Buoyant Force

Q.

According to which theorem/principle the sum of pressure head, velocity head and gravity head is constant:

Q. Statement 1: Consider an object that floats in water but sinks in oil. When the object floats in water with half submerged, if oil is slowly poured into the water till it completely covers the object, the object moves up.
Statement 2: As the oil is poured in the situation of Statement 1, pressure inside the water will increase everywhere resulting in an increase in upward force on the object.

1. Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement 1
2. Statement 1 is true, Statement 2 is true; Statement 2 is not correct explanation for Statement 1
3. Statement 1 is true, Statement 2 is false
4. Statement 1 is false, Statement 2 is true

Q. A sonometer wire is stretched by a hanging metal bob. Its fundamental frequency is $f_1$. When the bob is completely immersed in water, the frequency becomes $f_2$. The relative density of the metal is

1. $\frac{f_1^2}{f_1^2-f_2^2}$
2. $\frac{f_2^2}{f_1^2-f_2^2}$
3. $\frac{f_1}{f_1-f_2}$
4. $\frac{f_2}{f_1-f_2}$

Q. A cubical block is floating in a liquid with one fourth of its volume immersed in the liquid. If the whole of the system accelerates upwards with acceleration $g/4$, the fraction of volume immersed in the liquid will be

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

Q. A metal ball weighs $0.096\text{N}$ in air. If suspended in water, it has an apparent weight of $0.071\text{N}$. The density of the metal ball is

1. $3840{\text{kg/m}}^3$
2. $2040{\text{kg/m}}^3$
3. $2540{\text{kg/m}}^3$
4. $4838{\text{kg/m}}^3$

Q.

A cork is submerged in water by a spring attached to the bottom of a bowl. When the bowl is kept in an elevator moving with acceleration downwards, the length of spring

1. Increases

2. Decreases

3. Remains unchanged

4. None of these

Q. A beaker containing water is placed on the platform of a spring balance. Tha balance reads $1.5\text{kg}$. A stone of mass $0.5\text{kg}$ and density ${10}^4{\text{kg/m}}^3$ is floats in water without touching the walls of the beaker. What will be the balance reading now (in $\text{kg}$)?

Q. A wooden plank of length $1\text{m}$ and uniform cross-section is hinged at one end to the bottom of the tank. The tank is filled with water upto a height of $0.5\text{m}$. The specific gravity of the plank is $0.5$. The plank makes an angle of ${45}^{\circ }$ with the vertical in the equilibrium position. $A$ is the midpoint of the plank. $B$ is the middle point of the dipped part of the plank. Which of the following are correct?

1. The weight of the plank acts downwards at $A$
2. The buoyant force acts through $B$
3. $OB=\frac{1}{\sqrt{2}}\text{m}$
4. Both (b) and (c)

Q. Find the density of a metallic body which floats at the interface of mercury of specific gravity $13.6$ and water such that $31.75\%$ of its volume is submerged in mercury and $68.25\%$ in water.

1. $6000{\text{kg/m}}^3$
2. $5000{\text{kg/m}}^3$
3. $4000{\text{kg/m}}^3$
4. $7000{\text{kg/m}}^3$

Q.

A cubical block of steel of each side equal to $l$ is floating on mercury in a vessel. The density of steel is ${\rho }_s$ and that of mercury is ${\rho }_m$. The height of the block above the mercury level is given by

1. $l(1+\frac{{\rho }_s}{{\rho }_m})$

2. $l(1-\frac{{\rho }_s}{{\rho }_m})$

3. $l(1+\frac{{\rho }_m}{{\rho }_s})$

4. $l(1-\frac{{\rho }_m}{{\rho }_s})$

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

1. zero
2. $\frac{mg}{A}$
3. $\frac{mg\rho }{\sigma A}$
4. $\frac{m\sigma g}{\rho A}$

Q. Two non-mixing liquids of densities $1000{\text{kg/m}}^3$ and $2000{\text{kg/m}}^3$ having height of $1\text{m}$ each are taken in a container, as shown in the figure. When a solid cylinder of length $0.5\text{m}$ and density $\rho$ is inserted in this container, it is observed that the cylinder floats with its axis being vertical and a length $0.2\text{m}$ of it in the denser liquid. The density $\rho$ of the solid cyclinder will be equal to

1. $1500{\text{kg/m}}^3$
2. $1300{\text{kg/m}}^3$
3. $1600{\text{kg/m}}^3$
4. $1400{\text{kg/m}}^3$

Q. In a stable equilibrium for completely submerged bodies, what is the relation between forces? Given $F_B=\text{Buoyant force},W=\text{weight of body}$

1. $F_B=W$ and the centre of buoyancy is below the centre of gravity.
2. $F_B=W$ and the centre of buoyancy is above the centre of gravity.
3. $F_B<W$
4. None of the above mentioned statements are true.

Q. Ratio of the weights of a $1\text{kg}$ block of iron and $1\text{kg}$ block of wood as measured by a spring balance is:
Given: Density of iron $=7800{\text{kg/m}}^3$, density of wood $=800{\text{kg/m}}^3$ and density of air $=1.293{\text{kg/m}}^3$

1. $1.5$
2. $2.0015$
3. $1.0015$
4. $3.0015$

Q. If $M$ is the mass of water rises in a capillary tube of radius $r$, then what will be the mass of water which will rise in a capillary tube of same material having radius $2r$?

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

Q.

A cubical block of steel of each side equal to $l$ is floating on mercury in a vessel. The density of steel is ${\rho }_s$ and that of mercury is ${\rho }_m$. The height of the block above the mercury level is given by

1. $l(1+\frac{{\rho }_s}{{\rho }_m})$

2. $l(1-\frac{{\rho }_s}{{\rho }_m})$

3. $l(1+\frac{{\rho }_m}{{\rho }_s})$

4. $l(1-\frac{{\rho }_m}{{\rho }_s})$

Q. Ratio of the weights of a $1\text{kg}$ block of iron and $1\text{kg}$ block of wood as measured by a spring balance is:
Given: Density of iron $=7800{\text{kg/m}}^3$, density of wood $=800{\text{kg/m}}^3$ and density of air $=1.293{\text{kg/m}}^3$

1. $1.5$
2. $2.0015$
3. $1.0015$
4. $3.0015$

Q. Two solid bodies $X$ and $Y$ having same volume floats in a liquid. It is observed that $X$ floats with $50\%$ of its volume immersed in liquid and $Y$ floats with $25\%$ of its volume above the liquid level. The ratio of density of $X$ to $Y$ is

1. $3:2$
2. $2:3$
3. $2:5$
4. $5:2$

Q. The lower surface of the cube floating in water just touches the free end of a vertical spring fixed at the bottom of the vessel. The maximum weight that can be put on the block without wetting it is-

1. $0.20\text{N}$
2. $0.30\text{N}$
3. $0.35\text{N}$
4. $0.25\text{N}$

Q. If $M$ is the mass of water rises in a capillary tube of radius $r$, then what will be the mass of water which will rise in a capillary tube of same material having radius $2r$?

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

Q.

A fisherman hooks an old log of wood of weight 12N and volume 1000 $c{m}^3$. He pulls the log half way out of water. The tension in the string at this instant is

1. 12 N

2. 8 N

3. 10 N

4. 7 N

Q. Two non-mixing liquids of densities $1000{\text{kg/m}}^3$ and $2000{\text{kg/m}}^3$ having height of $1\text{m}$ each are taken in a container, as shown in the figure. When a solid cylinder of length $0.5\text{m}$ and density $\rho$ is inserted in this container, it is observed that the cylinder floats with its axis being vertical and a length $0.2\text{m}$ of it in the denser liquid. The density $\rho$ of the solid cyclinder will be equal to

1. $1500{\text{kg/m}}^3$
2. $1300{\text{kg/m}}^3$
3. $1600{\text{kg/m}}^3$
4. $1400{\text{kg/m}}^3$

Q. A $20\text{N}$ metal block is suspended by a spring balance. A beaker containing some water is placed on a weighing machine which reads $40\text{N}$. The spring balance is now lowered slowly into the beaker such that the block gets immersed in the water. The spring balance now reads $16\text{N}$. The reading of the weighing machine will be

1. $36\text{N}$
2. $60\text{N}$
3. $44\text{N}$
4. $56\text{N}$

Q. A solid sphere of radius $1\text{m}$ and density $1000{\text{kg/m}}^3$ is attached to one end of a massless spring of force constant $5000\text{N/m}$. The other end of the spring is connected to another solid sphere of radius $1\text{m}$ and density $3000{\text{kg/m}}^3$. The complete arrangement is placed in a liquid of density $2000{\text{kg/m}}^3$ and is allowed to reach equilibrium. The correct statement(s) is(are):

1. The net elongation of the spring is $\frac{8\pi }{3}\text{m}$
2. The net elongation of the spring is $\frac{4\pi }{3}\text{m}$
3. The light sphere is partially submerged
4. The light sphere is completely submerged

Q. A solid sphere of density $\eta (>1)$ times lighter than that of water is suspended in a water tank by a string tied to its base as shown in figure. If the mass of the sphere is $m$, then the tension in the string is given by

1. $\left(\frac{\eta -1}{\eta }\right)$
2. $\eta mg$
3. $\frac{mg}{\eta -1}$
4. $(\eta -1)mg$

Q. A cubical block is floating in a liquid with one fourth of its volume immersed in the liquid. If the whole of the system accelerates upwards with acceleration $g/4$, the fraction of volume immersed in the liquid will be

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

Q. A $20\text{N}$ metal block is suspended by a spring balance. A beaker containing some water is placed on a weighing machine which reads $40\text{N}$. The spring balance is now lowered slowly into the beaker such that the block gets immersed in the water. The spring balance now reads $16\text{N}$. The reading of the weighing machine will be

1. $36\text{N}$
2. $60\text{N}$
3. $44\text{N}$
4. $56\text{N}$

Q. A uniform rod of length $2.0\text{m}$, specific gravity $0.5$ and mass $2\text{kg}$ is hinged at one end to the bottom of a tank of water (specific gravity = $1.0$) filled upto a height of $1.0\text{m}$ as shown in the figure. Taking the case $\theta \ne 0^{\circ }$, the force exerted by the hinge on the rod is $(g=10{\text{m/s}}^2)$

1. $10.2\text{N}$ upwards
2. $4.2\text{N}$ downwards
3. $8.3\text{N}$ downwards
4. $6.2\text{N}$ upwards