Density:

❖ Definition: Mass per unit volume

❖ Formula: \(\rho=\frac{m}{V}\)

Table 1: Density Basics

Point	Value / Formula
Homogeneous isotropic substance	Density is scalar
Heterogeneous anisotropic substance	Density is tensor
Dimension	\([ML^{-3}]\)
SI unit	\(kg\ m^{-3}\)
CGS unit	\(g\ cm^{-3}\)
Conversion	\(1\ g\ cm^{-3}=10^3\ kg\ m^{-3}\)
Density of substance
Density of body

❖ Solid vs Hollow Body:

Table 1: Body Density

Body	Relation
Solid body	Density of body = density of substance
Reason	\(V{body}=V{substance}\)
Hollow body	Density of body < density of material
Reason	\(V{body}>V{substance}\)

❖ Immiscible Liquids:

•
Highest density liquid lies at bottom
•
Lowest density liquid lies at top
•
Interfaces are plane

❖ Average Density:

◉ General Formula: \(\rho=\frac{Total\ mass}{Total\ volume}=\frac{m_1+m_2+...+m_n}{\frac{m_1}{\rho_1}+\frac{m_2}{\rho_2}+...+\frac{m_n}{\rho_n}}\)

Table 1: Average Density

Condition	Mean	For 2 liquids	For n liquids
Equal masses	Harmonic mean
Equal volumes	Arithmetic mean	\(\rho=\frac{\rho_1+\rho_2}{2}\)

❖ Effect of Temperature and Pressure:

Table 1: Density Change

Factor	Effect	Formula
Temperature increases	Volume increases; mass constant; density decreases	\(\rho=\rho_0(1-\gamma\Delta\theta)\)
Pressure increases	Volume decreases; density increases
At depth \(h\)	Density increases

◉ Symbols:

•
•
\(K\) = bulk modulus
•
\(\Delta P\) = change in pressure
•
\(\rho_0\) = density at surface
•
\(\rho\) = average density

▢ Relative Density / Specific Gravity:

❖ Definition: Ratio of density of substance to density of water at 4°C

❖ Nature: Dimensionless and unitless

Table 1: Relative Density Formulae

Quantity	Formula / Point
Relative density
Density in \(g/cm^3\)	Numerically equal to relative density
Density in \(kg/m^3\)	\(1000\times R.D.\)
R.D. of substance	\(R.D.=\frac{W_1}{W_1-W_2}\)
R.D. of liquid	\(R.D.=\frac{W_1-W_3}{W_1-W_2}\)

❖ Symbols:

•
\(W_1\) = weight of body in air
•
\(W_2\) = weight of body in water
•

▢ Pressure:

❖ Definition: Force per unit area

❖ Formula: \(P=\frac{F}{A}\)

Table 1: Liquid Pressure

Quantity	Formula / Point
Pressure at depth \(h\)	\(P=h\rho g\)
Total pressure at depth \(h\)	\(P_T=P_0+h\rho g\)
Gauge pressure	\(P-P_0=h\rho g\)
Depends on	\(h,\rho,g\)
Independent of	Amount of liquid, shape of vessel, area considered

❖ Cylinder Filled with Liquid:

Table 1: Pressure in Cylindrical Vessel

Condition	Result
Mean pressure at bottom	\(h\rho g\)
Mean pressure at walls	\(\frac{h\rho g}{2}\)
Force on sides = force on bottom	\(h=r=\frac{d}{2}\)

❖ Instruments:

Table 1: Pressure Measuring Instruments

Instrument	Use
Barometer	Measures atmospheric pressure
Manometer	Measures liquid pressure with respect to atmospheric pressure
Pressure gauge	Measures static pressure of fluid flowing in pipe

▢ Buoyant Force / Upthrust:

❖ Definition: Upward force acting on a body immersed in liquid

❖ Formula: \(U=V\rho g\)

❖ Meaning: Upthrust = weight of liquid displaced by immersed part of body

Table 1: Upthrust

Depends on	Independent of
Volume of body inside fluid	Mass of body
Density of liquid	Density of body
Acceleration due to gravity	Shape and size except immersed volume

❖ Special Point: During free fall of vessel containing liquid, upthrust is zero

▢ Pascal's Law:

❖ Statement: External pressure applied to a closed liquid is transmitted equally in all directions

❖ Points:

•
Pressure acts equally in all possible directions
•
Liquid pressure is always perpendicular to surface
•
Hydraulic press is based on Pascal's law

▢ Archimedes' Principle:

❖ Statement: When a body is fully or partially immersed in liquid, it loses weight equal to upthrust

Table 1: Archimedes Formulae

Quantity	Formula
Loss in weight	\(W_1-W_2=U\)
Upthrust	\(U=V\rho g\)
Apparent weight	\(W_2=W_1-U\)
If body density = \(\sigma\)	\(W_2=V(\sigma-\rho)g\)

❖ Independent Support Case:

•
Sinking solid suspended independently in liquid
•
Weight of liquid increases by upthrust
•
Loss in weight of body = increase in weight of liquid
•
Due to Newton's third law reaction of upthrust acts downward on liquid

❖ Volume of Cavity:

Table 1: Volume of Cavity

Quantity	Formula
Volume of body with cavity	\(V_1=\frac{W_1-W_2}{\rho g}\)
Volume of material without cavity	\(V_2=\frac{W_1}{\sigma g}\)
Volume of cavity

◉ Symbols:

•
\(\sigma\) = density of material
•
\(\rho\) = density of liquid
•
\(W_1\) = weight in air
•
\(W_2\) = weight in water

▢ Principle of Floatation:

❖ Symbols:

•
\(\sigma\) = density of body/material
•
\(\rho\) = density of liquid
•
\(W\) = weight of body
•
\(U\) = upthrust
•
\(V\) = volume of body
•
\(V'\) = volume inside liquid

Table 1: Sinking and Floating

Condition	Weight vs Upthrust	Result
\(\sigma>\rho\)	\(W>U\)	Body sinks
\(\sigma=\rho\)	\(W=U\)	Body just floats / just sinks with entire volume under surface
\(\sigma<\rho\)	\(W	Body floats

❖ Law of Floatation:

Table 1: Floating Body Formulae

Quantity	Formula
Displaced liquid weight	Equal to weight of floating body
Law	\(V'\rho g=V\sigma g\)
Volume inside liquid	\(V'=\frac{\sigma}{\rho}V\)
Fraction inside liquid	\(\frac{V'}{V}=\frac{\sigma}{\rho}\)
% inside liquid	\(\frac{\sigma}{\rho}\times100\%\)
Apparent weight of floating body	Zero

❖ Floating Body SHM:

◉ Condition: Floating body pressed down and released

◉ Time Period: \(T=2\pi\sqrt{\frac{m}{A\rho g}}\)

▢ Equilibrium of Floating Body:

❖ Terms:

Table 1: Floating Body Terms

Term	Meaning
Metacenter	Point where vertical through centre of buoyancy intersects central line
Centre of buoyancy	Point through which buoyant force acts; C.G. of displaced liquid
Central line	Line joining centre of gravity and centre of buoyancy
Centre of gravity	Point through which weight of body acts

Table 1: Equilibrium of Floating Body

Type	Condition	Example / Point
Stable equilibrium	Metacenter lies above C.G.	Heavy bottomed body; ships and boats have heavy bottom
Neutral equilibrium	Metacenter coincides with C.G.	—
Unstable equilibrium	Metacenter lies below C.G.	Heavy topped body; passengers should not stand on moving boat

❖ Translational Equilibrium: Floating body is in translational equilibrium if C.G. and C.B. lie in same vertical line

▢ Melting of Ice and Liquid Level:

❖ Ice Floating in Liquid:

◉ Symbols:

•
\(\sigma\) = density of liquid
•
\(\rho\) = density of water
•
\(m\) = mass of ice
•
•

Table 1: Ice Melting in Liquid

Condition	Volume relation	Liquid level
\(\sigma>\rho\)	\(V_2>V_1\)	Increases
\(\sigma=\rho\)	\(V_2=V_1\)	Unchanged
\(\sigma<\rho\)	\(V_2	Decreases

❖ Ice Containing Substance Melts in Water:

Table 1: Ice with Embedded Substance

	Water level
\(\sigma>\rho\)	Decreases
\(\sigma=\rho\)	Unchanged
\(\sigma<\rho\)	Unchanged

◉ Examples:

•
Ice containing metal melts in water → water level falls
•
Boat carrying stones: stones unloaded into water → water level decreases
•
Lead shot embedded in ice melts → water level goes down
•
Cork embedded in ice melts → water level unchanged
•
Man in boat drinks pond water → water level unchanged

▢ Body Released Inside Liquid:

❖ Condition: Body of density \(\sigma\) held at depth \(h\) inside liquid of density \(\rho\), where \(\rho>\sigma\), then released

Table 1: Rising Body in Liquid

Quantity	Formula
Resultant upward force	\(F=V(\rho-\sigma)g\)
Acceleration inside liquid	\(a=\frac{\rho-\sigma}{\sigma}g\)
Velocity at liquid surface
Height raised in air	\(h'=\frac{\rho-\sigma}{\sigma}h\)

▢ Floating in Two Immiscible Liquids:

❖ Condition: Homogeneous block floats in two immiscible liquids of densities \(\sigma_1\) and \(\sigma_2\)

❖ Formula: \(\rho=x\sigma_1+(1-x)\sigma_2\)

❖ Symbols:

•
•
•
\(\rho\) = density of block

▢ Read and Digest:

Table 1: Important Hydrostatics Points

Fact	Point
Most characteristic property of liquid	Volume conservation
Stable floating object	Centre of buoyancy vertically above centre of gravity
Just floating body pressed down and released	Sinks
Buoyancy depends on	Mass of liquid displaced
Hydraulic press	Based on Pascal's law
Satellite orbiting Earth	Bodies weightless; upthrust zero
Temperature increases	Liquid density decreases; upthrust decreases; apparent weight increases
Gauge pressure	\(P-P_0=h\rho g\)
Gauge pressure	Independent of vessel shape
	Remains unchanged
Mercury barometer on Moon in normal air cabin	\(760\times6\ mm\)
Weightless rubber balloon with 50 g water in water	Weighs zero
Parrot in wire cage starts flying	Apparent weight of cage decreases
Parrot in airtight cage starts flying	Apparent weight remains unchanged
Barely floating air balloon pushed down in water	Sinks to bottom
Finger put into water without touching vessel	Scale pan sinks
Ice melts; water cools 25°C to 4°C	Water level falls
Ice melts; water cools 4°C to 2°C	Water level rises
Hydrogen balloon easiest to lift	1 kg lightly packed feathers due to large volume and greater buoyancy
Cotton and iron same mass in vacuum	Both weigh same; no buoyant force
Wooden rod in pond	Cannot float vertically because metacenter lies below C.G.
Sudden fall in atmospheric pressure	Predicts possibility of storm
Same upthrust in liquid	Same immersed volume
Hydrostatic pressure	Independent of area; depends on depth and density

▢ High-Yield Recall:

Table 1: Hydrostatics One-Liners

Fact	Answer
Density	\(\rho=\frac{m}{V}\)
Density dimension	\([ML^{-3}]\)
Density SI unit	\(kg\ m^{-3}\)
Density CGS unit	\(g\ cm^{-3}\)
Relative density
R.D. of substance	\(\frac{W_1}{W_1-W_2}\)
R.D. of liquid	\(\frac{W_1-W_3}{W_1-W_2}\)
Pressure	\(P=\frac{F}{A}\)
Liquid pressure	\(P=h\rho g\)
Total pressure at depth	\(P_T=P_0+h\rho g\)
Gauge pressure	\(h\rho g\)
Mean wall pressure	\(\frac{h\rho g}{2}\)
Upthrust	\(U=V\rho g\)
Upthrust in free fall	Zero
Pascal's law	Pressure transmitted equally in all directions
Archimedes principle	Loss in weight = upthrust
Apparent weight	\(W_2=W_1-U\)
Floating body law	\(V'\rho g=V\sigma g\)
Fraction immersed	\(\frac{V'}{V}=\frac{\sigma}{\rho}\)
Floating body apparent weight	Zero
Stable equilibrium	Metacenter above C.G.
Neutral equilibrium	Metacenter coincides with C.G.
Unstable equilibrium	Metacenter below C.G.
Ice melts in water	Water level unchanged
Ice with metal melts in water	Water level falls
Ice with cork melts in water	Water level unchanged
Body released in denser liquid acceleration	\(a=\frac{\rho-\sigma}{\sigma}g\)
Height raised in air	\(h'=\frac{\rho-\sigma}{\sigma}h\)
Density in two liquids	\(\rho=x\sigma_1+(1-x)\sigma_2\)
Hydraulic press	Pascal's law
Hydrostatic pressure depends on	Depth, density, gravity

Q1.

A block of wood floats with 2/3 of its volume submerged. The density of wood is:

📅MOE 2014

Q2.

A body weighs 60g in air and 40g in water. Its specific gravity is:

📅IOM

Q3.

A body weighs 160g in air, 130g in water and 136g in oil. The specific gravity of oil is:

Q4.

If g decreases by 2%, the barometric height of mercury:

Q5.

A beaker has 3cm oil (SG=1.2) and 10cm water. Pressure at bottom in cm Hg (SG=13.6):

Q6.

An iceberg (2100 cm³, ρ=0.5 g/cm³) floats in seawater (ρ=1.2 g/cm³). Volume immersed is:

Q7.

Fraction of wooden raft (ρ=0.8 g/cc) outside seawater (ρ=1.2 g/cc):

Q8.

Alloy with 75% metal (SG=10) and 25% metal (SG=5). Density of alloy (kg/m³):

Q9.

Wooden block floats with 40% volume outside liquid (ρ_liquid=1.2 g/cm³). Density of wood:

Q10.

2kg wooden block floats with 1/4 volume submerged. Downward force to fully submerge:

Q11.

Block weighs 24g in air, 21g in water. Its weight in liquid (SG=1.1):

Q12.

How much lead (SG=11) should be added to 10g cork (SG=0.2) to just float on water?

📅IOM 2010

Q13.

Air bubble radius doubles rising from lake bottom (atm pressure = H water column). Lake depth:

Q14.

Combination of bodies A (SG=p₁) and B (SG=p₂) neither floats nor sinks in liquid (SG=p). Mass ratio:

Q15.

120kg wooden block (ρ=600 kg/m³) floats. Additional mass to just sink:

Q16.

Hydrogen balloon (V=1000 m³, ρ_H=0.09 kg/m³) in air (ρ=1.29 kg/m³) can lift:

Q17.

Boat (3m×2m) sinks 1cm when man boards. Man's mass:

Q18.

Wooden cube sinks 2cm more when 200g added. Side length:

Q19.

Log (12N, 1000 cm³) pulled halfway out. Tension in line:

Q20.

Body floats with 1/3 outside water and 3/4 outside another liquid. Density of liquid:

Q21.

Ice (10m thick, ρ=0.9 g/cc) floating in lake. Minimum rope length to scoop water:

Q22.

Metallic sphere (200g in air, 120g in water, ρ_metal=5 g/cm³). Cavity volume:

Q23.

Metallic sphere with cavity floats in liquid (ρ_liquid=ρ_metal/8). Cavity to sphere radius ratio:

Q24.

Hydrometer reads SG=1.6. Where is mark 1.5?

📅IOM 2017

15Hydrostatics

Table 1: Density Basics

Table 1: Body Density

Table 1: Average Density

Table 1: Density Change

Table 1: Relative Density Formulae

Table 1: Liquid Pressure

Table 1: Pressure in Cylindrical Vessel

Table 1: Pressure Measuring Instruments

Table 1: Upthrust

Table 1: Archimedes Formulae

Table 1: Volume of Cavity

Table 1: Sinking and Floating

Table 1: Floating Body Formulae

Table 1: Floating Body Terms

Table 1: Equilibrium of Floating Body

Table 1: Ice Melting in Liquid

Table 1: Ice with Embedded Substance

Table 1: Rising Body in Liquid

Table 1: Important Hydrostatics Points

Table 1: Hydrostatics One-Liners

More CEE preparation resources