15Hydrostatics

📚
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
\(\frac{Mass\ of\ substance}{Volume\ of\ substance}\)
Density of body
\(\frac{Mass\ of\ body}{Volume\ of\ body}\)
❖ 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
\(\rho=\frac{2\rho_1\rho_2}{\rho_1+\rho_2}\)
\(\frac{n}{\rho}=\frac{1}{\rho_1}+\frac{1}{\rho_2}+...+\frac{1}{\rho_n}\)
Equal volumes
Arithmetic mean
\(\rho=\frac{\rho_1+\rho_2}{2}\)
\(\rho=\frac{\rho_1+\rho_2+...+\rho_n}{n}\)
❖ 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
\(\rho=\rho_0\left(1+\frac{\Delta P}{K}\right)\)
At depth \(h\)
Density increases
\(\rho_h=\rho_0\left(1+\frac{h\rho g}{K}\right)\)
◉ Symbols:
  • •
    \(\gamma\) = coefficient of cubical expansion
  • •
    \(K\) = 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
\(R.D.=\frac{Density\ of\ substance}{Density\ of\ water\ at\ 4^\circ C}\)
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
  • •
    \(W_3\) = weight of body in given liquid
▢ 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
\(V=V_1-V_2=\frac{W_1-W_2}{\rho g}-\frac{W_1}{\sigma g}\)
◉ 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
  • •
    \(V_1=\frac{m}{\sigma}\) = volume displaced before melting
  • •
    \(V_2=\frac{m}{\rho}\) = volume after melting

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

Density of substance \(\sigma\) vs water \(\rho\)
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
\(v=\sqrt{2\left(\frac{\rho-\sigma}{\sigma}\right)gh}\)
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:
  • •
    \(x\) = fraction of block in liquid of density \(\sigma_1\)
  • •
    \(1-x\) = fraction in liquid of density \(\sigma_2\)
  • •
    \(\rho\) = 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
Barometric height despite variation in \(g\)
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
\(\frac{Density\ of\ substance}{Density\ of\ water\ at\ 4^\circ C}\)
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 vessel contains oil (density 0.8 g/cm³) over mercury (density 13.6 g/cm³). A homogeneous sphere floats with half its volume immersed in mercury and the other half in oil. The density of the material of the sphere in g/cm³ is:
📅CEE MBBS 2024B
Q2.
Which of the following is correct for stable equilibrium of a floating body?
📅CEE MBBS 2023
Q3.
A cylindrical vessel of radius \(R\) is filled with a homogeneous liquid. The height at which the average force on the side becomes equal to the force on the bottom is:
📅CEE MBBS 2022
Q4.
A boat of length 4 m and breadth 2 m floats on a lake. It sinks by 1 cm when a man gets on it. The mass of the man is:
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Q5.
Density of a body is 10 g/cm³ then its specific gravity is:
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Q6.
A wooden block floats in water with \(\frac{2}{3}\) of its volume submerged. Find its density.
📅MOE 2014
Q7.
A body weighs \(60\ \mathrm{gf}\) in air and \(40\ \mathrm{gf}\) in water. Find its specific gravity.
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Q8.
A body weighs \(160\ \mathrm{gf}\) in air, \(130\ \mathrm{gf}\) in water and \(136\ \mathrm{gf}\) in oil. Find the specific gravity of oil.
Q9.
If gravitational acceleration decreases by \(2\%\) while atmospheric pressure and mercury density remain constant, the barometric height will:
Q10.
A beaker contains \(3\ \mathrm{cm}\) of oil of specific gravity \(1.2\) and \(10\ \mathrm{cm}\) of water. Express the gauge pressure at the bottom as an equivalent height of mercury (specific gravity \(13.6\)).
Q11.
An iceberg of volume \(2100\ \mathrm{cm^3}\) and density \(0.5\ \mathrm{g/cm^3}\) floats in seawater of density \(1.2\ \mathrm{g/cm^3} \). Find the submerged volume.
Q12.
A wooden raft of density \(0.8\ \mathrm{g/cm^3}\) floats in seawater of density \(1.2\ \mathrm{g/cm^3}\). What fraction remains above the surface?
Q13.
An alloy contains \(75\%\) by mass of a metal with specific gravity \(10\) and \(25\%\) by mass of a metal with specific gravity \(5\). Assuming additive volumes, find its density.
Q14.
A wooden block floats with \(40\%\) of its volume outside a liquid of density \(1.2\ \mathrm{g/cm^3}\). Find the density of wood.
Q15.
A wooden block of mass \(2\ \mathrm{kg}\) floats in water with one-quarter of its volume submerged. What downward force is needed to hold it just fully submerged? Take \(g=10\ \mathrm{m/s^2}\).
Q16.
A block weighs \(24\ \mathrm{gf}\) in air and \(21\ \mathrm{gf}\) in water. Find its apparent weight in a liquid of specific gravity \(1.1\).
Q17.
How much lead (specific gravity \(11\)) must be attached to \(10\ \mathrm g\) of cork (specific gravity \(0.2\)) so that the combination is just completely submerged and neutrally buoyant in water?
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Q18.
An air bubble doubles its radius while rising from the bottom of a lake to its surface. If atmospheric pressure equals the pressure of a water column of height \(H\), find lake depth (isothermal expansion).
Q19.
Two bodies of masses \(m_1,m_2\) and specific gravities \(p_1,p_2\) are joined and remain neutrally buoyant in a liquid of specific gravity \(p\). Find \(m_1/m_2\).
Q20.
A wooden block of mass \(120\ \mathrm{kg}\) and density \(600\ \mathrm{kg/m^3}\) floats in freshwater. What additional mass will make it just fully submerged?
Q21.
A hydrogen balloon has volume \(1000\ \mathrm{m^3}\). The densities of hydrogen and air are \(0.09\) and \(1.29\ \mathrm{kg/m^3}\), respectively. Neglecting balloon-envelope mass, what maximum payload mass can it lift?
Q22.
A boat has a horizontal waterline area of \(3\ \mathrm m\times2\ \mathrm m\). It sinks \(1\ \mathrm{cm}\) farther when a man boards. Find his mass in freshwater.
Q23.
A floating wooden cube sinks \(2\ \mathrm{cm}\) farther into water when a \(200\ \mathrm g\) mass is placed on it. Find the length of its side.
Q24.
A log weighs \(12\ \mathrm N\) and has volume \(1000\ \mathrm{cm^3}\). What upward tension holds it stationary when half its volume is immersed in freshwater? Use \(g=10\ \mathrm{m/s^2}\).
Q25.
A body floats with one-third of its volume above water and three-quarters of its volume above another liquid. Find the density of the other liquid.
Q26.
An ice sheet \(10\ \mathrm m\) thick with density \(0.9\ \mathrm{g/cm^3}\) floats in freshwater. What minimum rope length reaches the water surface from its top?
Q27.
A hollow metallic sphere weighs \(200\ \mathrm{gf}\) in air and \(120\ \mathrm{gf}\) when fully submerged in water. If the metal has density \(5\ \mathrm{g/cm^3}\), find the cavity volume.
Q28.
A hollow metallic sphere floats just fully submerged in a liquid whose density is one-eighth that of the metal. Find the ratio of cavity radius to outer radius.
Q29.
A hydrometer reads a specific gravity of \(1.6\). Where is the \(1.5\) marking relative to the \(1.6\) marking on its stem?
📅IOM 2017