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

Q 1 :

A spherical ball of radius $1 \times 10^{- 4}$ m and density $10^{5}$ $k g / m^{3}$ falls freely under gravity through a distance $h$ before entering a tank of water. If after entering in water the velocity of the ball does not change, then the value of $h$ is approximately

(The coefficient of viscosity of water is $9.8 \times 10^{- 6}$ $N s / m^{2}$ ) [2024]

2296 m
2249 m
2518 m
2396 m

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(3)

Terminal velocity, $v_{T} = \frac{2 g}{9} \frac{R^{2} [ρ_{B} - ρ_{L}]}{η}$

$\Rightarrow v_{T} = \frac{2}{9} \times \frac{9.8 \times {(10^{- 4})}^{2}}{9.8 \times 10^{- 6}} [10^{5} - 10^{3}] = 220 m / s$

When ball fall from height $(h), v = \sqrt{2 g h}$

Now $h = \frac{v^{2}}{2 g} = \frac{{(220)}^{2}}{2 \times 9.8} \approx 2518 m$

Q 2 :

A small spherical ball of radius $r$ , falling through a viscous medium of negligible density has terminal velocity $' v'$ . Another ball of the same mass but of radius $2 r$ , falling through the same viscous medium will have terminal velocity [2024]

$v / 2$
$v / 4$
$4 v$
$2 v$

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(1)

Since density is negligible hence Buoyancy force will be negligible

At terminal velocity,

$M g = 6 π η r v$

$v \propto \frac{1}{r}$ (as mass is constant)

Now, $\frac{v}{v^{'}} = \frac{r^{'}}{r}$

$r^{'} = 2 r$

So, $v^{'} = \frac{v}{2}$

Q 3 :

Small water droplets of radius 0.01 mm are formed in the upper atmosphere and falling with a terminal velocity of 10 cm/s. Due to condensation, if 8 such droplets are coalesced and formed a larger drop, the new terminal velocity will be _______ cm/s. [2024]

(40) $m = mass of small drop,$ $M = mass of bigger drop$

$V_{t} = \frac{2}{9} \frac{r^{2} (ρ - σ) g}{η} \Rightarrow V_{t} \propto r^{2}$

$8 m = M \Rightarrow 8 r^{3} = R^{3} \Rightarrow R = 2 r$

$Radius double so V_{t} becomes 4 times :$

$\Rightarrow V_{t} = 4 \times 10 = 40 cm / s$

Q 4 :

A small ball of mass m and density $ρ$ is dropped in a viscous liquid of density $ρ_{0}$ . After some time, the ball falls with constant velocity. The viscous force on the ball is: [2024]

$m g (\frac{ρ_{0}}{ρ} - 1)$
$m g (1 - ρ ρ_{0})$
$m g (1 - \frac{ρ}{ρ_{0}})$
$m g (1 - \frac{ρ_{0}}{ρ})$

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(4)

At constant velocity, $F_{v} + F_{B} = m g$

$F_{B} = ρ_{0 \cdot} (\frac{m}{ρ}) \cdot g$

$F_{v} = m g - F_{B} = m g - \frac{m}{ρ} ρ_{0} g \Rightarrow F_{v} = m g (1 - \frac{ρ_{0}}{ρ})$

Q 5 :

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity-time graph for the transit of the ball? [2024]

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(4)

$m g - F_{B} - F_{v} = m a$

$(ρ \frac{4}{3} π r^{3}) g - (ρ_{L} \frac{4}{3} π r^{3}) g - 6 π η r v = m \frac{d v}{d t}$

Let $\frac{4}{3 m} π R^{3} g (ρ - ρ_{L}) = K_{1} and \frac{6 π η r}{m} = K_{2}$

$\frac{d v}{d t} = K_{1} - K_{2} v \Rightarrow d v = (K_{1} - K_{2} v) d t$

$\int_{0}^{v} \frac{d v}{K_{1} - K_{2} v} = \int_{0}^{t} d t$

$- \frac{1}{K_{2}} \ln {[K_{1} - K_{2} v]}_{0}^{v} = t \Rightarrow \ln (\frac{K_{1} - K_{2} v}{K_{1}}) = - K_{2} t$

$K_{1} - K_{2} v = K_{1} e^{- K_{2} t} \Rightarrow v = \frac{K_{1}}{K_{2}} [1 - e^{- K_{2} t}]$

Q 6 :

A small rigid spherical ball of mass M is dropped in a long vertical tube containing glycerine. The velocity of the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the viscous force acting on the ball will be (consider g as acceleration due to gravity) [2025]

$\frac{3}{2} M g$
$\frac{M g}{2}$
Mg
2 Mg

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(2)

At terminal velocity a = 0

$F = \frac{V ρ}{2} g = \frac{M g}{2}$

$∴ f = M g - \frac{M g}{2} = \frac{M g}{2}$

Q 7 :

Given below are two statements:

Statement-I : The hot water flows faster than cold water.

Statement-II : Soap water has higher surface tension as compared to fresh water.

In the light above statements, choose the correct answer from the options given below: [2025]

Statement-I is false but Statement-II is true
Statement-I is true but Statement-II is false
Both Statement-I and Statement-II are true
Both Statement-I and Statement-II are false

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(2)

Hot water flows faster because of less viscosity and soap water has less surface tension because bubbles are easily formed.

Q 8 :

A solid steel ball of diameter 3.6 mm acquired terminal velocity $2.45 \times 10^{- 2} m / s$ while falling under gravity through an oil of density 925 $kg m^{- 3}$ . Take density of steel as 7825 $kg m^{- 3}$ and g as 9.8 $m / s^{2}$ . The viscosity of the oil in SI unit is [2025]

2.18
2.38
1.68
1.99

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(4)

$v_{T} \Rightarrow \frac{2}{9} \frac{(ρ_{0} - ρ_{l}) r^{2} g}{η}$

$η = \frac{2}{9} (\frac{7825 - 925}{2.45 \times 10^{- 2}}) \times {(1.8)}^{2} \times 10^{- 6} \times 9.8$

$η = 1.99$

Q 9 :

A small ball of mass M and density $ρ$ is dropped in a viscous liquid of density $ρ_{0}$ . After some time, the ball falls with a constant velocity. What is the viscous force on the ball? [2023]

$F = M g (1 - \frac{ρ_{0}}{ρ})$
$F = M g (1 + \frac{ρ}{ρ_{0}})$
$F = M g (1 + ρ ρ_{0})$
$F = M g (1 + \frac{ρ_{0}}{ρ})$

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(1)

$For constant velocity F_{net} = 0$

$F_{vis} + ρ_{0} v g = ρ v g$

$F_{vis} = (ρ - ρ_{0}) v g$

$= ρ v g (1 - \frac{ρ_{0}}{ρ}) = M g (1 - \frac{ρ_{0}}{ρ})$

Q 10 :

Eight equal drops of water are falling through air with a steady speed of 10 cm/s. If the drops coalesce, the new velocity is [2023]

10 cm/s
40 cm/s
16 cm/s
5 cm/s

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(2)

$v \propto r^{2}$

$\Rightarrow \frac{v_{1}}{v_{2}} = {(\frac{r}{R})}^{2}$

$8 \cdot \frac{4}{3} π r^{3} = \frac{4}{3} π R^{3}$

$R = 2 r$

$\Rightarrow \frac{10}{v_{2}} = {(\frac{1}{2})}^{2} \Rightarrow v_{2} = 40 cm/s$

Q 11 :

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: A spherical body of radius $(5 \pm 0.1)$ mm having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is 4%.

Reason R: The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius.

In the light of the above statements, choose the correct answer from the options given below [2023]

Both A and R are true but R is NOT the correct explanation of A
Both A and R are true and R is the correct explanation of A
A is false but R is true
A is true but R is false

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(4)

$Terminal velocity of a spherical body in liquid$

$\Rightarrow V_{t} \propto r^{2} \Rightarrow \frac{Δ V_{t}}{V_{t}} = 2 \cdot \frac{Δ r}{r}$

$\Rightarrow \frac{Δ V_{t}}{V_{t}} \times 100 % = 2 \frac{0.1}{5} \times 100 = 4 %$

$Also V_{t} \propto r^{2}$

$Reason R is false$

Q 12 :

A spherical ball of radius 1 mm and density 10.5 g/cc is dropped in glycerine of coefficient of viscosity 9.8 poise and density 1.5 g/cc. The viscous force on the ball when it attains constant velocity is $3696 \times 10^{- x} N$ . The value of $x$ is. (Given: $g = 9.8 {m/s}^{2}$ and $π = \frac{22}{7}$ ) ________ . [2023]

(7)

$When the ball attains terminal velocity$

$F_{v} = (m g - F_{B}) (∵ a = 0)$

$= V σ_{b} g - V ρ_{ℓ} g = V (σ_{b} - ρ_{ℓ})$

$= \frac{4}{3} π {(10^{- 3})}^{3} \times 9.8 (10.5 - 1.5) \times 10^{3}$

$= 3696 \times 10^{- 7} N$

$So, x = 7$

Q 13 :

A metal block of base area $0.20 m^{2}$ is placed on a table, as shown in the figure. A liquid film of thickness 0.25 mm is inserted between the block and the table. The block is pushed by a horizontal force of 0.1 N and moves with a constant speed. If the viscosity of the liquid is $5.0 \times 10^{- 3}$ poiseuille, the speed of the block is _______ $\times 10^{- 3} m/s$ . [2023]

(25)

$| F |$ $= η A \frac{Δ v}{Δ h}$ : $0.1 = 5 \times 10^{- 3} \times 0.2 \times \frac{v}{0.25 \times 10^{- 3}}$

$v = 0.025 m/s or v = 25 \times 10^{- 3} m/s$

Q 14 :

An air bubble of diameter 6 mm rises steadily through a solution of density $1750 {kg/m}^{3}$ at the rate of 0.35 cm/s. The coefficient of viscosity of the solution (neglect density of air) is _______ P as (given, $g = 10 {ms}^{- 2}$ ). [2023]

(10)

Since the bubble is moving at constant speed, the force acting on it is zero.

$B = F_{v}$

$\frac{4}{3} π R^{3} ρ g = 6 π η R v$

$η = \frac{2 R^{2} ρ g}{9 v} = \frac{2 \times {(3 \times 10^{- 3})}^{2} \times 1750 \times 10}{9 \times 0.35 \times 10^{- 2}} = 10 Pas$

Q 15 :

Sixty four rain drops of radius 1 mm each falling down with a terminal velocity of 10 cm/s coalesce to form a bigger drop. The terminal velocity of the bigger drop is ______ cm/s. [2026]

(160)

$V_{T} = \frac{2 r^{2} g}{9 η} (σ - ρ)$

$V_{T} \propto r^{2}$

$64 drops$

$64 (\frac{4}{3} π R_{1}^{3}) = \frac{4}{3} π R_{2}^{3}$

$R_{2} = 4 R_{1}$

$\frac{{(V_{T})}_{1}}{{(V_{T})}_{2}} = {(\frac{R_{1}}{R_{2}})}^{2} = {(\frac{1}{4})}^{2}$

$\frac{10}{{(V_{T})}_{2}} = \frac{1}{16}$

${(V_{T})}_{2} = 160 cm/sec$

Q 16 :

A small metallic sphere of diameter 2 mm and density $10.5 {g/cm}^{3}$ is dropped in glycerine having viscosity 10 Poise and density $1.5 {g/cm}^{3}$ respectively. The terminal velocity attained by the sphere is ____ cm/s.

$(π = \frac{22}{7} and g = 10 {m/s}^{2})$ [2026]

3.0
1.0
2.0
1.5

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(3)

$V_{T} = \frac{2 r^{2} g}{9 η} (ρ_{b} - ρ_{ℓ})$

$V_{T} = \frac{2}{9} \cdot \frac{{(1)}^{2} \times 10}{10} (10.5 - 1.5)$

$V_{T} = 2 cm/sec.$

Q 17 :

A ball of radius r and density $ρ$ dropped through a viscous liquid of density $σ$ and viscosity $η$ attains its terminal velocity at time t, given by $t = A ρ^{a} r^{b} η^{c} σ^{d}$ where A is a constant and a,b,c and d are integers. The value of $\frac{b + c}{a + d} is$ _______. [2026]

(1)

$T = {(M L^{- 3})}^{a} L^{b} {(M L^{- 1} T^{- 1})}^{c} (M L^{- 3})^{d}$

$T = M^{a + c + d} L^{- 3 a - c - 3 d + b} T^{- c}$

$on comparing$

$c = - 1; a + c + d = 0; - 3 a - c - 3 d + b = 0$

$b = 2; a + d = 1$

$b + c = 1$

Q 18 :

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm, in the same fluid will be ______ cm/s. [2026]

(5)

$We know:$

$Terminal velocity \propto {(radius)}^{2}$

$\frac{{(v_{T})}_{1}}{{(v_{T})}_{2}} = {(\frac{6}{3})}^{2}$

${(v_{T})}_{2} = \frac{{(v_{T})}_{1}}{4} = 5 cm/sec$

Topic Question Set

Q 2 :

A small spherical ball of radius $r$ , falling through a viscous medium of negligible density has terminal velocity $' v'$ . Another ball of the same mass but of radius $2 r$ , falling through the same viscous medium will have terminal velocity [2024]

Q 3 :

Small water droplets of radius 0.01 mm are formed in the upper atmosphere and falling with a terminal velocity of 10 cm/s. Due to condensation, if 8 such droplets are coalesced and formed a larger drop, the new terminal velocity will be _______ cm/s. [2024]

Q 4 :

A small ball of mass m and density $ρ$ is dropped in a viscous liquid of density $ρ_{0}$ . After some time, the ball falls with constant velocity. The viscous force on the ball is: [2024]

Q 5 :

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity-time graph for the transit of the ball? [2024]

Q 7 :

Given below are two statements:

Statement-I : The hot water flows faster than cold water.

Statement-II : Soap water has higher surface tension as compared to fresh water.

In the light above statements, choose the correct answer from the options given below: [2025]

Q 8 :

A solid steel ball of diameter 3.6 mm acquired terminal velocity $2.45 \times 10^{- 2} m / s$ while falling under gravity through an oil of density 925 $kg m^{- 3}$ . Take density of steel as 7825 $kg m^{- 3}$ and g as 9.8 $m / s^{2}$ . The viscosity of the oil in SI unit is [2025]

Q 9 :

A small ball of mass M and density $ρ$ is dropped in a viscous liquid of density $ρ_{0}$ . After some time, the ball falls with a constant velocity. What is the viscous force on the ball? [2023]

Q 10 :

Eight equal drops of water are falling through air with a steady speed of 10 cm/s. If the drops coalesce, the new velocity is [2023]

Q 14 :

An air bubble of diameter 6 mm rises steadily through a solution of density $1750 {kg/m}^{3}$ at the rate of 0.35 cm/s. The coefficient of viscosity of the solution (neglect density of air) is _______ P as (given, $g = 10 {ms}^{- 2}$ ). [2023]

Q 15 :

Sixty four rain drops of radius 1 mm each falling down with a terminal velocity of 10 cm/s coalesce to form a bigger drop. The terminal velocity of the bigger drop is ______ cm/s. [2026]

Q 16 :

A small metallic sphere of diameter 2 mm and density $10.5 {g/cm}^{3}$ is dropped in glycerine having viscosity 10 Poise and density $1.5 {g/cm}^{3}$ respectively. The terminal velocity attained by the sphere is ____ cm/s.

$(π = \frac{22}{7} and g = 10 {m/s}^{2})$ [2026]

Q 18 :

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm, in the same fluid will be ______ cm/s. [2026]

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Topic Question Set

Q 2 : A small spherical ball of radius r, falling through a viscous medium of negligible density has terminal velocity 'v'. Another ball of the same mass but of radius 2r, falling through the same viscous medium will have terminal velocity [2024]

Q 3 : Small water droplets of radius 0.01 mm are formed in the upper atmosphere and falling with a terminal velocity of 10 cm/s. Due to condensation, if 8 such droplets are coalesced and formed a larger drop, the new terminal velocity will be _______ cm/s. [2024]

Q 4 : A small ball of mass m and density ρ is dropped in a viscous liquid of density ρ0. After some time, the ball falls with constant velocity. The viscous force on the ball is: [2024]

Q 5 : A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity-time graph for the transit of the ball? [2024]

Q 7 : Given below are two statements: Statement-I : The hot water flows faster than cold water. Statement-II : Soap water has higher surface tension as compared to fresh water. In the light above statements, choose the correct answer from the options given below: [2025]

Q 8 : A solid steel ball of diameter 3.6 mm acquired terminal velocity 2.45×10–2 m/s while falling under gravity through an oil of density 925 kg m–3. Take density of steel as 7825 kg m–3 and g as 9.8 m/s2. The viscosity of the oil in SI unit is [2025]

Q 9 : A small ball of mass M and density ρ is dropped in a viscous liquid of density ρ0. After some time, the ball falls with a constant velocity. What is the viscous force on the ball? [2023]

Q 10 : Eight equal drops of water are falling through air with a steady speed of 10 cm/s. If the drops coalesce, the new velocity is [2023]

Q 12 : A spherical ball of radius 1 mm and density 10.5 g/cc is dropped in glycerine of coefficient of viscosity 9.8 poise and density 1.5 g/cc. The viscous force on the ball when it attains constant velocity is 3696×10-x N. The value of x is. (Given: g=9.8 m/s2 and π=227) ________ . [2023]

Q 14 : An air bubble of diameter 6 mm rises steadily through a solution of density 1750 kg/m3 at the rate of 0.35 cm/s. The coefficient of viscosity of the solution (neglect density of air) is _______ P as (given, g=10 ms-2). [2023]

Q 15 : Sixty four rain drops of radius 1 mm each falling down with a terminal velocity of 10 cm/s coalesce to form a bigger drop. The terminal velocity of the bigger drop is ______ cm/s. [2026]

Q 16 : A small metallic sphere of diameter 2 mm and density 10.5 g/cm3 is dropped in glycerine having viscosity 10 Poise and density 1.5 g/cm3 respectively. The terminal velocity attained by the sphere is ____ cm/s. (π=227 and g=10 m/s2) [2026]

Q 17 : A ball of radius r and density ρ dropped through a viscous liquid of density σ and viscosity η attains its terminal velocity at time t, given by t=A ρarbηcσd where A is a constant and a,b,c and d are integers. The value of b+ca+d is _______. [2026]

Q 18 : The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm, in the same fluid will be ______ cm/s. [2026]

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Q 2 :

A small spherical ball of radius $r$ , falling through a viscous medium of negligible density has terminal velocity $' v'$ . Another ball of the same mass but of radius $2 r$ , falling through the same viscous medium will have terminal velocity [2024]

Q 3 :

Small water droplets of radius 0.01 mm are formed in the upper atmosphere and falling with a terminal velocity of 10 cm/s. Due to condensation, if 8 such droplets are coalesced and formed a larger drop, the new terminal velocity will be _______ cm/s. [2024]

Q 4 :

A small ball of mass m and density $ρ$ is dropped in a viscous liquid of density $ρ_{0}$ . After some time, the ball falls with constant velocity. The viscous force on the ball is: [2024]

Q 5 :

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity-time graph for the transit of the ball? [2024]

Q 7 :

Given below are two statements:

Statement-I : The hot water flows faster than cold water.

Statement-II : Soap water has higher surface tension as compared to fresh water.

In the light above statements, choose the correct answer from the options given below: [2025]

Q 8 :

A solid steel ball of diameter 3.6 mm acquired terminal velocity $2.45 \times 10^{- 2} m / s$ while falling under gravity through an oil of density 925 $kg m^{- 3}$ . Take density of steel as 7825 $kg m^{- 3}$ and g as 9.8 $m / s^{2}$ . The viscosity of the oil in SI unit is [2025]

Q 9 :

A small ball of mass M and density $ρ$ is dropped in a viscous liquid of density $ρ_{0}$ . After some time, the ball falls with a constant velocity. What is the viscous force on the ball? [2023]

Q 10 :

Eight equal drops of water are falling through air with a steady speed of 10 cm/s. If the drops coalesce, the new velocity is [2023]

Q 14 :

An air bubble of diameter 6 mm rises steadily through a solution of density $1750 {kg/m}^{3}$ at the rate of 0.35 cm/s. The coefficient of viscosity of the solution (neglect density of air) is _______ P as (given, $g = 10 {ms}^{- 2}$ ). [2023]

Q 15 :

Sixty four rain drops of radius 1 mm each falling down with a terminal velocity of 10 cm/s coalesce to form a bigger drop. The terminal velocity of the bigger drop is ______ cm/s. [2026]

Q 16 :

A small metallic sphere of diameter 2 mm and density $10.5 {g/cm}^{3}$ is dropped in glycerine having viscosity 10 Poise and density $1.5 {g/cm}^{3}$ respectively. The terminal velocity attained by the sphere is ____ cm/s.

$(π = \frac{22}{7} and g = 10 {m/s}^{2})$ [2026]

Q 18 :

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm, in the same fluid will be ______ cm/s. [2026]