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 $= g$):

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

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 $\rho$):

$A$ tank with a square base of area $1.0 \; m^{2}$ is divided by a vertical partition in the middle. The bottom of the partition has a small hinged door of area $20 \; cm^{2}$. The tank is filled with water in one compartment and an acid (of relative density $1.7$) in the other,both to a height of $4.0 \; m$. Compute the force (in $N$) necessary to keep the door closed.

$Assertion :$ $A$ thin stainless steel needle can float on a still water surface.
$Reason :$ Any object floats when the buoyancy force balances the weight of the object.

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $\rho$. 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: