When an air bubble rises from the bottom of a lake to the surface,its radius is doubled. The atmospheric pressure is equal to that of a column of water of height $H$. The depth of the lake is:

State whether the following statements are True or False:
$(i)$ In incompressible flow,the density of a fluid changes with time and position.
$(ii)$ Non-viscous flow can be a streamline flow.
$(iii)$ In a steady flow,the velocity of the fluid at each point remains identical with time.

Two cylindrical vessels of equal cross-sectional area of $2 \ m^2$ contain water up to heights of $10 \ m$ and $6 \ m$,respectively. If the vessels are connected at their bottom,then the work done by the force of gravity is (Density of water is $10^3 \ kg/m^3$ and $g = 10 \ m/s^2$):

At a speed of .......... $m/s$,the velocity head of water is equal to the pressure head of $40\, cm$ of $Hg$.

$A$ cylindrical vessel of height $500 \ mm$ has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height $H$. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with the height of the water column being $200 \ mm$. Find the fall in height (in $mm$) of the water level due to the opening of the orifice.
[Take atmospheric pressure $= 1.0 \times 10^5 \ N/m^2$,density of water $= 1000 \ kg/m^3$ and $g = 10 \ m/s^2$. Neglect any effect of surface tension.]

Vapour is injected at a uniform rate into a closed vessel which was initially evacuated. The pressure in the vessel: