Fluid Mechanics | Surface Tension And Viscosity | JEE Main previous year questions with Solutions

Q 1 :

Correct Bernoulli's equation is (symbols have their usual meaning) [2024]

$P + ρ g h + \frac{1}{2} ρ v^{2} = c o n s \tan t$
$P + ρ g h + ρ v^{2} = c o n s \tan t$
$P + m g h + \frac{1}{2} m v^{2} = c o n s \tan t$
$P + \frac{1}{2} ρ g h + \frac{1}{2} ρ v^{2} = c o n s \tan t$

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(1)

$P + ρ g h + \frac{1}{2} ρ v^{2} = c o n s \tan t$

Energy per unit volume is conserved.

Q 2 :

The reading of pressure metre attached with a closed pipe is $4.5 \times 10^{4} N / m^{2}$ . On opening the valve, water starts flowing and the reading of pressure metre falls to $2.0 \times 10^{4} N / m^{2}$ . The velocity of water is found to be $\sqrt{V} m / s$ . The value of V is ___________ . [2024]

(50) $Change in pressure = \frac{1}{2} ρ v^{2}$

$4.5 \times 10^{4} - 2 \times 10^{4} = \frac{1}{2} \times 10^{3} \times v^{2}$

$2.5 \times 10^{4} = \frac{1}{2} \times 10^{3} \times v^{2}$

$v^{2} = 50$

$v = \sqrt{50}$

Q 3 :

A plane is in level flight at constant speed and each of its two wings has an area of $40$ $m^{2}$ . If the speed of the air is 180 km/h over the lower wing surface and 252 km/h over the upper wing surface, the mass of the plane is __________ kg.

$(T a k e a i r d e n s i t y t o b e 1 k g$ $m^{- 3}$ $a n d$ $g = 10$ $m s^{- 2})$ [2024]

(9600) By Bernoulli's equation

$P_{1} + \frac{1}{2} ρ v_{1}^{2} = P_{2} + \frac{1}{2} ρ v_{2}^{2}$

$\frac{1}{2} \times ρ \times [4900 - 2500] = \frac{m g}{A}$

$\frac{1}{2} \times 1 \times 2400 = \frac{m \times 10}{80}$

$m = 9600 kg$

Q 4 :

Given below are two statements:

Statement I: When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $P_{1} - P_{2} = ρ g (h_{2} - h_{1}) .$

Statement II: In venturi tube shown, $2 g h = v_{1}^{2} - v_{2}^{2}$

In the light of the above statements, choose the most appropriate answer from the options given below. [2024]

Both Statement I and Statement II are correct.
Both Statement I and Statement II are incorrect.
Statement I is correct but Statement II is incorrect.
Statement I is incorrect but Statement II is correct.

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(3)

From Bernoulli’s equation

$P_{1} + ρ g h_{1} + \frac{1}{2} ρ v_{1}^{2} = P_{2} + ρ g h_{2} + \frac{1}{2} ρ v_{2}^{2}$

[ $h_{1}$ and $h_{2}$ are height of points from any reference level]

For statement 1:

When speed of liquid is zero everywhere, $v_{1} = v_{2} = 0$

$∴$ $P_{1} - P_{2} = ρ g (h_{2} - h_{1}),$ Hence, statement I is correct.

For statement 2, $h_{1} = h_{2}$

$P_{1} + \frac{1}{2} ρ v_{1}^{2} = P_{2} + \frac{1}{2} ρ v_{2}^{2} \Rightarrow P_{1} - P_{2} = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2})$

But, $P_{1} - P_{2} = ρ g h \Rightarrow ρ g h = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2})$

Hence, $2 g h = v_{2}^{2} - v_{1}^{2}$ , Hence statement II is incorrect.

Q 5 :

In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 ${ms}^{- 1}$ and 65 ${ms}^{- 1}$ respectively. If the wing area is 2 $m^{2}$ the lift of the wing is ______ N.

(Given density of air = 1.2 kg $m^{- 1}$ ) [2024]

(810)

$F = \frac{1}{2} ρ (v_{1}^{2} - v_{2}^{2}) A$

$F = \frac{1}{2} \times 1.2 \times (70^{2} - 65^{2}) \times 2 = 810 N$

Q 6 :

A tube of length L is shown in the figure. The radius of cross section at the point (1) is 2 cm and at the point (2) is 1 cm, respectively. If the velocity of water entering at point (1) is 2 m/s, then velocity of water leaving the point (2) will be: [2025]

2 m/s
4 m/s
6 m/s
8 m/s

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(4)

$A_{1} V_{1} = A_{2} V_{2}$

$\Rightarrow 2 π {(2 R)}^{2} = V_{2} π R^{2}$

$∴ V_{2} = 8 m / s$

Q 7 :

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure guage attached to the pipe is $P_{1}$ . The reading of the pressure gauge falls to $P_{2}$ when the valve is opened. The speed of water flowing in the pipe is proportional to [2025]

$\sqrt{P_{1} - P_{2}}$
${(P_{1} - P_{2})}^{2}$
${(P_{1} - P_{2})}^{4}$
$P_{1} - P_{2}$

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(1)

Using Bernoulli's theorem

$P_{1} - P_{2} = \frac{1}{2} ρ v^{2}$

$v \propto \sqrt{P_{1} - P_{2}}$

Q 8 :

Consider a completely full cylindrical water tank of height 1.6 m and cross-sectional area 0.5 $m^{2}$ . It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load 50 kg is applied at the top surface of the water in the tank them the velocity of the water coming out at the instant when the hole is opened is: ( $g = 10 m / s^{2}$ ) [2025]

3 m/s
5 m/s
2 m/s
4 m/s

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(4)

Apply Bernoulli equation between points 1 and 2

$P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h = P_{2} + \frac{1}{2} ρ v_{2}^{2} + 0$

$P_{0} + \frac{m g}{A} + ρ g \frac{70}{100} = P_{0} + \frac{1}{2} ρ v_{2}^{2}$

As the tank area is large $v_{1}$ is negligible compared to $v_{2}$

$\frac{5000}{5} + 10^{3} \times 10 \frac{70}{100} = \frac{1}{2} \times 10^{3} v_{2}^{2}$

$10^{3} + 10^{3} \times 7 = \frac{10^{3}}{2} v_{2}^{2} \Rightarrow v_{2}^{2} = 16$

$\Rightarrow v_{2} = 4 m / s$

Q 9 :

A fully loaded Boing aircraft has a mass of $5.4 \times 10^{5} kg$ . Its total wing area is $500 m^{2}$ . It is in level flight with a speed of $1080 km/h$ . If the density of air $ρ$ is $1.2 {kg m}^{- 3}$ , the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface in percentage will be ( $g = 10 {m/s}^{2}$ ) [2023]

16
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10

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(4)

$P_{2} A - P_{1} A = 5.0 \times 10^{5} \times g$

$P_{2} - P_{1} = \frac{5.4 \times 10^{6}}{500} = 5.4 \times 2 \times 10^{2} \times 10 = 10.8 \times 10^{3}$

$P_{2} + 0 + \frac{1}{2} ρ V_{2}^{2} = P_{1} + 0 + \frac{1}{2} ρ V_{1}^{2}$

$P_{2} - P_{1} = \frac{1}{2} ρ (V_{1}^{2} - V_{2}^{2}) = \frac{1}{2} ρ (V_{1} - V_{2}) (V_{1} + V_{2})$

$10.8 \times 10^{3} = \frac{1}{2} \times 1.2 (V_{1} - V_{2}) \times 2 \times 3 \times 10^{2}$

$10.8 \times 10 = 3.6 (V_{1} - V_{2})$

$V_{1} - V_{2} = 30$

$(\frac{V_{1} - V_{2}}{V}) \times 100 = \frac{30}{300} \times 100 = 10 %$

Q 10 :

The figure shows a liquid of given density flowing steadily in a horizontal tube of varying cross-section. Cross-sectional areas at A is $1.5 {cm}^{2}$ , and B is $25 {mm}^{2}$ . If the speed of liquid at B is $60 cm/s$ , then $(P_{A} - P_{B})$ is

(Given $P_{A}$ and $P_{B}$ are liquid pressures at A and B points. Density $ρ = 1000 {kg m}^{3}$ .

A and B are on the axis of the tube [2023]

175 Pa
27 Pa
135 Pa
36 Pa

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(1)

$From continuity theorem A_{1} V_{1} = A_{2} V_{2}$

$1.5 \times V_{1} = 25 \times 10^{- 2} \times 60$

$V_{1} = \frac{25 \times 60 \times 10^{- 2} \times 10}{1.5} = 10 cm/s$

$By Bernoulli's theorem$

$P_{1} + \frac{1}{2} \times 1000 \times {(0.1)}^{2} = P_{2} + \frac{1}{2} \times 1000 \times {(0.6)}^{2}$

$P_{1} + 5 = P_{2} + \frac{1}{2} \times 1000 \times 36 \times 10^{- 2}$

$\Rightarrow P_{1} - P_{2} = 175 Pa$

Q 11 :

The surface of water in a water tank of cross-sectional area $750 {cm}^{2}$ on the top of a house is $h$ m above the tap level. The speed of water coming out through the tap of cross-sectional area $500 {mm}^{2}$ is $30 cm/s$ . At that instant, $\frac{d h}{d t}$ is $x \times 10^{- 3} m/s$ . The value of $x$ will be _______. [2023]

(2)

$a_{1} v_{1} = a_{2} v_{2}$

$750 \times 10^{- 4} v_{1} = 500 \times 10^{- 6} \times 0.3$

$v_{1} = \frac{500 \times 3 \times 10^{- 3}}{750} m/s = 2 \times 10^{- 3} m/s$

$\Rightarrow \frac{d h}{d t} = - 2 \times 10^{- 3} m/s$

Q 12 :

Figure below shows a liquid being pushed out of the tube by a piston having area of cross section $2.0 {cm}^{2}$ . The area of cross section at the outlet is $10 {mm}^{2}$ . If the piston is pushed at a speed of ${4 cm s}^{- 1}$ , the speed of outgoing fluid is _________ ${cm s}^{- 1}$ . [2023]

(80)

$By equation of continuity$

$A_{1} V_{1} = A_{2} V_{2}$

$\Rightarrow V_{2} = \frac{2 \times 4}{10 \times 10^{- 2}} = 80 cm/s$

Q 13 :

Glycerine of density $1.25 \times 10^{3} {kg m}^{- 3}$ is flowing through the conical section of a pipe. The area of cross-section of the pipe at its ends is $10 {cm}^{2}$ and $5 {cm}^{2}$ and the pressure drop across its length is ${3 N m}^{- 2}$ . The rate of flow of glycerine through the pipe is $x \times 10^{- 5} m^{3} s^{- 1}$ . The value of $x$ is _________. [2023]

(4)

$Δ P = P_{1} - P_{2} = {3 N/m}^{2} (given)$

${By continuity eq}^{n}, A_{1} v_{1} = A_{2} v_{2}$

$∴$ $v_{1} = \frac{A_{2}}{A_{1}} v_{2}$ ...(i)

${By Bernoulli's eq}^{n},$

$P_{1} + \frac{1}{2} ρ v_{1}^{2} = P_{2} + \frac{1}{2} ρ v_{2}^{2}$

$P_{1} - P_{2} = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2})$

$\Rightarrow Δ P = \frac{1}{2} ρ [1 - {(\frac{A_{2}}{A_{1}})}^{2}] v_{2}^{2}$

$3 = \frac{1}{2} \times 1.25 \times 10^{3} [1 - {(\frac{5}{10})}^{2}] v_{2}^{2}$

$∴$ $v_{2} = 8 \times 10^{- 2} m/s$

$So discharge rate = A_{2} v_{2} = 5 \times 10^{- 4} \times 8 \times 10^{- 2} = 4 \times 10^{- 5} m^{3} / s$

$Correct answer is x = 4$

Q 14 :

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at A and B are $6 {cm}^{2}$ and $3 {cm}^{2}$ respectively. The rate of flow will be ______ ${cm}^{3} / s$ . (take g = 10 m/s²) [2026]

$\frac{200}{\sqrt{3}}$
$100 \sqrt{3}$
$200 \sqrt{3}$
$200 \sqrt{6}$

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(3)

From continuity equation

$A_{A} V_{A} = A_{B} V_{B} \Rightarrow 6 V_{A} = 3 V_{B} \Rightarrow V_{B} = 2 V_{A}$

Applying Bernoulli's equation between A & B,

$P_{A} + \frac{1}{2} ρ V_{A}^{2} = P_{B} + \frac{1}{2} ρ V_{B}^{2}$

$\Rightarrow ρ g \times 0.05 = \frac{1}{2} ρ [V_{B}^{2} - V_{A}^{2}] = \frac{1}{2} ρ (3 V_{A}^{2})$

$\Rightarrow V_{A} = \sqrt{\frac{2 g \times 0.05}{3}} m/s = \frac{1}{\sqrt{3}} m/s = \frac{100}{\sqrt{3}} cm/s$

$\Rightarrow Volume flow rate = A_{A} V_{A} = \frac{6 \times 100}{\sqrt{3}} {cm}^{3} / sec$

$= 200 \sqrt{3} {cm}^{3} / sec$

Topic Question Set

Q 1 :

Correct Bernoulli's equation is (symbols have their usual meaning) [2024]

Q 5 :

In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 ${ms}^{- 1}$ and 65 ${ms}^{- 1}$ respectively. If the wing area is 2 $m^{2}$ the lift of the wing is ______ N.

(Given density of air = 1.2 kg $m^{- 1}$ ) [2024]

Q 6 :

A tube of length L is shown in the figure. The radius of cross section at the point (1) is 2 cm and at the point (2) is 1 cm, respectively. If the velocity of water entering at point (1) is 2 m/s, then velocity of water leaving the point (2) will be: [2025]

Q 7 :

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure guage attached to the pipe is $P_{1}$ . The reading of the pressure gauge falls to $P_{2}$ when the valve is opened. The speed of water flowing in the pipe is proportional to [2025]

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Topic Question Set

Q 1 : Correct Bernoulli's equation is (symbols have their usual meaning) [2024]

Q 2 : The reading of pressure metre attached with a closed pipe is 4.5×104N/m2. On opening the valve, water starts flowing and the reading of pressure metre falls to 2.0×104N/m2. The velocity of water is found to be Vm/s. The value of V is ___________ . [2024]

Q 3 : A plane is in level flight at constant speed and each of its two wings has an area of 40 m2. If the speed of the air is 180 km/h over the lower wing surface and 252 km/h over the upper wing surface, the mass of the plane is __________ kg. (Take air density to be 1kg m-3 and g=10 ms-2) [2024]

Q 5 : In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 ms-1 and 65 ms-1 respectively. If the wing area is 2 m2 the lift of the wing is ______ N. (Given density of air = 1.2 kg m-1) [2024]

Q 6 : A tube of length L is shown in the figure. The radius of cross section at the point (1) is 2 cm and at the point (2) is 1 cm, respectively. If the velocity of water entering at point (1) is 2 m/s, then velocity of water leaving the point (2) will be: [2025]

Q 7 : Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure guage attached to the pipe is P1. The reading of the pressure gauge falls to P2 when the valve is opened. The speed of water flowing in the pipe is proportional to [2025]

Q 11 : The surface of water in a water tank of cross-sectional area 750 cm2 on the top of a house is h m above the tap level. The speed of water coming out through the tap of cross-sectional area 500 mm2 is 30 cm/s. At that instant, dhdt is x×10-3 m/s. The value of x will be _______. [2023]

Q 12 : Figure below shows a liquid being pushed out of the tube by a piston having area of cross section 2.0 cm2. The area of cross section at the outlet is 10 mm2. If the piston is pushed at a speed of 4 cm s-1, the speed of outgoing fluid is _________ cm s-1. [2023]

Q 14 : Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at A and B are 6cm2 and 3cm2 respectively. The rate of flow will be ______ cm3/s. (take g = 10 m/s²) [2026]

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Q 1 :

Correct Bernoulli's equation is (symbols have their usual meaning) [2024]

Q 5 :

In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 ${ms}^{- 1}$ and 65 ${ms}^{- 1}$ respectively. If the wing area is 2 $m^{2}$ the lift of the wing is ______ N.

(Given density of air = 1.2 kg $m^{- 1}$ ) [2024]

Q 6 :

A tube of length L is shown in the figure. The radius of cross section at the point (1) is 2 cm and at the point (2) is 1 cm, respectively. If the velocity of water entering at point (1) is 2 m/s, then velocity of water leaving the point (2) will be: [2025]

Q 7 :

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure guage attached to the pipe is $P_{1}$ . The reading of the pressure gauge falls to $P_{2}$ when the valve is opened. The speed of water flowing in the pipe is proportional to [2025]