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 ....... $m$.

The Karman line is a theoretical construct that separates the Earth's atmosphere from outer space. It is defined as the height at which the lift on an aircraft flying at the speed of a polar satellite $(8 \, km/s)$ is equal to its weight. Taking a fighter aircraft of wing area $30 \, m^2$ and mass $7500 \, kg$,the height of the Karman line above the ground will be in the range of .............. $km$. (Assume the density of air at height $h$ above the ground to be $\rho(h) = 1.2 e^{-h/10} \, kg/m^3$,where $h$ is in $km$,and the lift force to be $\frac{1}{2} \rho v^2 A$,where $v$ is the speed of the aircraft and $A$ is its wing area.)

An air bubble of volume $1\,cm^3$ rises from the bottom of a lake $40\,m$ deep to the surface at a temperature of $12^{\circ}C$. The atmospheric pressure is $1 \times 10^5\,Pa$,the density of water is $1000\,kg/m^3$,and $g = 10\,m/s^2$. There is no difference in the temperature of water at the depth of $40\,m$ and on the surface. The volume of the air bubble when it reaches the surface will be $..........\,cm^3$. (in $,cm^3$)

$A$ balloon is made of a material of surface tension $S$ and its inflation outlet (from where gas is filled in it) has small area $A$. It is filled with a gas of density $\rho$ and takes a spherical shape of radius $R$. When the gas is allowed to flow freely out of it,its radius changes from $R$ to $0$ in time $T$. If the speed $\psi(r)$ of gas coming out of the balloon depends on $r$ as $r^\alpha$ and $T \propto S^a A^\beta \rho^\gamma R^\delta$,then:

The vertical limbs of a $U$ shaped tube are filled with a liquid of density $\rho$ up to a height $h$ on each side. The horizontal portion of the $U$ tube having length $2h$ contains a liquid of density $2\rho$. The $U$ tube is moved horizontally with an acceleration $g/2$ parallel to the horizontal arm. The difference in heights in liquid levels in the two vertical limbs,at steady state,will be:

$Assertion :$ $A$ 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.