Objective Problems

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends $90°$angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical. Ratio $\frac{d_1}{d_2}$ is

On heating water, bubbles beings formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius $R$ and making a circular contact of radius $r$ with the bottom of the vessel. If $r<<R$ and the surface tension of water is $T$, value of $r$ just before bubbles detach is (density of water is $\rho$)

If a ball of steel (density $\rho =7.8 \mathrm{g} {\mathrm{cm}}^{-3}$) attains a terminal velocity of $10 \mathrm{cm} {\mathrm{s}}^{-1}$ when falling in a tank of water (coefficient of viscosity ${\eta }_{\mathrm{water} }=8.5\times {10}^{-4} \mathrm{Pa}-\mathrm{s}$) then its terminal velocity in glycerine $\left(\rho =1.2 \mathrm{g} {\mathrm{cm}}^{-3},\eta =13.2 \mathrm{Pa}-\mathrm{s}\right)$ would be nearly 

A large open tank has two holes in its wall. One is a square hole of side $a$ at a depth of $x$ from the top and the other is a circular hole of radius $r$ at a depth $4x$ from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then, $r$ is equal to 

An object weighs $m_1$ in a liquid of density $d_1$ and that in liquid of density $d_2$ is $m_2$. The density $d$ of the object is

A body floats in water with $40\%$ of its volume outside water. When the same body floats in an oil, $60\%$ of its volume remains outside oil. The relative density of oil is

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

Three liquids of equal masses are taken in three identical cubical vessels $A,B$ and $C$. Their densities are ${\rho }_{\mathrm{A}},{\rho }_{\mathrm{B}}$ and ${\rho }_{\mathrm{C}}$ respectively but ${\rho }_{\mathrm{A}}<{\rho }_{\mathrm{B}}<{\rho }_{\mathrm{C}}$. The force exerted by the liquid on the base of the cubical vessel is