Bernoullis Principle

8 MCQs1 revision card9-step worked example

Source: NCERT Properties of Bulk MatterPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

The high-frequency trap with Bernoulli's equation is forgetting the height term. When a pipe changes both cross-section and elevation, aspirants who drop the ρgh term get a plausible-looking but wrong pressure or velocity — and lose marks to negative marking.

Bernoulli's principle states that for steady, incompressible, non-viscous flow along a streamline:

P + ½ρv² + ρgh = constant

This is energy conservation per unit volume. P is the pressure energy density, ½ρv² is the kinetic energy density, and ρgh is the gravitational potential energy density (NCERT Class 11 Physics Chapter 9, page 6).

Three conditions must hold for the equation to apply: (1) the flow is steady (no turbulence), (2) the fluid is incompressible (constant ρ), and (3) the fluid is non-viscous (no energy loss to friction). If any condition breaks — turbulent flow, compressible gas at high speed, or significant viscosity — Bernoulli's equation gives incorrect results.

How NEET uses this: Questions typically give two points along a pipe system (often a Venturi meter or a pipe with a constriction at a different height) and ask for the pressure difference or the velocity at one point. The continuity equation A₁v₁ = A₂v₂ is almost always needed alongside Bernoulli's equation to eliminate one unknown velocity.

Watch out: When both points are at the same height, the ρgh terms cancel and you get P₁ + ½ρv₁² = P₂ + ½ρv₂². Many aspirants memorise only this simplified form and then apply it to problems where heights differ — that is exactly the distractor NTA exploits.

Practice MCQs

Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.

MCQ 1Easy RecallPractice

Bernoulli's theorem is a statement of conservation of which quantity for a fluid in streamline flow?

MCQ 2Easy RecallPractice

Which of the following is NOT a necessary condition for Bernoulli's equation to be valid?

MCQ 3Easy RecallPractice

In Bernoulli's equation P + ½ρv² + ρgh = constant, the term ½ρv² represents:

MCQ 4Direct ApplicationPractice

Water flows through a horizontal pipe that narrows from cross-sectional area 4.0 × 10⁻² m² to 2.0 × 10⁻² m². If the speed of water in the wider section is 2.0 m/s, what is the speed in the narrower section?

MCQ 5Direct ApplicationPractice

Water flows steadily through a horizontal pipe. At point 1, the pressure is 3.0 × 10⁵ Pa and the speed is 2.0 m/s. At point 2 (same height), the speed is 4.0 m/s. Taking ρ = 1.0 × 10³ kg/m³, the pressure at point 2 is:

MCQ 6Direct ApplicationPractice

A Venturi meter has a wide section of area 4.0 × 10⁻² m² and a constriction of area 1.0 × 10⁻² m², both at the same height. If the pressure difference between the two sections is 4.5 × 10³ Pa and ρ = 1.0 × 10³ kg/m³, the speed of flow in the wide section is:

MCQ 7Concept TrapPractice

An ideal fluid flows through a pipe that rises vertically by height h while maintaining the same cross-sectional area throughout. Compared to the bottom, the pressure at the top is:

MCQ 8CalculationPractice

Water (ρ = 1.0 × 10³ kg/m³) flows through a pipe that narrows from area 8.0 × 10⁻² m² to 4.0 × 10⁻² m² while rising 5.0 m. The speed in the wider lower section is 2.0 m/s and the pressure there is 2.0 × 10⁵ Pa. Taking g = 10 m/s², the pressure in the narrower upper section is:

Quick recall before you leave

Worked Example

1
Given
A pipe carries water (ρ = 1.00 × 10³ kg/m³) from ground level to a point 3.0 m higher. At the ground-level section, the pipe area is 6.0 × 10⁻² m², the speed is 3.0 m/s, and the gauge pressure is 2.50 × 10⁵ Pa. At the upper section the pipe area is 3.0 × 10⁻² m². Take g = 10 m/s² (exact, problem-defined).
2
Required
Find the gauge pressure at the upper section.
3
Concept
Bernoulli's equation along a streamline connects pressure, speed, and height. The continuity equation connects speeds at different cross-sections.
4
Formula
Continuity: A₁v₁ = A₂v₂ Bernoulli: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
5
Substitution
From continuity: v₂ = (6.0 × 10⁻² / 3.0 × 10⁻²) × 3.0 = 6.0 m/s Set h₁ = 0, h₂ = 3.0 m. P₂ = P₁ + ½ρ(v₁² − v₂²) − ρgΔh P₂ = 2.50 × 10⁵ + ½(1.00 × 10³)(9.0 − 36.0) − (1.00 × 10³)(10)(3.0)
6
Calculation
½(1.00 × 10³)(−27.0) = −1.35 × 10⁴ Pa (1.00 × 10³)(10)(3.0) = 3.00 × 10⁴ Pa P₂ = 2.50 × 10⁵ − 1.35 × 10⁴ − 3.00 × 10⁴ P₂ = 2.50 × 10⁵ − 4.35 × 10⁴ = 2.065 × 10⁵ Pa Note: g = 10 m/s² is stated as exact (problem-defined constant) and the integer 2 in the area ratio is a counting ratio. Neither limits significant figures. The given data (three significant figures) governs the final answer.
7
Final answer
P₂ = 2.07 × 10⁵ Pa (to three significant figures). The pressure drops because both the increased speed and the increased height drain pressure energy from the fluid.
8
Common trap
Forgetting the ρgΔh term. If you drop it, you get P₂ = 2.50 × 10⁵ − 1.35 × 10⁴ = 2.365 × 10⁵ Pa — a plausible-looking but wrong answer. This is the distractor NTA builds into the options (per pattern description: "drops ρgh term").
9
Similar NEET-style question
Oil (ρ = 8.0 × 10² kg/m³) flows through a horizontal pipe that widens from area 2.0 × 10⁻² m² to 8.0 × 10⁻² m². If the speed in the narrow section is 8.0 m/s and the pressure there is 1.0 × 10⁵ Pa, find the pressure in the wide section. (Answer: Apply continuity to get v₂ = 2.0 m/s, then Bernoulli at constant height gives P₂ = 1.0 × 10⁵ + ½(800)(64 − 4) = 1.0 × 10⁵ + 2.4 × 10⁴ = 1.24 × 10⁵ Pa.) ---

Before solving, remember these

Law

Bernoulli's principle

Along a streamline of an ideal incompressible fluid: P + ½ ρ v² + ρ g h = constant. Higher speed → lower pressure. Basis of aerofoil lift, venturi meter, atomiser.

-- NCERT, p. 6

Formulas

12 formulas — click to collapse

Bernoulli's equation

P + \frac{1}{2} ρ v^{2} + ρ g h = const

Conservation of energy along a streamline of incompressible non-viscous flow.

Symbol	Quantity	SI Unit
P	pressure	Pa
rho	density	kg/m^3
v	speed	m/s
g	gravity	m/s^2
h	height	m

Valid when

Steady, non-viscous, incompressible flow
Along a single streamline
No work added/removed

Bulk modulus

K = - V \frac{d P}{d V}

Resistance of a material to uniform compression. Inverse: compressibility.

Symbol	Quantity	SI Unit
K	bulk modulus	Pa
V	volume	m^3
P	pressure	Pa

Valid when

Isotropic compression
Within elastic regime

Capillary rise/depression

h = \frac{2 T c o s θ}{ρ g r}

Height a liquid rises (or falls) in a capillary tube. cos(theta) > 0: rises (wetting); < 0: depresses.

Symbol	Quantity	SI Unit
h	capillary height	m
T	surface tension	N/m
theta	contact angle	rad
rho	density	kg/m^3
r	tube radius	m

Valid when

Narrow tube (capillary regime)
Constant theta

Pressure in static fluid

P = P_{0} + ρ g h

Pressure at depth h below free surface of fluid of density rho.

Symbol	Quantity	SI Unit
P	total pressure	Pa
P0	atmospheric/surface pressure	Pa
rho	density	kg/m^3
h	depth	m

Valid when

Static fluid (no flow)
Constant g
Constant rho (incompressible)

Latent heat

Q = m L

Heat absorbed/released during phase change at constant T. L_fusion or L_vaporisation.

Symbol	Quantity	SI Unit
Q	heat	J
m	mass	kg
L	latent heat	J/kg

Valid when

Phase transition (constant T during)
All mass m undergoes the transition

Specific heat / heat capacity

Q = m c Δ T

Heat required to raise mass m by temperature Delta_T. Specific heat c is material property.

Symbol	Quantity	SI Unit
Q	heat	J
m	mass	kg
c	specific heat	J/kg/K
Delta_T	temp change	K

Valid when

No phase change during heating
c approximately constant in temp range

Stefan-Boltzmann radiation law

P = σ ε A T^{4}

Radiation power from a body. Black body epsilon=1; net to surroundings P = sigma*epsilon*A*(T^4 - T_s^4).

Symbol	Quantity	SI Unit
sigma	Stefan-Boltzmann = 5.67e-8	W/m^2/K^4
epsilon	emissivity (0-1)	-
A	surface area	m^2
T	absolute temp	K

Valid when

Body in radiative equilibrium
T in kelvins

Stokes' law (viscous drag on sphere)

F = 6 π η r v

Drag force on a sphere of radius r moving with velocity v through viscous fluid (low Reynolds number).

Symbol	Quantity	SI Unit
F	drag force	N
eta	viscosity	Pa*s
r	sphere radius	m
v	velocity	m/s

Valid when

Smooth, slow flow (low Reynolds number)
Spherical body
Newtonian fluid

Excess pressure inside drop/bubble

Δ P_{drop} = \frac{2 T}{r}; Δ P_{bubble} = \frac{4 T}{r}

Excess internal pressure due to surface tension. Bubble has 2 surfaces, hence factor 4.

Symbol	Quantity	SI Unit
Delta_P	excess pressure	Pa
T	surface tension	N/m
r	radius	m

Valid when

Spherical drop or bubble
Constant T (one fluid pair)

Terminal velocity of sphere in viscous fluid

v_{t} = \frac{2 r ^{2} g ( ρ _{s} - ρ _{f} )}{9 η}

Constant velocity reached when net force is zero (gravity balanced by buoyancy + viscous drag).

Symbol	Quantity	SI Unit
v_t	terminal velocity	m/s
r	sphere radius	m
rho_s	sphere density	kg/m^3
rho_f	fluid density	kg/m^3
eta	viscosity	Pa*s

Valid when

Steady state (net force zero)
Stokes regime applicable

Thermal expansion (linear/area/volume)

\frac{Δ L}{L} = α Δ T; \frac{Δ A}{A} = 2 α Δ T; \frac{Δ V}{V} = β Δ T

Fractional change in length, area, volume per degree temperature change.

Symbol	Quantity	SI Unit
alpha	linear coefficient	1/K
beta	volume coefficient	1/K
Delta_T	temperature change	K

Valid when

Isotropic material
Modest temperature range (alpha ~ constant)

Young's modulus

Y = \frac{F L}{A Δ L}

Ratio of longitudinal stress to longitudinal strain in a stretched wire/rod within elastic limit.

Symbol	Quantity	SI Unit
Y	Young's modulus	Pa
F	applied force	N
A	cross-section area	m^2
L	original length	m
Delta_L	extension	m

Valid when

Within elastic limit (Hooke's law region)
Uniform cross-section
Force along length

Exam Traps & Common Mistakes

These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.

5 items — click to collapse

Trap

Soap bubble vs drop excess pressure

Category: Similar Terms

Student uses 2T/r for soap bubble (drop formula). Bubble has 2 surfaces → 4T/r.

When it triggers

Question mentions soap bubble OR liquid drop OR air bubble in liquid.

How to avoid

Drop in air: 1 surface → 2T/r. Soap bubble in air: 2 surfaces → 4T/r. Air bubble in liquid: 1 surface → 2T/r.

Trap

Young's vs bulk modulus confusion

Category: Similar Terms

Student uses Y formula when problem is about volumetric compression (use K) or vice versa.

When it triggers

Problem describes longitudinal stretching (use Y), volumetric pressure (use K), or shear (use G).

How to avoid

Y: longitudinal stress/strain. K: volumetric. G: shear. Match modulus to deformation type.

Mistake

Uses 2T/r excess-pressure formula for soap bubble. Bubble has 2 surfaces → 4T/r.

Root cause: concept gap

Correction

Drop in air: 1 surface → ΔP = 2T/r. Soap bubble: 2 surfaces (inner + outer) → ΔP = 4T/r. Air bubble inside liquid: 1 surface → 2T/r.

Mistake

Treats v_terminal as linear in r instead of r².

Root cause: formula misuse

Correction

v_t ∝ r² (because Stokes drag ∝ r v, gravity ∝ r³ - r³ = r³_diff). Doubling radius quadruples terminal velocity, not doubles.

Mistake

Uses Young's modulus formula for volumetric compression problem (or vice versa).

Root cause: formula misuse

Correction

Y for longitudinal stretch (FL/A·ΔL); K for volumetric compression (-V·dP/dV); G for shear. Match modulus type to deformation type before computing.

Past Year Questions

15 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse

NEET 2025

A balloon is made of a material of surface tension S and its inflation outlet (from where gas is filled in it) has small area A. It is filled with a gas of density ρ and takes a spherical shape of radius R. When the gas is allowed to flow freely out of it, its radius r changes from R to 0 (zero) in time T. If the speed v(r) of gas coming out of the balloon depends on r as ra and T ∝ Sα Aβ ργ Rδ then 1 1 1 1 7 1 1 3

1a= ,α= ,β=− ,γ= ,δ=

2a= ,α= ,β=−1,γ=+1,δ= 2 2 2 2 2 2 2 2 1 1 1 5 1 1 1 7

3a=− ,α=– ,β=−1,γ=− ,δ=

4a=− ,α=− ,β=−1,γ= ,δ= 2 2 2 2 2 2 2 2

NTA Answer: Option 4(final)

NEET 2025

Consider a water tank shown in the figure. It has one wall at x = L and can be taken to be very wide in the z direction. When filled with a liquid of surface tension S and density ρ, the liquid surface makes angle θ 0 (θ 0 << 1) with the x-axis at x = L. If y(x) is the height of the surface then the equation for y(x) is: dy (takeθ(x)=sinθ(x)=tanθ(x)= ,g is the acceleration due to gravity) dx dy ρg d2y ρg

1= x

2= x dx S dx2 S d2y ρg d2y ρg

3= y

4= dx2 S dx2 S

NTA Answer: Option 3(final)

NEET 2024Revised key

The maximum elongation of a steel wire of 1 m length if the elastic limit of steel and its Young’s modulus, respectively, are 8 × 108 N m–2 and 2 × 1011 N m–2, is:

14 mm

20.4 mm

340 mm

48 mm

NTA Answer: Option 1(revised_final)

NEET 2024Revised key

A thin flat circular disc of radius 4.5 cm is placed gently over the surface of water. If surface tension of water is 0.07 Nm–1, then the excess force required to take it away from the surface is

119.8 mN

2198 N

31.98 mN

499 N

NTA Answer: Option 1(revised_final)

NEET 2024Revised key

A metallic bar of Young’s modulus, 0.5 × 1011 N m–2 and coefficient of linear thermal expansion 10–5 °C–1, length 1 m and area of cross-section 10–3 m2 is heated from 0°C to 100°C without expansion or bending. The compressive force developed in it is :

15 × 103 N

250 × 103 N

3100 × 103 N

42 × 103 N

NTA Answer: Option 2(revised_final)

NEET 2023

The amount of energy required to form a soap bubble of radius 2 cm from a soap solution is nearly (surface tension of soap solution = 0.03 N m–1)

15.06 × 10–4 J

23.01 × 10–4 J

350.1 × 10–4 J

430.16 × 10–4 J

NTA Answer: Option 2(final)

NEET 2023

The venturi-meter works on

1Bernoulli’s principle

2The principle of parallel axes

3The principle of perpendicular axes

4Huygen’s principle

NTA Answer: Option 1(final)

NEET 2023

Let a wire be suspended from the ceiling (rigid support) and stretched by a weight W attached at its free end. The longitudinal stress at any point of cross-sectional area A of the wire is

1W/A

2W/2A

3Zero

42W/A

NTA Answer: Option 1(final)

NEET 2022

If a soap bubble expands, the pressure inside the bubble

1Is equal to the atmospheric pressure

2Decreases

3Increases

4Remains the same

NTA Answer: Option 2(final)

NEET 2022

A spherical ball is dropped in a long column of a highly viscous liquid. The curve in the graph shown, which represents the speed of the ball (v) as a function of time (t) is

NTA Answer: Option 3(final)

NEET 2022

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): The stretching of a spring is determined by the shear modulus of the material of the spring. Reason (R): A coil spring of copper has more tensile strength than a steel spring of same dimensions. In the light of the above statements, choose the most appropriate answer from the options given below

1(A) is false but (R) is true

2Both (A) and (R) are true and (R) is the correct explanation of (A)

3Both (A) and (R) are true and (R) is not the correct explanation of (A)

4(A) is true but (R) is false

NTA Answer: Option 4(final)

NEET 2021

The velocity of a small ball of mass M and density d, when dropped in a container filled with glycerine becomes constant after some time. If the density of d glycerine is , then the viscous force acting on the 2 ball will be Mg

12Mg

22 3 Mg

3Mg

NTA Answer: Option 2(final)

NEET 2021

A cup coffee cools from 90°C to 80°C in t minutes, when the room temperature is 20°C. The time taken by a similar cup of coffee to cool from 80°C to 60°C at a room temperature same at 20°C is 5 13 t t

1(2) 13 10 13 10 t t

2d hc   2m c  λ2 λ =  

3(4) 5 13

4d h  

NTA Answer: Option 3(final)

NEET 2020

r U U 17. A capillary tube of radius r is immersed in water ° h – ÷ U and water rises in it to a height h. The mass of the water in the capillary is 5 g. Another capillary 5 g – 2r tube of radius 2r is immersed in water. The mass U – U — of water that will rise in this tube is :

22.5 g

35.0 g

NTA Answer: Option 4(final)

NEET 2020

r U r (r =1.5 r ) U U 45. The quantities of heat required to raise the 1 2 1 2 1 K fl h U fl c temperature of two solid copper spheres of radii — r and r (r =1.5 r ) through 1 K are in the 1 2 1 2 5 ratio :

127 3

29 8

33 4

NTA Answer: Option 2(final)

How NEET usually asks this

5 recurring patterns from past papers — click to collapse

Apply Bernoulli's equation to flow through pipes of varying cross-section / heights / Venturi-like geometries.

InterpretationMedium

Common distractors

forgets height term

Drops rho*g*h term

equates pressures incorrectly

Picks wrong reference points

Two bodies of given material and dimensions; find ratio of heat needed for same temp change. Q ∝ m c.

Direct ApplicationEasy

Common distractors

ignores mass difference

Compares only specific heats, not masses

Energy to form a bubble of given radius from soap solution; or pressure inside vs outside. Bubble has 2 surfaces (factor 4T/r).

Direct ApplicationEasy

Common distractors

uses 2T r instead of 4T r for bubble

Treats soap bubble like a drop

Sphere falling in viscous liquid; find terminal velocity or shape of v(t) curve. v_t ∝ r^2.

InterpretationMedium

Common distractors

expects linear acceleration

Default to constant acceleration without recognising drag-induced terminal v

Wire stretching: given Y, length, area, force or stress, find elongation or stress at limit. Y = FL/(A delta_L).

Multi StepMedium

Common distractors

uses bulk modulus formula

Confuses Y with K

Sources

NCERT refs: Class 11 Physics Chapter 9, p.6

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