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 ....... $m$.

On opposite sides of a wide vertical vessel filled with water (density $\rho$), two identical holes are drilled, each having cross-section area $A$. The height difference between holes is $x$. The resultant force of reaction of water flowing out of the vessel is ($g$ = acceleration due to gravity).

$A$ liquid kept in a cylindrical vessel is rotated about a vertical axis passing through the center of the circular base. The difference in the heights of the liquid at the center of the vessel and its edge is ($R=$ radius of vessel,$\omega=$ angular velocity of rotation,$g=$ acceleration due to gravity).

$A$ cylindrical furnace has height $(H)$ and diameter $(D)$ both $1 \ m$. It is maintained at temperature $T=360 \ K$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \ K$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \ m$ and height $h=9 \ m$ above the furnace and exits the chimney. As a result,atmospheric air of density $\rho_a=1.2 \ kg \ m^{-3}$,pressure $P_a$ and temperature $T_a=300 \ K$ enters the furnace. Assume air as an ideal gas,neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects. [Given: The acceleration due to gravity $g=10 \ ms^{-2}$ and $\pi=3.14$]
$(1)$ Considering the air flow to be streamline,the steady mass flow rate of air exiting the chimney is . . . . .$g \ s^{-1}$.
$(2)$ When the chimney is closed using a cap at the top,a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air,due to the stoppage of air flow,are negligible then the value of $\Delta P$ is . . . . .$N \ m^{-2}$.

$A$ liquid drop of density $\rho$ is floating half-immersed in a liquid of density $d$. If $T$ is the surface tension,then the diameter of the liquid drop is ($g$ = acceleration due to gravity).

$A$ liquid is kept in a closed vessel. If a glass plate (negligible mass) with a small hole is kept on top of the liquid surface,then the vapour pressure of the liquid in the vessel is