JEE Advanced Pressure, Density, Pascal’s Law & Archimedes’ Principle PYQ

Q 1 :

An open-ended U-tube of uniform cross-sectional area contains water (density $10^{3} kg m^{- 3}$ ). Initially the water level stands at 0.29 m from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800 kg m^{- 3}$ is added to the left arm until its length is 0.1 m, as shown in the schematic figure below. The ratio $(\frac{h_{1}}{h_{2}})$ of the heights of the liquid in the two arms is [2020]

[IMAGE 378]

$\frac{15}{14}$
$\frac{35}{33}$
$\frac{7}{6}$
$\frac{5}{4}$

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(2)

[IMAGE 379]

$Pressure will be same at the same horizontal level.$

$∴$ $P_{A} = P_{B}$

$\Rightarrow P_{0} + ρ_{k} g \times 0.1 + ρ_{w} g \times (h_{1} - 0.1) = ρ_{w} g h_{2} + p_{0}$

$\Rightarrow 80 + 1000 (h_{1} - 0.1) = 1000 h_{2}$

$\Rightarrow 80 + 1000 h_{1} - 100 = 1000 h_{2}$

$\Rightarrow h_{1} - h_{2} = \frac{20}{1000}$

$\Rightarrow h_{1} - h_{2} = 0.02 \dots (i)$

$Also, h_{1} - 0.1 + h_{2} = 2 \times 0.29$

$\Rightarrow h_{1} + h_{2} = 0.68 \dots (i i)$

$Solving (i) & (ii), we get$

$2 h_{1} = 0.70$

$\Rightarrow h_{1} = 0.35 m$

$and h_{2} = 0.33 m$

$So, \frac{h_{1}}{h_{2}} = \frac{35}{33}$

Q 2 :

A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring such that it is half submerged in a liquid of density $σ$ at equilibrium position. The extension $x_{0}$ of the spring when it is in equilibrium is: [2012]

$\frac{M g}{k}$
$\frac{M g}{k} (1 - \frac{L A σ}{M})$
$\frac{M g}{k} (1 - \frac{L A σ}{2 M})$
$\frac{M g}{k} (1 + \frac{L A σ}{M})$

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[IMAGE 380]

$From figure, k x_{0} + F_{B} = M g$

$k x_{0} + σ \frac{L}{2} A g = M g$

$[∵ mass = density \times volume]$

$\Rightarrow k x_{0} = M g - σ \frac{L}{2} A g$

$\Rightarrow x_{0} = \frac{M g - \frac{σ L A g}{2}}{k} = \frac{M g}{k} (1 - \frac{L A σ}{2 M})$

$Hence, extension of the spring when it is in equilibrium is,$

$x_{0} = \frac{M g}{k} (1 - \frac{L A σ}{2 M})$

Q 3 :

A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance $ℓ$ and $h$ are shown here. After some time the coin falls into the water. Then [2002]

[IMAGE 381]

$ℓ$ decreases and $h$ increases
$ℓ$ increases and $h$ decreases
both $ℓ$ and $h$ increase
both $ℓ$ and $h$ decrease

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(4)

When the coin falls from the top of block into water the block moves upwards because the weight of floating body becomes less and hence $l$ decreases. When the coin was floating, it displaces water equal to its own weight. When the coin is inside the water, it displaces water equal to its own volume. As its density is greater than that of water, it displaces more water in first case. Hence, $h$ decreases when coin fall into the water.

Q 4 :

A hemispherical portion of radius R is removed from the bottom of a cylinder of radius R. The volume of the remaining cylinder is V and its mass M. It is suspended by a string in a liquid of density $ρ$ where it stays vertical. The upper surface of the cylinder is at a depth $h$ below the liquid surface. The force on the bottom of the cylinder by the liquid is [2001]

[IMAGE 382]

$M g$
$M g - V ρ g$
$M g + π R^{2} h ρ g$
$ρ g (V + π R^{2} h)$

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[IMAGE 383]

$From Archimedes principle$

$Upthrust = wt. of fluid displaced$

$Now, F_{bottom} - F_{top} = V ρ g$

$\Rightarrow F_{bottom} = F_{top} + V ρ g$

$= P_{1} \times A + V ρ g$

$= (h ρ g) \times (π R^{2}) + V ρ g$

$= ρ g [π R^{2} h + V]$

Q 5 :

A cubical solid aluminium (bulk modulus $= - V \frac{d P}{d V} = 70 GPa$ ) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^{3} {kg m}^{- 3}$ and $10 {m s}^{- 2}$ , respectively, the change in the edge length of the block in mm is ______. [2020]

(0.24)

$As we know, bulk modulus B = \frac{P}{(- \frac{d V}{V})}$

$\Rightarrow \frac{d V}{V} = - \frac{P}{B} \Rightarrow 3 (\frac{Δ ℓ}{ℓ}) = - \frac{P}{B}$

$Δ ℓ = (\frac{- P}{B}) \frac{ℓ}{3} = (\frac{ρ g h}{B}) \frac{ℓ}{3}$

$= \frac{10^{3} \times 10 \times 5 \times 10^{3}}{70 \times 10^{9}} \times \frac{1}{3} = 0.238 \times 10^{- 3}$

$∴$ $Δ ℓ = 0.238 mm ≃ 0.24 mm$

Q 6 :

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $θ = 45 °$ . $P_{1}$ and $P_{2}$ are pressures at points 1 and 2, respectively, located at the base of the tube. Let $β = \frac{(P_{1} - P_{2})}{ρ g d},$ where $ρ$ is density of water, $d$ is the inner diameter of the tube and $g$ is the acceleration due to gravity. Which of the following statement(s) is(are) correct? [2021]

[IMAGE 384]

$β = 0$ when $a = \frac{g}{\sqrt{2}}$
$β > 0$ when $a = \frac{g}{\sqrt{2}}$
$β = \frac{\sqrt{2} - 1}{\sqrt{2}}$ when $a = \frac{g}{2}$
$β = \frac{1}{\sqrt{2}}$ when $a = \frac{g}{2}$

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[IMAGE 385]

$P_{1} = P_{2} - ρ a \cos 45 ° d + ρ (g - a \sin 45 °) d$

$\Rightarrow \frac{P_{1} - P_{2}}{ρ g d} = 1 - \frac{\sqrt{2} a}{g}$

$∴$ $β = 0 for a = \frac{g}{\sqrt{2}}$

$and β = \frac{\sqrt{2} - 1}{\sqrt{2}} for a = \frac{g}{2}$

Q 7 :

A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If $P (r)$ is the pressure at $r (r < R)$ , then the correct option(s) is(are) [2015]

$P (r = 0) = 0$
$\frac{P (r = 3 R / 4)}{P (r = 2 R / 3)} = \frac{63}{80}$
$\frac{P (r = 3 R / 5)}{P (r = 2 R / 5)} = \frac{16}{21}$
$\frac{P (r = R / 2)}{P (r = R / 3)} = \frac{20}{27}$

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[IMAGE 386]

$Let us consider an elemental mass d m shown in the shaded portion of the figure.$

$Here, P (4 π r^{2}) - (P + d P) (4 π r^{2}) = \frac{G M r}{R^{3}} ρ (4 π r^{2}) d r$

$∴$ $- \int_{0}^{P} d P = \frac{G M ρ}{R^{3}} \int_{r}^{R} r d r$

$∴$ $P = \frac{G M ρ}{2 R^{3}} [R^{2} - r^{2}]$

$∴$ $\frac{P (r = 3 R / 4)}{P (r = 2 R / 3)} = \frac{[R^{2} - \frac{9 R^{2}}{16}]}{[R^{2} - \frac{4 R^{2}}{9}]} = \frac{\frac{7 R^{2}}{16}}{\frac{5 R^{2}}{9}} = \frac{63}{80}$

$and \frac{P (r = 3 R / 5)}{P (r = 2 R / 5)} = \frac{[R^{2} - \frac{9 R^{2}}{25}]}{[R^{2} - \frac{4 R^{2}}{25}]} = \frac{16}{21}$

Q 8 :

A solid sphere of radius R and density $ρ$ is attached to one end of a mass-less spring of force constant $k$ . The other end of the spring is connected to another solid sphere of radius R and density $3 ρ$ . The complete arrangement is placed in a liquid of density $2 ρ$ and is allowed to reach equilibrium. The correct statement(s) is(are) [2013]

The net elongation of the spring is $\frac{4 π R^{3} ρ g}{3 k}$
The net elongation of the spring is $\frac{8 π R^{3} ρ g}{3 k}$
The light sphere is partially submerged
The light sphere is completely submerged

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[IMAGE 387]

$The complete system is as shown in figure.$

$Let x be the net elongation of the spring.$

$At equilibrium, for upper sphere$

$W + F_{S} = F_{B}$

$\frac{4}{3} π R^{3} ρ g + k x = \frac{4}{3} π R^{3} (2 ρ) g$

$\Rightarrow k x = \frac{4}{3} π R^{3} (2 ρ) g - \frac{4}{3} π R^{3} ρ g$

$\Rightarrow k x = \frac{4 π R^{3} ρ g}{3}$ or $x = \frac{4 π R^{3} ρ g}{3 k}$

Q 9 :

A cylindrical tank has a hole of diameter $2 r$ in its bottom. The hole is covered by a wooden cylindrical block of diameter $4 r$ , height $h$ and density $ρ / 3$ .

[IMAGE 388]

Situation I: Initially, the tank is filled with water of density $ρ$ to a height such that the height of water above the top of the block is $h_{1}$ (measured from the top of the block).

Situation II: The water is removed from the tank to a height $h_{2}$ (measured from the bottom of the block), as shown in the figure. The height $h_{2}$ is smaller than $h$ (height of the block) and thus the block is exposed to the atmosphere.

Q. Find the minimum value of height $h_{1}$ (in situation 1), for which the block just starts to move up? [2006]

$\frac{2 h}{3}$
$\frac{5 h}{4}$
$\frac{5 h}{3}$
$\frac{5 h}{2}$

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(3)

[IMAGE 389]

$Consider the equilibrium of wooden cylindrical block.$

$Forces acting in the downward direction are$

$(i) Weight of wooden cylinder$

$W = π {(2 r)}^{2} \times h \times \frac{ρ}{3} \times g$

$= π \times 4 r^{2} \times \frac{h ρ}{3} g$

$(i i) Force due to pressure (P_{1}) created by liquid of height h_{1} above the wooden block$

$F_{1} = P_{1} \times π {(2 r)}^{2} = [P_{0} + h_{1} ρ g] \times π {(2 r)}^{2}$

$(i i i) Force acting on the upward direction due to pressure P_{2} exerted from below the wooden block and atmospheric pressure$

$F_{2} = P_{2} \times π [{(2 r)}^{2} - r^{2}] + P_{0} \times π {(r)}^{2}$

$= [P_{0} + (h_{1} + h) ρ g] \times π \times 3 r^{2} + P_{0} π r^{2}$

$In situation 1, at the verge of rising the block, F_{2} = F_{1} + W$

$[P_{0} + (h_{1} + h) ρ g] \times (π \times 3 r^{2}) + π r^{2} P_{0}$

$= [P_{0} + h_{1} ρ g] \times 4 π r^{2} + \frac{π \times 4 r^{2} h ρ g}{3}$ or, $h_{1} = \frac{5 h}{3}$

Q 10 :

A cylindrical tank has a hole of diameter $2 r$ in its bottom. The hole is covered by a wooden cylindrical block of diameter $4 r$ , height $h$ and density $ρ / 3$ .

[IMAGE 390]

Situation I: Initially, the tank is filled with water of density $ρ$ to a height such that the height of water above the top of the block is $h_{1}$ (measured from the top of the block).

Situation II: The water is removed from the tank to a height $h_{2}$ (measured from the bottom of the block), as shown in the figure. The height $h_{2}$ is smaller than $h$ (height of the block) and thus the block is exposed to the atmosphere.

Q. Find the height of the water level $h_{2}$ (in situation II), for which the block remains in its original position without the application of any external force. [2006]

$\frac{h}{3}$
$\frac{4 h}{9}$
$\frac{2 h}{3}$
$h$

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(2)

$Again considering equilibrium of wooden block.$

$Total downward force = total force upwards$

$Wt. of block + force due to atmospheric pressure = force due to pressure of liquid + Force due to atmospheric pressure$

$π (16 r^{2}) \frac{ρ}{3} \times g + P_{0} π \times 16 r^{2} = [h_{2} ρ g + P_{0}] π [(16 - 4) r^{2}] + P_{0} \times 4 r^{2}$

$where h_{2} = height of the water level in situation-2 for$ $which the block remains in its original position.$

$∴$ $h_{2} = \frac{4}{9} h$

Q 11 :

A cylindrical tank has a hole of diameter $2 r$ in its bottom. The hole is covered by a wooden cylindrical block of diameter $4 r$ , height $h$ and density $ρ / 3$ .

[IMAGE 391]

Situation I: Initially, the tank is filled with water of density $ρ$ to a height such that the height of water above the top of the block is $h_{1}$ (measured from the top of the block).

Situation II: The water is removed from the tank to a height $h_{2}$ (measured from the bottom of the block), as shown in the figure. The height $h_{2}$ is smaller than $h$ (height of the block) and thus the block is exposed to the atmosphere.

Q. In situation 2, if $h_{2}$ is further decreased, then [2006]

cylinder will not move up and remains at its original position
for $h_{2} = \frac{h}{3}$ , cylinder again starts moving up
for $h_{2} = \frac{h}{4}$ , cylinder again starts moving up
for $h_{2} = \frac{h}{5}$ , cylinder again starts moving up

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(1)

In situation - 2 when the height $h_{2}$ of water level is further decreased, then the upward force acting on the wooden block decreases. The total force downward remains the same. This difference will be compensated by the normal reaction by the tank wall on the wooden block. Hence the block does not moves up and remains at its original position.

Topic Question Set

Q 3 :

A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance $ℓ$ and $h$ are shown here. After some time the coin falls into the water. Then [2002]

[IMAGE 381]

Q 7 :

A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If $P (r)$ is the pressure at $r (r < R)$ , then the correct option(s) is(are) [2015]

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Topic Question Set

Q 3 : A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance ℓ and h are shown here. After some time the coin falls into the water. Then [2002] [IMAGE 381]

Q 7 : A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If P(r) is the pressure at r (r<R), then the correct option(s) is(are) [2015]

Want Us to Email you About Special Offers & Updates?

Q 3 :

A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance $ℓ$ and $h$ are shown here. After some time the coin falls into the water. Then [2002]

[IMAGE 381]

Q 7 :

A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If $P (r)$ is the pressure at $r (r < R)$ , then the correct option(s) is(are) [2015]