NDA Physics · Fluid Mechanics and Properties of Matter

Pressure and Surface Tension

Pressure is force spread over an area (P = F/A); inside a liquid it grows only with depth and density (P = rho g h) and is transmitted undiminished through an enclosed fluid (Pascal). Surface tension is the elastic skin of a liquid surface, and it weakens as temperature rises.

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Why this matters

Eight PYQs and the foundation the whole chapter rests on. The recurring tests are: pressure = force / area (smaller area means more pressure), liquid pressure depends on height and density but NOT on the base area or container shape, the SI unit (pascal = N/m²), what a barometer measures, and the one surface-tension fact the bank loves — it falls when temperature increases. Two of the eight are HARD, both pressure-versus-area reasoning, so get the P = F/A intuition airtight.

Concept 1 of 5: Pressure — force spread over an area

Pressure measures how concentrated a force is. The same weight pressing on a tiny area bites hard (a drawing pin); spread over a large area it barely registers (a flat shoe on snow). Push the same force through a smaller area and the pressure goes up.

Definition

Pressure is the normal force acting per unit area: $P = F/A$ .

For a fixed force, smaller contact area means larger pressure (P proportional to 1/A).
It is a scalar — pressure has magnitude but no single direction; in a fluid it pushes outward on every surface it touches.
SI unit: the pascal (Pa), where $1\,\text{Pa} = 1\,\text{N/m}^2$ .

Pressure

P = \dfrac{F}{A}

Ppressure (Pa = N/m²)
Fforce normal to the surface (N)
Aarea over which the force acts (m²)

Gauge pressure depends only on depth h and density rho (P = rho g h) — not on the shape of the container or the area of its base. The arrows grow with depth.

Worked example

A force of 30 N is applied normally to a surface of area 0.5 m². What pressure does it exert?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Apr 2022 · Q99Hard

Example 1 · Fluid Mechanics and Properties of Matter · Pressure and Surface Tension

A wooden box of mass 2 kg and dimensions (30 cm

\times

15 cm

\times

10 cm) is placed on a table with sides 30 cm and 10 cm touching the tabletop. Which one of the following is the approximate pressure exerted on the table?

Pressure rises when area shrinks

For a fixed force, P is inversely proportional to A. Resting a block on its SMALLEST face gives the GREATEST pressure. A common slip is to think the largest face presses hardest — it presses softest, because the same weight is spread over more area.

Drill 1 more on pressure — force spread over an area

Concept 2 of 5: Pressure in a liquid — P = rho g h

Dive deeper into water and you feel your ears squeeze: the deeper you go, the more liquid sits above you, and its weight presses down harder. The pressure at a point depends only on how deep it is and how dense the liquid is — not on the shape of the container or how wide its base is.

Definition

The gauge pressure at depth $h$ below a liquid surface is $P = \rho g h$ .

It depends on depth $h$ and density $\rho$ — and on $g$ .
It does NOT depend on the base area or the shape of the container (the hydrostatic paradox).
Pressure is the same at all points on the same horizontal level in a connected liquid at rest — but it INCREASES with depth, so it is not the same at every point.

Pressure due to a liquid column

P = \rho g h

Pgauge pressure at depth h (Pa)
$\rho$ density of the liquid (kg/m³)
gacceleration due to gravity (m/s²)
hdepth below the free surface (m)

Worked example

Find the gauge pressure at a depth of 2 m in water (density 1000 kg/m³, g = 10 m/s²).

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Apr 2020 · Q74Moderate

Example 2 · Fluid Mechanics and Properties of Matter · Pressure and Surface Tension

A liquid is kept in a glass beaker. Which one of the following statements is correct regarding the pressure exerted by the liquid column at the base of the beaker?

Pressure is NOT the same at all points

A frequent statement-MCQ trap claims 'pressure is the same at all points in a fluid at rest.' False — pressure is equal only at the same horizontal LEVEL; it grows with depth. What IS true: pressure exists everywhere in the fluid and presses on the walls.

Shape and base area do not matter

The hydrostatic paradox: a thin tall column and a wide shallow tank filled to the same height give the same pressure at the base. Only depth and density count, not the volume or the container shape.

Drill 1 more on pressure in a liquid — p = rho g h

Concept 3 of 5: Pascal's principle — the hydraulic press

Squeeze an enclosed liquid anywhere and the extra pressure shows up everywhere, equally. That is how a small push on a narrow piston lifts a heavy car on a wide piston: the same pressure acting on a bigger area gives a bigger force.

Definition

Pascal's principle: a pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls. In a hydraulic press the pressure is the same under both pistons, so $F_1/A_1 = F_2/A_2$ .

The wide piston multiplies force by the area ratio $A_2/A_1$ .
It is a force multiplier, not an energy creator — the wide piston moves a shorter distance.

Hydraulic press

\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}

F_1force on the small piston (N)
A_1area of the small piston (m²)
F_2force on the large piston (N)
A_2area of the large piston (m²)

Pressure applied to an enclosed liquid is transmitted undiminished to every point. A wider output piston multiplies the force by the area ratio A₂/A₁.

Worked example

In a hydraulic press the small piston has area 0.01 m² and the large piston 0.5 m². A force of 20 N is applied to the small piston. What force appears on the large piston?

Practice this conceptself-check · 3 quick reps

Force is multiplied, energy is not

The hydraulic press gains force but loses distance — the big piston moves a smaller distance than the small piston, so work in equals work out. Do not claim it 'creates' energy. It only redistributes force and displacement.

Concept 4 of 5: Atmospheric pressure, gauge vs absolute, and the pascal

The air above us has weight, and it presses on everything at about 100 kPa at sea level — that is atmospheric pressure, measured by a barometer. A pressure gauge usually reads the EXTRA pressure above the atmosphere; add the atmosphere back to get the true (absolute) pressure.

Definition

Atmospheric pressure is the pressure of the air column above a point; it is measured with a barometer (a mercury barometer reads about 76 cm of Hg at sea level).
Gauge pressure is pressure measured RELATIVE to the atmosphere; absolute pressure = gauge pressure + atmospheric pressure.
Units: the SI unit is the pascal, and $1\,\text{Pa} = 1\,\text{N/m}^2$ — they are the SAME unit. (1 bar = 100000 Pa; 1 atm is about 101325 Pa.)

Absolute pressure in a liquid open to air

P_{\text{abs}} = P_{\text{atm}} + \rho g h

$P_{\text{abs}}$ absolute (true) pressure (Pa)
$P_{\text{atm}}$ atmospheric pressure (Pa)
$\rho g h$ gauge pressure due to the liquid column (Pa)

Worked example

A pressure gauge on a tank reads 30 kPa. If atmospheric pressure is 100 kPa, what is the absolute pressure inside the tank?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Sep 2025 · Q57Easy

Example 4 · Fluid Mechanics and Properties of Matter · Pressure and Surface Tension

Which one among the following statements is correct ?

Barometer vs manometer vs thermometer

A barometer measures ATMOSPHERIC pressure. A manometer measures the pressure of an enclosed gas. A thermometer measures temperature. The bank often lists all three as distractors — pick the barometer for atmospheric pressure.

Drill 1 more on atmospheric pressure, gauge vs absolute, and the pascal

Concept 5 of 5: Surface tension — the skin of a liquid

A liquid surface behaves like a stretched elastic skin. Molecules at the surface are pulled inward by their neighbours, so the surface tries to shrink to the smallest area — that is why droplets are round and a needle can rest on water. Heat the liquid and the skin weakens.

Definition

Surface tension is the force per unit length acting along a liquid surface, arising from the inward attraction on surface molecules. Consequences and facts:

It makes drops spherical (smallest area for a given volume) and lets small dense objects rest on water.
It drives capillary rise — a liquid that wets a narrow tube climbs higher in a thinner bore (rise proportional to 1/r) and forms a concave meniscus.
It DECREASES as temperature increases (it becomes zero at the critical temperature). Adding detergent also lowers it.

Surface tension

T = \dfrac{F}{L}

Tsurface tension (N/m)
Fforce along the surface (N)
Llength over which the force acts (m)

Surface tension pulls water up a narrow tube (capillary rise) and forms a concave meniscus. The thinner the bore, the higher it climbs.

Worked example

Why do free liquid drops take a spherical shape in the absence of other forces?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Sep 2025 · Q66Easy

Example 5 · Fluid Mechanics and Properties of Matter · Pressure and Surface Tension

Which one of the following statements about surface tension is correct ?

Temperature LOWERS surface tension

The bank tests this directly. A tempting wrong option says surface tension 'increases with temperature' — it does the opposite. More heat means weaker surface attraction, so surface tension falls (and vanishes at the critical temperature).

Drill 1 more on surface tension — the skin of a liquid

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 (5)

Pressure — force spread over an area
Pressure
$P = \dfrac{F}{A}$
Pressure in a liquid — P = rho g h
Pressure due to a liquid column
$P = \rho g h$
Pascal's principle — the hydraulic press
Hydraulic press
$\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}$
Atmospheric pressure, gauge vs absolute, and the pascal
Absolute pressure in a liquid open to air
$P_{\text{abs}} = P_{\text{atm}} + \rho g h$
Surface tension — the skin of a liquid
Surface tension
$T = \dfrac{F}{L}$

Watch out for (6)

Pressure rises when area shrinks
→ Pressure — force spread over an area
Pressure is NOT the same at all points
→ Pressure in a liquid — P = rho g h
Shape and base area do not matter
→ Pressure in a liquid — P = rho g h
Force is multiplied, energy is not
→ Pascal's principle — the hydraulic press
Barometer vs manometer vs thermometer
→ Atmospheric pressure, gauge vs absolute, and the pascal
Temperature LOWERS surface tension
→ Surface tension — the skin of a liquid

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