Buoyancy and Archimedes' Principle

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JEE Main 2020

Consider a solid sphere of radius $R$ and mass density $ρ (r) = ρ_{0} (1 - \frac{r ^{2}}{R ^{2}})$ , $0 < r \leq R$ . The min. density of a liquid in which it will float is

\frac{ρ _{0}}{3}

(B)

\frac{2 ρ _{0}}{5}

(C)

\frac{2 ρ _{0}}{3}

(D)

\frac{ρ _{0}}{5}

JEE Main 2020

LEVELJEE Main

A leak proof cylinder of length $1 m$ , made of a metal which has very low coefficient of expansion is floating vertically in water at $0^{\circ} C$ such that its height above the water surface is $20 cm$ . When the temperature of water is increased to $4^{\circ} C$ , the height of the cylinder above the water surface becomes $21 cm$ . The density of water at $T = 4^{\circ} C$ , relative to the density at $T = 0^{\circ} C$ is close to

A jar is filled with two non-mixing liquids 1 and 2 having densities $ρ_{1}$ and $ρ_{2}$ , respectively. A solid ball, made of a material of density $ρ_{3}$ , is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for $ρ_{1}, ρ_{2}$ and $ρ_{3}$ ?

(A)

ρ_{3} < ρ_{1} < ρ_{2}

(B)

ρ_{1} > ρ_{3} > ρ_{2}

(C)

ρ_{1} > ρ_{2} > ρ_{3}

(D)

ρ_{1} < ρ_{3} < ρ_{2}

JEE Main 2020

LEVELJEE Main

An air bubble of radius $1 cm$ in water has an upward acceleration $9.8 cm s^{- 2}$ . The density of water is $1 g/cm^{3}$ and water offers negligible drag force on the bubble. The mass of the bubble is (Take, $g = 980 cm/s^{2}$ )

A hollow spherical shell at outer radius $R$ floats just submerged under the water surface. The inner radius of the shell is $r$ . If the specific gravity of the shell material is $\frac{27}{8}$ with respect to water, value of $r$ is

(A)

\frac{8}{9} R

(B)

\frac{1}{3} R

(C)

\frac{2}{3} R

(D)

\frac{4}{9} R

JEE Main 2019

LEVELJEE Main

A cubical block of side $0.5 m$ floats on water with $30%$ of its volume under water. What is the maximum weight that can be put on the block without fully submerging it under water? [Take, density of water = $1 0^{3} kg / m^{3}$ ]

A ball is made of a material of density $ρ$ , where $ρ_{oil} < ρ < ρ_{water}$ with $ρ_{oil}$ and $ρ_{water}$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?

A wooden block floating in a bucket of water has $\frac{4}{5}$ of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in oil. The density of oil relative to that of water is

A barometer made of a very narrow tube (see figure) is placed at normal temperature and pressure. The coefficient of volume expansion of mercury is $0.00018 /^{\circ} C$ and that of the tube is negligible. The temperature of mercury in the barometer is now raised by $1^{\circ} C$ but the temperature of the atmosphere does not change. Then, the mercury height in the tube remains unchanged.

(A)

w_{1} > w_{2}

(B)

w_{1} = w_{2}

(C)

w_{1} < w_{2}

A homogeneous solid cylinder of length $L$ and cross-sectional area $A /5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L /4$ in the denser liquid as shown in the figure. The lower density liquid is open to atmosphere having pressure $p_{0}$ . Then, density $D$ of solid is given by

(A)

\frac{5}{4} d

(B)

\frac{4}{5} d

(C)

4 d

(D)

\frac{d}{5}

JEE Advanced 1979

LEVELJEE Main

A boat floating in a water tank is carrying a number of large stones. If the stones are unloaded into water, what will happen to the water level?

JEE Advanced 1988

LEVELJEE Main

A vessel contains oil (density $= 0.8 g/cm^{3}$ ) over mercury (density $= 13.6 g/cm^{3}$ ). A homogeneous sphere floats with half its volume immersed in mercury and the other half in oil. The density of the material of the sphere in $g/cm^{3}$ is

A body floats in a liquid contained in a beaker. The whole system as shown in figure falls freely under gravity. The upthrust on the body is

(A)

zero

(B)

equal to the weight of liquid displaced

(C)

equal to the weight of the body in air

(D)

equal to the weight of the immersed portion of the body

JEE Advanced 1992

LEVELJEE Advanced

A ball of density $d$ is dropped on to a horizontal solid surface. It bounces elastically from the surface and returns to its original position in a time $t_{1}$ . Next, the ball is released and it falls through the same height before striking the surface of a liquid of density $d_{L}$ . (a) If $d < d_{L}$ , obtain an expression (in terms of $d$ , $t_{1}$ and $d_{L}$ ) for the time $t_{2}$ the ball takes to come back to the position from which it was released. (b) Is the motion of the ball simple harmonic? (c) If $d = d_{L}$ , how does the speed of the ball depend on its depth inside the liquid? Neglect all frictional and other dissipative forces. Assume the depth of the liquid to be large.

JEE Advanced 1995

LEVELJEE Advanced

A container of large uniform cross-sectional area $A$ resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities $d$ and $2 d$ , each of height $H /2$ as shown in figure. The lower density liquid is open to the atmosphere having pressure $p_{0}$ . (a) A homogeneous solid cylinder of length $L$ ( $L < H /2$ ), cross-sectional area $A /5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L /4$ in the denser liquid. Determine (i) the density $D$ of the solid, (ii) the total pressure at the bottom of the container. (b) The cylinder is now removed and the original arrangement is restored. A tiny hole of area $s$ ( $s ≪ A$ ) is punched on the vertical side of the container at a height $h$ ( $h < H /2$ ). Determine: (i) the initial speed of efflux of the liquid at the hole, (ii) the horizontal distance $x$ travelled by the liquid initially, and (iii) the height $h_{m}$ at which the hole should be punched so that the liquid travels the maximum distance $x_{m}$ initially. Also calculate $x_{m}$ . (Neglect the air resistance in these calculations)

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

A small spherical monoatomic ideal gas bubble (

γ = 5/3

) is trapped inside a liquid of density

ρ_{l}

(see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains

n

moles of gas. The temperature of the gas when the bubble is at the bottom is

T_{0}

, the height of the liquid is

H

and the atmospheric pressure is

p_{0}

(Neglect surface tension)

Question 1:

As the bubble moves upwards, besides the buoyancy force the following forces are acting on it.

(A)

Only the force of gravity

(B)

The force due to gravity and the force due to the pressure of the liquid

(C)

The force due to gravity, the force due to the pressure of the liquid and the force due to viscosity of the liquid

(D)

The force due to gravity and the force due to viscosity of the liquid.

Question 2:

When the gas bubble is at a height y from the bottom, its temperature is

(A)

T_{0} [\frac{p _{0} + r h o _{l} g H}{p _{0} + r h o _{l} g y}]^{(} 2/5)

(B)

T_{0} [\frac{p _{0} + r h o _{l} g ( H - y )}{p _{0} + r h o _{l} g H}]^{(} 2/5)

(C)

T_{0} [\frac{p _{0} + r h o _{l} g H}{p _{0} + r h o _{l} g y ) ] ^{(} 3/5}

(D)

T_{0} [\frac{p _{0} + r h o _{l} g ( H - y )}{p _{0} + r h o _{l} g H}]^{(} 3/5)

Question 3:

The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant)

(A)

r h o_{l} n R g T_{0} \frac{( p _{0} + r h o _{l} g H ) ^{(} 2/5 )}{( p _{0} + r h o _{l} g y ) ^{(} 2/5 )}

(B)

\frac{r h o _{l} n R g T _{0}}{[( p _{0} + r h o _{l} g H ) ^{(} 2/5 ) [ p _{0} + r h o _{l} g ( H - y ) ] ^{(} 3/5 )}

(C)

r h o_{l} n R g T_{0} \frac{( p _{0} + r h o _{l} g H ) ^{(} 3/5 )}{( p _{0} + r h o _{l} g y ) ^{(} 8/5 )}

(D)

\frac{r h o _{l} n R g T _{0}}{[( p _{0} + r h o _{l} g H ) ^{(} 3/5 ) [ p _{0} + r h o _{l} g ( H - y ) ] ^{(} 2/5 )]}

JEE Advanced (2002)

LEVELJEE Advanced

A uniform solid cylinder of density $0.8 g/cm^{3}$ floats in equilibrium in a combination of two non-mixing liquids A and B with its axis vertical. The densities of the liquids A and B are $0.7 g/cm^{3}$ and $1.2 g/cm^{3}$ , respectively. The height of liquid A is $h_{A} = 1.2 cm$ . The length of the part of the cylinder immersed in liquid B is $h_{B} = 0.8 cm$ . (a) Find the total force exerted by liquid A on the cylinder. (b) Find $h$ , the length of the part of the cylinder in air. (c) The cylinder is depressed in such a way that its top surface is just below the upper surface of liquid A and is then released. Find the acceleration of the cylinder immediately after it is released.

JEE Advanced (2013)

LEVELJEE Main

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

(A)

\frac{M g}{k}

(B)

\frac{M g}{k} (1 - \frac{L A σ}{M})

(C)

\frac{M g}{k} (1 - \frac{L A σ}{2 M})

(D)

\frac{M g}{k} (1 + \frac{L A σ}{M})

JEE Advanced (2012)

LEVELJEE Advanced

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $ρ_{c}$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

(A)

more than half-filled if

ρ_{c}

is less than 0.5

(B)

more than half-filled if

ρ_{c}

is more than 1.0

(C)

half-filled if

ρ_{c}

is more than 0.5

(D)

less than half-filled if

ρ_{c}

is less than 0.5

JEE Advanced 2001

LEVELJEE Advanced

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 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

(A)

M g

(B)

M g - V ρ g

(C)

M g + π R^{2} h ρ g

(D)

$\rho g (V + \pi R^2 h)

JEE Advanced (1985)

LEVELJEE Main

The spring balance $A$ reads $2 kg$ with a block $m$ suspended from it. A balance $B$ reads $5 kg$ when a beaker with liquid is put on the pan of the balance. The two balances are now so arranged that the hanging mass is inside the liquid in the beaker as shown in the figure. In this situation

* Multiple Correct Options

(A)

the balance

A

will read more than

2 kg

(B)

the balance

B

will read more than

5 kg

(C)

the balance

A

will read less than

2 kg

and

B

will read more than

5 kg

(D)

the balances

A

and

B

will read

2 kg

and

5 kg

respectively

JEE Advanced (2002)

LEVELJEE Advanced

A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance $l$ and $h$ are shown there. After sometime the coin falls into the water. Then

(A)

l

decreases and

h

increases

(B)

l

increases and

h

decreases

(C)

Both

l

and

h

increase

(D)

Both

l

and

h

decrease

JEE Advanced (1984)

LEVELJEE Advanced

A wooden plank of length $1 m$ and uniform cross-section is hinged at one end to the bottom of a tank as shown in figure. The tank is filled with water upto a height $0.5 m$ . The specific gravity of the plank is $0.5$ . Find the angle $θ$ that the plank makes with the vertical in the equilibrium position (exclude the case $θ = 0^{\circ}$ ).

JEE Advanced 2020

LEVELJEE Main

A hot air balloon is carrying some passengers, and a few sandbags of mass 1 kg each so that its total mass is 480 kg. Its effective volume giving the balloon its buoyancy is V. The balloon is floating at an equilibrium height of 100 m. When N number of sandbags are thrown out, the balloon rises to a new equilibrium height close to 150 m with its volume V remaining unchanged. If the variation of the density of air with height h from the ground is $ρ (h) = ρ_{0} e^{- \frac{h}{h _{0}}}$ , where $ρ_{0} = 1.25 kg m^{- 3}$ and $h_{0} = 6000 m$ , the value of N is _________.

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

A soft plastic bottle, filled with water of density

1 gm/cc

, carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of

5 gm

, and it is made of a thick glass of density

2.5 gm/cc

. Initially the bottle is sealed at atmospheric pressure

p_{0} = 1 0^{5} Pa

so that the volume of the trapped air is

v_{0} = 3.3 cc

. When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure

P_{0} + Δ p

without changing its orientation. At this pressure, the volume of the trapped air is

v_{0} - Δ v

. Let

Δ v = X cc

and

Δ p = Y \times 1 0^{3} Pa

Question 1:

The value of $X$ is _______.

Question 2:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider a solid sphere of radius R and mass density ρ(r)=ρ0​(1−R2r2​), 0<r≤R. The min. density of a liquid in which it will float is

A boat floating in a water tank is carrying a number of large stones. If the stones are unloaded into water, what will happen to the water level?

A body floats in a liquid contained in a beaker. The whole system as shown in figure falls freely under gravity. The upthrust on the body is

Comprehension Passage

As the bubble moves upwards, besides the buoyancy force the following forces are acting on it.

When the gas bubble is at a height y from the bottom, its temperature is

The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance l and h are shown there. After sometime the coin falls into the water. Then

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of X is _______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of Y is _______.

Consider a solid sphere of radius $R$ and mass density $ρ (r) = ρ_{0} (1 - \frac{r ^{2}}{R ^{2}})$ , $0 < r \leq R$ . The min. density of a liquid in which it will float is

A wooden block, with a coin placed on its top, floats in water as shown in figure. The distance $l$ and $h$ are shown there. After sometime the coin falls into the water. Then

The value of $X$ is _______.

The value of $Y$ is _______.