State whether the following statements are True or False:
$(i)$ If the weight of a body is greater than the buoyant force of the liquid,the body floats on the surface of the liquid.
$(ii)$ The velocity of water at the bottom of a river is always slower than at the surface.
$(iii)$ The angle of contact increases with an increase in the temperature of the liquid.

$(a)$ It is known that the density $\rho$ of air decreases with height $y$ as $\rho = \rho_{0} e^{-y / y_{0}}$,where $\rho_{0} = 1.25 \; kg \, m^{-3}$ is the density at sea level,and $y_{0}$ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of the atmosphere remains constant (isothermal conditions). Also,assume that the value of $g$ remains constant.
$(b)$ $A$ large $He$ balloon of volume $1425 \; m^{3}$ is used to lift a payload of $400 \; kg$. Assume that the balloon maintains a constant radius as it rises. How high does it rise?
[Take $y_{0} = 8000 \; m$ and $\rho_{He} = 0.18 \; kg \, m^{-3}$]

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 $g = 10 \ m/s^2$):

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

The rate of diffusion is

When an air bubble of radius $r$ rises from the bottom to the surface of a lake,its radius becomes $5r/4$ (the pressure of the atmosphere is equal to the $10 \, m$ height of water column). If the temperature is constant and the surface tension is neglected,the depth of the lake is .... $m$ (in $.53$)