JEE Advanced Mechanical Properties of Fluids Mixed Concepts PYQ

Q 1 :

Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $ρ$ remains uniform throughout the volume. The rate of fractional change in density $(\frac{1}{ρ} \frac{d ρ}{d t})$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to [2017]

$R$
$R^{3}$
$\frac{1}{R}$
$R^{2 / 3}$

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(1)

$Given : \frac{1}{ρ} \frac{d ρ}{d t} = constant$

$∴$ $\frac{4 π R^{3}}{3 m} \frac{d}{d t} [\frac{m}{\frac{4}{3} π R^{3}}] = constant$

$\Rightarrow R^{3} \frac{d}{d t} (R^{- 3}) = constant$

$\Rightarrow R^{3} (- 3 R^{- 4}) \frac{d R}{d t} = constant$ $∴$ $| \frac{d R}{d t} |$ $\propto R$

Q 2 :

A glass tube of uniform internal radius $(r)$ has a valve separating the two identical ends. Initially, the valve is in a tightly closed position.

[IMAGE 406]

End 1 has a hemispherical soap bubble of radius $r$ . End 2 has sub-hemispherical soap bubble as shown in figure. Just after opening the valve, [2008]

air from end 1 flows towards end 2. No change in the volume of the soap bubbles
air from end 1 flows towards end 2. Volume of the soap bubble at end 1 decreases
no changes occurs
air from end 2 flows towards end 1. volume of the soap bubble at end 1 increases

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(2)

[IMAGE 407]

We know that excess pressure in a soap bubble is inversely proportional to its radius, i.e., $P = \frac{4 T}{r}$

The soap bubble at end 1 has smaller radius as compared to the soap bubble at end 2 (given). Therefore excess pressure at 1 is more.

Hence, air flows from end 1 to end 2 and the volume of soap bubble at end 1 decreases.

Q 3 :

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $μ_{0}$ . Which of the following statements is (are) true?} [2018]

The resistive force of liquid on the plate is inversely proportional to $h$
The resistive force of liquid on the plate is independent of the area of the plate
The tangential (shear) stress on the floor of the tank increases with $μ_{0}$
The tangential (shear) stress on the plate varies linearly with the viscosity $η$ of the liquid

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4

Select one or more options

(1, 3, 4)

$F = - η A (\frac{d v}{d x})$ or $| F |$ $= η A \frac{u_{0}}{h}$

where $\frac{u_{0}}{h} = velocity gradient (\frac{d v}{d x})$

$F \propto η; F \propto A; F \propto u_{0} and F \propto \frac{1}{h}$

Q 4 :

A person in lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance $d$ of 1.2 m from the person. In the following, state of the lift's motion is given in List-I and the distance where the water jet hits the floor of the lift is given in List-II. Match the statements from List-I with those in List-II and select the correct answer using the code given below the lists. [2014]

List - I List - II

P. Lift is accelerating vertically up 1. $d = 1.2 m$

Q. Lift is accelerating vertically down with an acceleration less than the gravitational acceleration 2. $d > 1.2 m$

R. Lift is moving vertically up with constant speed 3. $d < 1.2 m$

S. Lift is falling freely 4. No water leaks out of the jar

Code:

P-2, Q-3, R-2, S-4
P-2, Q-3, R-1, S-4
P-1, Q-1, R-1, S-4
P-2, Q-3, R-1, S-1

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(3)

[IMAGE 408]

Horizontal distance,

$d = v \times t$

$= \sqrt{2 g h_{2}} \times \sqrt{\frac{2 h_{1}}{g}}$

$= 2 \sqrt{h_{1} h_{2}}$

If $g_{eff} > g$

$g_{eff} = g$

$g_{eff} < g$

In all the three cases, $d = 2 \sqrt{h_{1} h_{2}} = 1.2 m$

If $g_{eff} = 0,$ then no water leaks out as there will be no pressure difference.

Topic Question Set

Q 3 :

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $μ_{0}$ . Which of the following statements is (are) true?} [2018]

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	List - I		List - II
P.	Lift is accelerating vertically up	1.	$d = 1.2 m$
Q.	Lift is accelerating vertically down with an acceleration less than the gravitational acceleration	2.	$d > 1.2 m$
R.	Lift is moving vertically up with constant speed	3.	$d < 1.2 m$
S.	Lift is falling freely	4.	No water leaks out of the jar

Topic Question Set

Q 3 : Consider a thin square plate floating on a viscous liquid in a large tank. The height h of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity μ0. Which of the following statements is (are) true?} [2018]

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Q 3 :

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $μ_{0}$ . Which of the following statements is (are) true?} [2018]