The diagram shows a cup of tea seen from above. The tea has been stirred and is now rotating without turbulence. $A$ graph showing the speed $v$ with which the liquid is crossing points at a distance $X$ from $O$ along a radius $XO$ would look like:

If the heart pushes $1 \ cc$ of blood in one second under pressure $20000 \ N/m^2$,the power of the heart is ....... $W$.

$A$ long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis,the liquid rises up near the wall. If the radius of the vessel is $5 \, cm$ and its rotational speed is $2$ rotations per second,then the difference in the heights between the centre and the sides,in $cm$,will be

An air bubble of volume $V_0$ is released by a fish at a depth $h$ in a lake. The bubble rises to the surface. Assume constant temperature and standard atmospheric pressure $P$ above the lake. The volume of the bubble just before touching the surface will be (density of water is $\rho$):

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

As shown in the figure,a liquid is at the same level in both arms of a $U$-tube of uniform cross-section when at rest. If the $U$-tube moves with an acceleration '$f$' towards the right,the difference between the liquid heights in the two arms of the $U$-tube will be (acceleration due to gravity $= g$):