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]

0%

(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)

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

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

$η = 1.99$

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]

0%

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]

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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]

0%

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]

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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]