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

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

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

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

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

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]

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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]

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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]