Fluid Statics: theory | Physics Circle

CONCEPT

Pressure in a Fluid

In one line
Pressure at a point in a still fluid depends only on its depth, the liquid and g.

PRESSURE

1.1.1 What is pressure?

Thrust is the normal force of a liquid at rest on a surface.
Pressure is thrust per unit area.

P = FA

SI unit: N/m² or pascal (Pa). CGS unit: dyne/cm².
Dimensions: [ML⁻¹T⁻²].
Pressure at a point acts in all directions, with no definite direction. So it is not a vector.
Other units: atmosphere, bar, torr (mm of Hg).

1 atm = 1.01 × 10⁵ Pa = 1.01 bar = 760 torr

Atmospheric pressure is largest at the earth’s surface and falls as we go up.

DEPTH

1.1.2 Pressure at a depth

Point at depth h

P = P₀ + hρg

P₀ is the atmospheric pressure on the free surface.
Gauge pressure is the difference between P and P₀.

P − P₀ = hρg

SHAPE

1.1.3 Shape of the vessel does not matter

P_A = P_B = P_C, but the weights differ

P depends on h, ρ and g.
It is independent of the amount of liquid and the shape or area of the vessel.
Same height, same liquid: same pressure at the base, though the weight of liquid differs.

SAME LEVEL

1.1.4 Same level, same pressure

Connected vessels of different shapes

At one horizontal level in a liquid at rest, pressure is the same at all points.
If it were not, the liquid would flow.
So connected vessels fill to the same height.

U-TUBE

1.1.5 U-tube with two liquids

Take the level of the common surface

Choose the interface level; both sides have equal pressure there.
Both open ends are at P₀, so P₀ cancels.

h₁ρ₁ = h₂ρ₂

BAROMETER

1.1.6 Barometer

Column balances the atmosphere

P₀ = hρg

A lighter liquid needs a taller column.
Mercury: h = 76 cm. A sudden fall in the barometer height warns of a storm.

BUBBLE

1.1.7 Air bubble rising in a lake

Radius grows as pressure falls

At constant temperature, PV = constant.
Write pressures as heights of water: atmosphere = H.

(H + d) r₁³ = H r₂³

Radius doubles: volume is 8 times, so d = 7H.

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Corner

Force on a vertical wall

Pressure grows with depth

Mean pressure on a wall of height h is hρg2.
Force = mean pressure × area of the wall.

F = ρgh2 × A

Levelling of connected vessels

Water moves from the taller to the shorter column until the levels match.
Layer of height h₁ − h₂2 crosses over; its centre falls by the same amount.

W_gravity = 14 ρgA(h₁ − h₂)²

Practise PYQs on this topicPressure in a Fluid · 26 questions with solutions

CONCEPT

Density, Compressibility & Pascal’s Law

In one line
Density tells how much mass sits in a volume. Pascal’s law says pressure on an enclosed fluid spreads equally.

DENSITY

1.2.1 Density

ρ = lim_ΔV→0 ΔmΔV = dmdV

It is a scalar with dimensions [ML⁻³].
SI unit kg/m³; 1 gcc = 10³ kg/m³.
Immiscible liquids settle with the densest at the bottom, with plane interfaces.

RELATIVE DENSITY

1.2.2 Relative density

RD = Density of bodyDensity of water

Also called specific gravity. If it is x, density = x g/cm³ = x × 10³ kg/m³.

MIXTURES

1.2.3 Mixture of two liquids

Mixed	Density of mixture	If equal
masses m₁, m₂	total mass ÷ total volume	2ρ₁ρ₂ρ₁ + ρ₂ (harmonic mean)
volumes V₁, V₂	ρ₁V₁ + ρ₂V₂V₁ + V₂	ρ₁ + ρ₂2 (arithmetic mean)

COMPRESSION

1.2.4 Effect of temperature and pressure

Heating expands the volume, so density falls: ρ ≈ ρ₀(1 − γΔθ).
Raising pressure reduces volume, so density rises. Compressibility is 1B.

B = ΔPΔV ÷ V ⇒ ΔVV = ΔPB

ρ ≈ ρ₀(1 + ΔPB)

At depth h in a liquid, ΔP = hρg.
Smaller B means the liquid compresses more.

PASCAL

1.2.5 Pascal’s law

An increase in pressure at one point of an enclosed liquid is transmitted equally to all points and to the walls.
Uses: hydraulic lift, hydraulic press, hydraulic brakes.

HYDRAULIC LIFT

1.2.6 Hydraulic lift

Small force on a small piston lifts a heavy load

fa = FA ⇒ F = f (Aa)

Since A ≫ a, F ≫ f.
Circular pistons: Aa = D²d².
Force is multiplied, work is not.

Practise PYQs on this topicDensity, Compressibility & Pascal's Law · 23 questions with solutions

CONCEPT

Archimedes’ Principle & Floatation

In one line
A body in a fluid is pushed up by a force equal to the weight of the fluid it displaces.

PRINCIPLE

1.3.1 Archimedes’ principle

Pressure on the lower face is larger

Side forces cancel.
F₁ = A(h₁σg + P₀) acts down on the top face.
F₂ = A(h₂σg + P₀) acts up on the bottom face.

F = F₂ − F₁ = Aσg(h₂ − h₁) = Vσg

This net upward force is upthrust or buoyancy.
It acts through the centre of gravity of the displaced fluid, the centre of buoyancy.

UPTHRUST

1.3.2 What upthrust depends on

Only on the volume inside the fluid and the density of the fluid.
Not on mass, density or shape of the body.
It is larger in sea water than in fresh water.
Equal upthrust on two bodies means equal volumes immersed.

APPARENT WEIGHT

1.3.3 Apparent weight

W_app = W − Vσg = W(1 − σρ)

W₁ = weight in air, W₂ = weight in water.

W₁ − W₂ = Vσg ⇒ V = W₁ − W₂σg

RELATIVE DENSITY

1.3.4 Finding relative density

R.D. = Weight of bodyLoss of weight in water = W₁W₁ − W₂

σ_Lσ_W = W_air − W_liquidW_air − W_water

MOTION

1.3.5 Body dropped into a liquid

Body of density ρ falls through h, reaching v² = 2gh.
In a denser liquid (σ > ρ), retardation a = (σ − ρ)gρ.

d_max = v²2a = hρσ − ρ

CONDITIONS

1.3.6 Sink, stay or float

Body of density ρ in liquid of density σ

A body floats only if ρ ≤ σ.
Floating: W_app = 0.
Floating is unaffected by g, as both thrust and weight depend on it.

FLOATING

1.3.7 Floating body

Vρg = V_inσg ⇒ V_inV = ρσ

V_out = V(1 − ρσ)

For a block of height H: submerged height h = H × ρσ.
The same body in two liquids: V_Aσ_A = V_Bσ_B.

PLATFORM

1.3.8 Loaded platform

Platform of mass M and area A

Floating: Mg = hAσg. A mass m sinks it further by y.

mg = Aσyg ⇒ W ∝ y

TWO LIQUIDS

1.3.9 Floating between two liquids

Weight = sum of both upthrusts

Vd = V₁ρ₁ + V₂ρ₂

A body at an interface has a density between the two liquids.
Half in each: d = ρ₁ + ρ₂2.

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Corner

Rotational equilibrium: metacentre

Tilted floating body

When a floating body tilts, the centre of buoyancy shifts to B′.
Metacentre M: where the vertical through B′ meets the initial vertical line.
M above G: the couple restores. Stable.
M below G: the couple topples the body. Unstable.
A wooden log cannot float vertically; a boat may capsize if passengers stand.

Sphere with a cavity

It just floats when weight of shell = weight of water displaced by the whole sphere.

σ(R³ − r³) = R³

Practise PYQs on this topicArchimedes' Principle & Floatation · 24 questions with solutions

EXAM-DAY CARD

Just see this before the exam

P = P₀ + hρg

depth

fa = FA

Pascal

ΔVV = ΔPB

compression

F = Vσg

upthrust

V_inV = ρσ

floating

R.D. = W₁W₁ − W₂

relative density

FORMULA SHEET

Whole chapter on one page

Pressure

P = FA, P = P₀ + hρg

gauge = hρg

1 atm = 1.01 × 10⁵ Pa

U-tube, bubble

h₁ρ₁ = h₂ρ₂

P₀ = hρg

(H + d)r₁³ = H r₂³

Density

equal masses: 2ρ₁ρ₂ρ₁ + ρ₂

equal volumes: ρ₁ + ρ₂2

ρ ≈ ρ₀(1 − γΔθ)

Compression, Pascal

B = ΔPΔV ÷ V

ρ ≈ ρ₀(1 + ΔPB)

fa = FA, F = f D²d²

Upthrust

F = Vσg

W_app = W(1 − σρ)

R.D. = W₁W₁ − W₂

Floating

V_inV = ρσ

interface: Vd = V₁ρ₁ + V₂ρ₂

d_max = hρσ − ρ

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Using hρg as the total pressure

Upthrust with the density of the body

Hydraulic force going with d

Bubble radius doubled: d = 2H

CORRECT ✓

P = P₀ + hρg; hρg is gauge pressure

Upthrust = Vσg, σ of the liquid

Force goes with d²

Volume × 8, so d = 7H