Floating

Q. The velocity of a small ball of mass $M$ and density $\rho$, when dropped in a container filled with glycerine, becomes constant after some time. If the density of glycerine is $\frac{\rho }{2},$ then the viscous force acting on the ball will be

1. $2Mg$
2. $\frac{Mg}{2}$
3. $Mg$
4. $\frac{3}{2}Mg$

Q. A vessel contains oil (density $0.8{\text{g/cm}}^2$) over mercury (density $13.6{\text{g/cm}}^3$). 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 ${\text{g/cm}}^3$ is

1. $3.3$
2. $6.4$
3. $7.2$
4. $12.8$

Q. A block of wood floats in water $\left(4/5\right)th$ of its volume submerged. If the same block just floats in a liquid, the density of liquid in ${\text{kg/m}}^3$ is
(Take density of water as $1000{\text{kg/m}}^3$)

1. $900{\text{kg/m}}^3$
2. $400{\text{kg/m}}^3$
3. $1250{\text{kg/m}}^3$
4. $800{\text{kg/m}}^3$

Q. A raft of wood $(\rho =600{\text{kg/m}}^3)$ of mass $120\text{kg}$ floats on water. The mass which can be put on the raft to make it just sink is

1. $100\text{kg}$
2. $80\text{kg}$
3. $20\text{kg}$
4. $40\text{kg}$

Q. A wooden cube just floats inside water with a $200\text{gm}$ mass placed on it. When the mass is removed, the cube floats with its top surface $2\text{cm}$ above the water level. What is the side of the cube? [Take $g=10{\text{m/s}}^2$]

1. $8\text{cm}$
2. $10\text{cm}$
3. $12\text{cm}$
4. $6\text{cm}$

Q. A concrete sphere of radius $R$ has a spherical cavity of radius $r$ which is packed with sawdust. The specific gravities of concrete and sawdust are respectively $2.4$ and $0.3$. For this sphere to float with its entire volume submerged under water, ratio of mass of concrete to the mass of sawdust will be

1. $8$
2. $4$
3. $3$
4. $1$

Q. An ice berg of density 900 Kg/$m^3$ is floating in water of density 1000 Kg/$m^3$. The percentage of volume of ice-cube outside the water is

1. 20%
2. 35%
3. 10%
4. 25%

Q.

A ferry boat has internal volume 1 $m^3$ and weight 50 kg.

(a) Neglecting the thickness of the wood, find the fraction of the volume of the boat immersed in water.

(b) If a leak develops in the bottom and water starts coming in, what fraction of the boat's volume will be filled with water before water starts coming in from the sides ?

Q. A spherical solid ball of volume $V$ is made of a material of density ${\rho }_1$. It is falling through a liquid of density ${\rho }_1({\rho }_2<{\rho }_1)$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F_{viscous}=-k{v}^2(k>0)$. The terminal speed of the ball is

1. $\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$
2. $\frac{Vg{\rho }_1}{k}$
3. $\sqrt{\frac{Vg{\rho }_1}{k}}$
4. $\frac{Vg({\rho }_1-{\rho }_2)}{k}$

Q.

What will be the fraction of volume of a floating object of volume V and density d above the surface of a liquid of density D?

Q. A hemispherical bowl just floats without sinking in a liquid of density $1.2\times {10}^3{\text{kg/m}}^3.$ If the outer diameter and the density of the bowl are $1\text{m}$ and $2\times {10}^4{\text{kg/m}}^3$ respectively, then the inner diameter of the bowl will be

1. $0.92\text{m}$
2. $0.94\text{m}$
3. $0.98\text{m}$
4. $0.96\text{m}$

Q. A solid uniform ball of volume $V$ floats on the interface of two immiscible liquids (see the figure). The specific gravity of the upper liquid is ${\rho }_1$ and that of lower one is ${\rho }_2$ and the specific gravity of ball is $\rho ({\rho }_1<\rho <{\rho }_2)$. The fraction of the volume of the ball in the upper liquid is

1. $\frac{{\rho }_2}{{\rho }_1}$
2. $\frac{\rho -{\rho }_2}{{\rho }_1-{\rho }_2}$
3. $\frac{\rho -{\rho }_1}{{\rho }_2-{\rho }_1}$
4. $\frac{\rho }{{\rho }_2}$

Q. A wooden ball of density $\sigma$ is released from the bottom of a tank which is filled with a liquid of density $\rho (\rho >\sigma )$ up to a height $h_1$. The ball rises in the liquid , emerges from its surface and attains a height $h_2$ in air. If viscous effects are neglected, the ratio $\frac{h_2}{h_1}$ is

1. $(\frac{\rho }{\sigma }+1)$
2. $(\frac{\rho }{\sigma }-1)$
3. $\frac{\rho }{\sigma }$
4. $\frac{\sigma }{\rho }$

Q.

An object of weight W and density $\rho$ is dipped in a fluid of density ${\rho }_1$. Its apparent weight will be

1. $W(\rho -{\rho }_1)$

2. $W(1-\frac{{\rho }_1}{\rho })$

3. $\left(\frac{\rho -{\rho }_1}{W}\right)$

4. $W({\rho }_1-\rho )$

Q. Two cubical blocks identical in dimensions float in water in such a way that the first block floats with half part immersed in water and the second block floats with $3/4\text{th}$ of its volume immersed inside the water. The ratio of density of first block to that of second block is

1. $1:3$
2. $1:4$
3. $2:3$
4. $3:4$

Q. A metallic sphere floats in an immiscible mixture of water and a liquid such that its $\frac{4}{5}\text{th}$ volume is in water and $\frac{1}{5}\text{th}$ volume is in the liquid. Then, density of the metal is

1. $3.5\times {10}^3{\text{kg/m}}^3$
2. $1.5\times {10}^3{\text{kg/m}}^3$
3. $4\times {10}^3{\text{kg/m}}^3$
4. $2\times {10}^3{\text{kg/m}}^3$

Q. A sphere of radius $R$ made up of material of relative density $\sigma$ has a concentric spherical cavity of radius $r$. It just floats when placed in a tank full of water. The value of the ratio $\frac{R}{r}$ will be

1. ${\left(\frac{\sigma }{\sigma -1}\right)}^{1/3}$
2. ${\left(\frac{\sigma -1}{\sigma }\right)}^{1/3}$
3. ${\left(\frac{\sigma +1}{\sigma -1}\right)}^{1/3}$
4. ${\left(\frac{\sigma -1}{\sigma +1}\right)}^{1/3}$

Q. A cube of ice floats partly in water and partly in kerosene oil. Specific gravity of kerosene oil and that of ice is $0.8$ and $0.9$ respectively. Ratio of the volume of ice immersed in water to that in kerosene oil is

1. $2:1$
2. $3:1$
3. $1:2$
4. $1:1$

Q. A cubical block of ice floating in water has to support a metal piece of mass $0.5\text{kg}$. If specific gravity of ice is $0.9$, then minimum edge length (in $\text{m}$) of the block so that it does not sink in water is

1. ${\left(\frac{1}{200}\right)}^{\frac{1}{3}}\text{m}$
2. ${\left(\frac{1}{150}\right)}^{\frac{1}{3}}\text{m}$
3. ${\left(\frac{1}{100}\right)}^{\frac{1}{3}}\text{m}$
4. ${\left(\frac{1}{250}\right)}^{\frac{1}{3}}\text{m}$

Q. Two cylinders of same cross-section and length $L$ but made of two different materials of densities $d_1$ and $d_2$ are cemented together to form a cylinder of length $2L$. The combination floats in a liquid of density $d$ with a length $\frac{L}{2}$ above the surface of the liquid. If $d_1<d_2$ then

1. $d_1<\frac{3}{4}d$
2. $\frac{d}{2}>d_1$
3. $\frac{d}{4}>d_1$
4. $d<d_1$

Q. A rectangular block with dimensions $10\text{cm}\times 10\text{cm}\times 15\text{cm}$ is floating in a water container with its $10\text{cm}$ side in vertical position. If it floats with its $15\text{cm}$ side in vertical position, then the level of water in the container will

1. rise
2. fall
3. changes according to the density of block
4. remain constant

Q. A jar is filled with two non - mixing liquid 1 and 2 having densities ${\rho }_1$ and ${\rho }_2$ respectively. A solid ball, made of a material of density ${\rho }_3$, is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for ${\rho }_1,{\rho }_2$ and ${\rho }_3$?

1. ${\rho }_1>{\rho }_2>{\rho }_3$
2. ${\rho }_3<{\rho }_1<{\rho }_2$
3. ${\rho }_1<{\rho }_3<{\rho }_2$
4. ${\rho }_1>{\rho }_3>{\rho }_2$

Q.

A piece of stone displaces 25 ml of water. Its volume is……

Q.

A wooden cube of side 10 cm and specific gravity 0.8 floats in water with its upper surface horizontal. What depth of the cube remains immersed? what mass of aluminum of specific gravity 2.7 must be attached to

(i)The upper surface

(ii) The lower surface so that the cube will be just immersed

Q. A ball is made of a material of density $\rho$ where, ${\rho }_{\text{oil}}<\rho <{\rho }_{\text{water}}$ with ${\rho }_{\text{oil}}$ and ${\rho }_{\text{water}}$ 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 positions?

1. 
2. 
3. 
4. 

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

1. Zero
2. Equal to the weight of the liquid displaced
3. Equal to the weight of the body in air.
4. Equal to the weight of the immerssed portion of the body

Q.

Solve the previous problem if the lead piece is fastened on the top surface of the block and the block is to float with its upper surface just dipping into water.

Q. An iceberg is floating in an ocean. What fraction of its volume is above the water? (Given: density of ice$=900{\text{kg/m}}^3$ and density of ocean water $=1030{\text{kg/m}}^3$)

1. $\frac{90}{103}$
2. $\frac{13}{103}$
3. $\frac{10}{103}$
4. $\frac{1}{103}$

Q. A block of wood floats in water with ${\frac{4}{5}}^{\text{th}}$ of its volume submerged but in another liquid, it floats fully submerged. The density of the unknown liquid is (in ${\text{kg/m}}^3$)

1. $750$
2. $800$
3. $1000$
4. $1250$

Q. A wooden cube just floats inside water with a $400\text{gm}$ mass placed on it. When the mass is removed, the cube floats with its top surface $4\text{cm}$ above the water level. What is the side of the cube?

1. $8\text{cm}$
2. $10\text{cm}$
3. $6\text{cm}$
4. $12\text{cm}$