$A$ liquid is kept in a cylindrical vessel which is rotated along its axis. The liquid rises at the sides. If the radius of the vessel is $0.05\,m$ and the speed of rotation is $2\,rev/s$,the difference in the height of the liquid at the center of the vessel and its sides will be .............. $cm$ $(\pi^2 = 10)$.

Drops of liquid of density $d$ are floating half immersed in a liquid of density $\rho$. If the surface tension of the liquid is $T$, then the radius of the drop is

The volume of an air bubble becomes three times as it rises from the bottom of a lake to its surface. Assuming atmospheric pressure to be $75 \ cm$ of $Hg$ and the density of water to be $1/10$ of the density of mercury,the depth of the lake is ....... $m$.

Two solid spheres $A$ and $B$ of equal volumes but of different densities $d_A$ and $d_B$ are connected by a string. They are fully immersed in a fluid of density $d_F$. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if:
$(A)$ $d_A < d_F$
$(B)$ $d_B > d_F$
$(C)$ $d_A + d_B = 2d_F$
$(D)$ $d_A > d_F$

The vessel shown in the figure has two sections. The lower part is a rectangular vessel with area of cross-section $A$ and height $h$. The upper part is a conical vessel of height $h$ with base area $A$ and top area $a$,and the walls of the vessel are inclined at an angle $30^o$ with the vertical. $A$ liquid of density $\rho$ fills both the sections up to a height $2h$. Neglecting atmospheric pressure:

$A$ motor is used to deliver water at a certain rate through a given horizontal pipe. To deliver $n$-times the water through the same pipe in the same time, the power of the motor must be increased as follows: