An open cubical tank was initially fully filled with water. When the tank was accelerated on a horizontal plane along one of its sides,it was found that one-third of the volume of water spilled out. The acceleration was:

$A$ liquid is taken in a long vertical cylindrical vessel and the cylinder is rotated about its vertical axis as shown in the figure. During rotation, the liquid rises along its sides. If the radius of the vessel is $0.05 \,m$ and the speed of rotation is $10 \,rad \,s^{-1}$, then the height difference between the liquid at the centre of the vessel and its sides is $\left(g=10 \,m \,s^{-2}\right)$.

The cross-sectional area of a horizontal tube increases along its length linearly as we move in the direction of flow. The variation of pressure, as we move along its length in the direction of flow ($x$-direction), is best depicted by which of the following graphs?

Two identical cylindrical vessels with their bases at the same level, each contains a liquid of density $d$. The height of the liquid in one vessel is $h_1$ and that in the other vessel is $h_2$. The area of either base is $A$. The work done by gravity in equalizing the levels when the two vessels are connected is

$A$ 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:

$A$ 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 . . . . .$N \ m^{-2}$.