Archimedes' Principle

Q. Two non-mixing liquids of densities $\rho$ and $n\rho (n>1)$ are put in a container. The height of each liquid is $h$. A solid cylinder of length $L$ and density $d$ is put in this container. The cylinder floats with its axis vertical and length $pL(p<1)$ in the denser liquid. The density $d$ is equal to:

1. $\left[1+(n+1)p\right]\rho$
2. $\left[2+(n-1)p\right]\rho$
3. $\left[1+(n-1)p\right]\rho$
4. $\left[2+(n+1)p\right]\rho$

Q. In the situation shown in figure, the block is in equilibrium and the vessel is placed in a lift. If the lift starts moving down with constant acceleration $a$, then at equilibrium

1. the buoyant force will decrease
2. the compression in the spring will be more
3. the compression in the spring will be less
4. the buoyant force will remain same

Q. A body floats in water with one-third of its volume above the surface of water. If it is placed in oil, it floats with half of its volume above the surface of the oil. The relative density of the oil is

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

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

1. $l$ decreases and $h$ increases
2. $l$ increases and $h$ decreases
3. Both $l$ and $h$ increase
4. Both $l$ and $h$ decrease

Q. A solid body floating in water has $\frac{1}{5}$th of its volume immersed in it. What fraction of its volume will be immersed if it floats in a liquid of specific gravity 1.2?

Q. A body floats with $1/3$ of its volume outside when immersed in water. The same body floats with $3/4$ of its volume inside when immersed in some other liquid. The density of the other liquid in $gc{m}^{-3}$ is (density of water $=1000kg/m^3$)

1. $\frac{8}{9}$
2. $\frac{9}{4}$
3. $\frac{8}{3}$
4. $\frac{4}{9}$

Q. Two bodies are in equilibrium when suspended in water from the arms of a balance. The mass of one body is 36 g and its density is 9 g / $c{m}^3$. If the mass of the other is 48 g, its density in g / $c{m}^3$ is

1. 4/3
2. 3/2
3. 3
4. 5

Q. Equal masses of three liquids A, B and C have temperatures ${10}^{\circ }C$, ${25}^{\circ }C$ and ${40}^{\circ }C$ respectively. If A and B are mixed, the mixture has a temperature of 15°C. If B and C are mixed the mixture has a temperature of ${30}^{\circ }C$. If A and C are mixed, the mixture will have a temperature of

1. 
2. 
3. 
4. 

Q. The relative strengths of gravitational, columb's and nuclear forces of protons are in the ratio (nearly)

1. $1:1:1$
2. $1:{10}^{36}:{10}^{38}$
3. ${10}^{38}:{10}^{36}:1$
4. $1:{10}^{30}:{10}^{34}$

Q. A square plate of 0.1 m side moves parallel to a second plate with a velocity of 0.1 m/s, both plates being immersed in water. If the viscous force is 0.002 N and the coefficient of viscosity is 0.01 poise, distance between the plates in m is

1. 0.1
2. 0.05
3. 0.005
4. 0.0005

Q. The weight of an object in a coal mine, at sea level, and at top of moutain are $W_1,W_2$ and $W_3$ respectively, then

1. $W_1<W_2>W_3$
2. $W_1=W_2=W_3$
3. $W_1<W_2<W_3$
4. $W_1>W_2>W_3$

Q.

A candle of diameter $d$ is floating on a liquid in a cylindrical container of diameter $D(D>>d)$ as shown in figure. If it is burning at the rate of $\frac{2cm}{hour}$ then the top of the candle will :

1. remain at the same height
2. fall at the rate of $1cm{hour}^{-1}$
3. fall at the rate of $2cmhou{r}^{-1}$
4. go up at the rate of $1cm{hour}^{-1}$

Q.

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

1. The net elongation of the spring is $\frac{\left(4\pi R^3\rho g\right)}{3k}$

2. The net elongation of the spring is $\frac{\left(8\pi R^3\rho g\right)}{3k}$

3. The light sphere is partially submerged.

4. The light sphere is completely submerged.

Q. A $700\text{g}$ solid cube having an edge of length $10\text{cm}$ floats in water. How much volume of the cube is outside the water? Density of water $=1{\text{g/cm}}^3.$

1. $700{\text{cm}}^3$
2. $300{\text{cm}}^3$
3. $1000{\text{cm}}^3$
4. $30{\text{cm}}^3$

Q. A cubical block of wood of edge $3\text{cm}$ floats in water. The lower surface of the cube just touches the free end of a vertical spring fixed at the bottom of the pot. Find the maximum weight that can be put on the block without wetting it.
(Take density of wood $=800{\text{kg/m}}^3$, spring constant of the spring $=50\text{N/m}$ and $g=10{\text{m/s}}^2$.)

1. $0.35\text{N}$
2. $0.6\text{N}$
3. $0.27\text{N}$
4. $0.78\text{N}$

Q. A small block of wood of specific gravity $0.5$ is submerged at a depth of $1.2\text{m}$ in a vessel filled with water. The vessel is accelerated upwards with an acceleration $a_0=\frac{g}{2}$. Time taken by the block to reach the surface, if it is released with zero initial velocity is $\frac{x}{10}s$. Then find the value of $x$ :

Q. A rod of length $6\text{m}$ has specific gravity $\rho (=25/36)$. One end of the rod is tied to a $5\text{m}$ long rope, which in turn is tied to the floor of a pool $10\text{m}$ deep, as shown. Find the length (in $\text{m}$) of the part of rod which is out of water.

Q. A solid sphere of radius R has a volume charge density $\rho ={\rho }_0{r}^3$ (where ${\rho }_0$ is a constant and r is the distance from center). At a distance x from its centre (for x < R), the electric field is directly proportional to:

1. $x^4$
2. $x$
3. $x^2$
4. $x^3$

Q. A cube made of material having a density of $900{\text{kg m}}^{-3}$ floats between water of density $1000{\text{kgm}}^{-3}$ and a liquid of density $700{\text{kgm}}^{-3}$ which is immiscible in water. What part of cube is inside the water?

1. 3/4
2. 3/7
3. 1/2
4. 2/3

Q. 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 some time the coin falls into the water. Then

1. $l$ decreases and $h$ increases
2. both $l$ and $h$ decreases
3. both $l$ and $h$ increases
4. $l$ increases and $h$ decreases

Q. A sphere of solid material of specific gravity $8$ has a concentric spherical cavity and just sinks in water. The ratio of radius of cavity to that of outer radius of the sphere must be

1. $\frac{7^{1/3}}{2}$
2. $\frac{5^{1/3}}{2}$
3. $\frac{9^{1/3}}{2}$
4. $\frac{3^{1/3}}{2}$

Q. A piece of wood of mass $6\text{kg}$ is floating in water with its $\frac{1}{3}$rd part inside the water. On this floating piece of wood, what maximum mass can be put such that the whole of the piece of wood gets drowned in the water?

1. $14\text{kg}$
2. $15\text{kg}$
3. $10\text{kg}$
4. $12\text{kg}$

Q.

A body of density $d_1$ is counterpoised by M g of weights of density $d_2$ in air of density d. Then the true mass of the body is

1. M

2. 

3. 

4. 

Q. A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it extends by $l_a$ and when the weight is immersed completely in water, the extension is reduced to $l_w$. Then the relative density of material of the weight is

1. $\frac{l_a}{l_w}$
2. $\frac{l_a}{l_a-l_w}$
3. $\frac{l_w}{l_a-l_w}$
4. $\frac{l_w}{l_a}$

Q.

A small spherical droplet of density $d$ is floating exactly half immersed in a liquid of density $\rho$ and surface tension $T$. The radius of the droplet is (take note that the surface tension applied an upward force on the droplet).

1. $r=\sqrt{\frac{2T}{\left(3\rho +d\right)g}}$

2. $r=\sqrt{\frac{T}{\left(\rho +d\right)g}}$

3. $r=\sqrt{\frac{T}{\left(\rho -d\right)g}}$

4. $r=\sqrt{\frac{3T}{\left(2\rho -d\right)g}}$

Q. The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and lengths are $3\%$ and $2\%$ respectively. The maximum $\%$ error in the measurement of density would be

1. $12\%$
2. $9\%$
3. $14\%$
4. $7\%$

Q. A raft of wood (density $600kg/m^3$) of mass $120kg$ floats in water. How much weight can be put on the raft to make it just sink?

1. $120kg$
2. $40kg$
3. $80kg$
4. $200kg$

Q. A uniform solid sphere of radius R is in equilibrium inside a liquid whose density varies with depth from free surface as $\rho ={\rho }_0(1+\frac{h}{h_0})$, where h is the depth from free surface. If the density of sphere is $\sigma ={\rho }_0(1+\frac{xd}{y{h}_0})$ then find (x + y).

Q. Iceberg floats in water with part of it submerged. What is the fraction of the volume of iceberg submerged if the density of ice is $p_i=0.917\frac{g}{c{m}^3}$?

Q. A cylinder of mass $m$ and density $\rho$ hanging from a string is lowered into a vessel of cross-sectional area $s$ containing a liquid of density $\sigma (<\rho )$ until it is fully immersed. The increase in pressure at the bottom of the vessel is :

1. $\frac{m\rho g}{\sigma s}$
2. $\frac{mg}{s}$
3. $\frac{m\sigma g}{\rho s}$
4. Zero