CDS Physics · Fluid Mechanics and Properties of Matter

Pressure, Buoyancy and Elasticity

Pressure as force per area and how it grows with depth, the buoyant force and why bodies float, the density of a mixture, and Hooke's law for springs and wires.

All Fluid Mechanics and Properties of Matter notes

Why this matters

Nine CDS questions from 2016 (II) to 2026 (II): four on buoyancy and floating, two on pressure, two on elasticity and springs, and one on the density of a mixture. The one HARD question compares a gold and a silver piece weighed in water.

Concept 1 of 4: Pressure and pressure at a depth

Pressure is how concentrated a force is: the same push spread over a big area presses gently, squeezed onto a small area it presses hard. That is why a sharp knife cuts and why snow shoes are wide. In a liquid, the weight of the liquid above adds pressure, so the deeper you go the more it presses, by the same amount in every direction.

Definition

Pressure in solids and fluids:

Pressure = force (thrust) ÷ area, $P = F/A$ , in pascals. Same force, smaller area, greater pressure.
In a liquid of density $\rho$ , the pressure at depth $h$ is $P = P_0 + \rho g h$ , where $P_0$ is the pressure at the surface (often the atmosphere).
So all points at the same depth in a liquid at rest have the same pressure, and pressure does not depend on the shape of the vessel.
Pascal's law: pressure applied to an enclosed liquid is passed on undiminished to every part of it. This is the idea behind the hydraulic lift and brakes, and the $P_0$ term above.
Bernoulli's principle is about liquids in motion; Archimedes' principle is about the upward (buoyant) force.

Pressure

P = \dfrac{F}{A} \qquad P = P_0 + \rho g h

F, Aforce and the area it acts on
P₀pressure at the liquid's surface
ρdensity of the liquid
hdepth below the surface

Worked example

A diver is 10 m below the surface of a lake. The atmospheric pressure is

1.0 \times 10^{5}

Pa. Find the total pressure on the diver. (Water density 1000 kg/m³,

g = 10

m/s².)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — General Knowledge · Q5Easy

Example 1 · Fluid Mechanics and Properties of Matter · Pressure, Buoyancy and Elasticity

What is the effect of pressure of a human body on sand ?

The weight stays the same; the area changes

Standing or lying, a body presses down with the same weight. What changes is the area of contact, so the pressure is larger on the small area of the feet.

Moving liquid is Bernoulli; still liquid is Pascal

The depth rule

P = P_0 + \rho g h

is about a liquid at rest, and goes with Pascal's law. Bernoulli's principle is for flowing liquids, and Archimedes' principle for the buoyant force.

Drill 1 more on pressure and pressure at a depth

Concept 2 of 4: Buoyancy and flotation

Push a ball under water and you feel the water push back up. That upward push is the buoyant force, and it equals the weight of the water the body pushes aside. A body floats when that push can match its weight before it is fully under. Since a floating body's weight and the buoyant force both contain g, g cancels: a body floats just as deep on the Moon as on the earth.

Definition

Archimedes' principle and floating:

Buoyant force = weight of liquid displaced = $\rho_{\text{liquid}} V_{\text{sub}} g$ . For a body fully under, $V_{\text{sub}}$ is its whole volume; its own mass does not enter.
Apparent weight in a liquid = true weight − buoyant force.
A body floats when the buoyant force equals its weight. It sinks if its density is greater than the liquid's.
Fraction submerged of a floating body = $\dfrac{\rho_{\text{body}}}{\rho_{\text{liquid}}}$ . g cancels, so this fraction is the same on the Moon.
For pieces with the same apparent weight in water, the less dense one loses a larger fraction of its weight, so it must be heavier in air.

Buoyant force and flotation

F_b = \rho_{\text{liquid}}\,V_{\text{sub}}\,g \qquad W_{\text{apparent}} = W - F_b \qquad \dfrac{V_{\text{sub}}}{V} = \dfrac{\rho_{\text{body}}}{\rho_{\text{liquid}}}

F_bbuoyant (upthrust) force
V_subvolume of the body under the liquid
Wtrue weight of the body

Worked example

A block of wood of density 600 kg/m³ floats in water (density 1000 kg/m³). What fraction of its volume is under water? Would the fraction change on the Moon?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — General Knowledge · Q27Moderate

Example 2 · Fluid Mechanics and Properties of Matter · Pressure, Buoyancy and Elasticity

A metallic sphere of mass 1 kg and volume

2 \times 10^{-4}\ \text{m}^3

is completely immersed in water. The buoyant force exerted by water on the sphere is : (Given : density of water =

1000\ \text{kg/m}^3

g = 10\ \text{m/s}^2

)

The body's own mass does not set the buoyant force

The buoyant force depends only on the volume under the liquid and the liquid's density. A heavy and a light ball of the same volume, fully under water, feel the same upthrust.

Floating means equal, not greater

A floating body is in equilibrium, so the buoyant force equals its weight. If it were greater, the body would rise further out of the liquid until they matched.

g cancels in floating

Weaker gravity reduces the weight and the buoyant force by the same factor, so a floating body sits at the same depth on the Moon.

Drill 3 more on buoyancy and flotation

Concept 3 of 4: Density of a mixture

Density is mass per volume. When two liquids are mixed, add the masses and add the volumes, then divide; never just average the densities without thinking. Equal volumes make the plain average come out right. Equal masses do not, because the lighter liquid then takes up more of the volume.

Definition

Density and mixtures:

Density $\rho = m/V$ ; 1 g/cm³ = 1000 kg/m³.
Mixture density = total mass ÷ total volume (for liquids that do not change volume on mixing).
Equal volumes of densities $\rho_1$ and $\rho_2$ : $\rho = \dfrac{\rho_1 + \rho_2}{2}$ .
Equal masses: $\rho = \dfrac{2\rho_1\rho_2}{\rho_1 + \rho_2}$ , which is smaller.

Density of a mixture

\rho = \dfrac{m_1 + m_2}{V_1 + V_2} \qquad \rho_{\text{equal volumes}} = \dfrac{\rho_1 + \rho_2}{2} \qquad \rho_{\text{equal masses}} = \dfrac{2\rho_1\rho_2}{\rho_1 + \rho_2}

ρ₁, ρ₂densities of the two liquids
m, Vmass and volume of each part

Worked example

Equal masses of two liquids of densities 600 kg/m³ and 900 kg/m³ are mixed. Find the density of the mixture.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — General Knowledge · Q8Moderate

Example 3 · Fluid Mechanics and Properties of Matter · Pressure, Buoyancy and Elasticity

If two miscible liquids of same volume but different densities

P_1

and

P_2

are mixed, then the density of the mixture is given by

Equal volumes average, equal masses do not

With equal volumes the mixture density is the plain average. With equal masses it is

\dfrac{2\rho_1\rho_2}{\rho_1 + \rho_2}

, always a little smaller. Read which one the question fixes.

Concept 4 of 4: Hooke's law and springs

Stretch a wire a little and it springs back: it is elastic. For small stretches the stretch grows in step with the pull, so the graph of stress against strain is a straight line. That straight part is where Hooke's law holds. Pull harder and the line bends; harder still and the wire stays stretched for good.

Definition

The stress-strain curve and springs:

Stress = force ÷ area (in Pa); strain = change in length ÷ original length (no unit).
Hooke's law: stress is proportional to strain. It holds only up to the proportional limit, the straight part of the curve.
Between the proportional limit and the elastic limit, the material still returns to its shape, but stress and strain are no longer proportional.
Beyond the elastic limit the material is plastic: it keeps a permanent stretch, and later breaks.
For a spring, $F = kx$ . The spring constant $k$ depends on the thickness of the wire, the diameter of the coils and the material: thicker wire, smaller coils and a stiffer material give a larger $k$ .

Part of the stress-strain curve	What happens
Up to the proportional limit	Straight line; Hooke's law holdsQ CDS 2019 (I): Hooke's law is valid only in the proportional region of the stress-strain curve.
Proportional limit to elastic limit	Returns to shape, but not in proportion
Beyond the elastic limit	Permanent stretch (plastic), then fracture
Spring constant k	Set by wire thickness, coil diameter and materialQ CDS 2016 (II): the spring constant depends on the thickness, the diameter and the material.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — General Knowledge · Q85Moderate

Example 4 · Fluid Mechanics and Properties of Matter · Pressure, Buoyancy and Elasticity

The Hooke's law is valid for

Proportional limit comes before elastic limit

Hooke's law holds only on the straight part of the curve. A material can still be elastic a little beyond that, but there stress is no longer proportional to strain, so 'the entire elastic region' is too much.

No single factor fixes the spring constant

A spring's stiffness depends on the wire thickness, the coil diameter and the material together. An option naming only the length, or only the material, is incomplete.

Drill 1 more on hooke's law and springs

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Pressure and pressure at a depth
Pressure
$P = \dfrac{F}{A} \qquad P = P_0 + \rho g h$
Buoyancy and flotation
Buoyant force and flotation
$F_b = \rho_{\text{liquid}}\,V_{\text{sub}}\,g \qquad W_{\text{apparent}} = W - F_b \qquad \dfrac{V_{\text{sub}}}{V} = \dfrac{\rho_{\text{body}}}{\rho_{\text{liquid}}}$
Density of a mixture
Density of a mixture
$\rho = \dfrac{m_1 + m_2}{V_1 + V_2} \qquad \rho_{\text{equal volumes}} = \dfrac{\rho_1 + \rho_2}{2} \qquad \rho_{\text{equal masses}} = \dfrac{2\rho_1\rho_2}{\rho_1 + \rho_2}$

Reference tables (1)

Hooke's law and springs4 rows

Part of the stress-strain curve	What happens
Up to the proportional limit	Straight line; Hooke's law holdsQ CDS 2019 (I): Hooke's law is valid only in the proportional region of the stress-strain curve.
Proportional limit to elastic limit	Returns to shape, but not in proportion
Beyond the elastic limit	Permanent stretch (plastic), then fracture
Spring constant k	Set by wire thickness, coil diameter and materialQ CDS 2016 (II): the spring constant depends on the thickness, the diameter and the material.

Watch out for (8)

The weight stays the same; the area changes
→ Pressure and pressure at a depth
Moving liquid is Bernoulli; still liquid is Pascal
→ Pressure and pressure at a depth
The body's own mass does not set the buoyant force
→ Buoyancy and flotation
Floating means equal, not greater
→ Buoyancy and flotation
g cancels in floating
→ Buoyancy and flotation
Equal volumes average, equal masses do not
→ Density of a mixture
Proportional limit comes before elastic limit
→ Hooke's law and springs
No single factor fixes the spring constant
→ Hooke's law and springs

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