The pressure at a point $10 \ m$ below the surface of water is (take atmospheric pressure $P_0 = 1.01 \times 10^5 \ Nm^{-2}$ and $g = 10 \ ms^{-2}$):

During a blood transfusion,a needle is inserted into a vein where the gauge pressure is $2000\,Pa$. At what height must the blood container be placed so that blood may just enter the vein (in $,m$)? (Density of blood $= 1.06 \times 10^3\,kg/m^3$ and take $g = 9.8\,m/s^2$).

$(a)$ Pressure decreases as one ascends the atmosphere. If the density of air is $\rho$,what is the change in pressure $dp$ over differential height $dh$?
$(b)$ Considering the pressure $P$ to be proportional to the density,find the pressure $P$ at a height $h$ if the pressure on the surface of the earth is $P_{0}$.
$(c)$ If $P_{0} = 1.03 \times 10^5 \text{ N/m}^2$,$\rho_0 = 1.29 \text{ kg/m}^3$,and $g = 9.8 \text{ m/s}^2$,at what height will the pressure drop to $\frac{1}{10}$ of the value at the surface of the earth?
$(d)$ This model of the atmosphere works for relatively small distances. Identify the underlying assumption that limits the model.

Why are two holes made to empty an oil tin?

$A$ tank is filled with a liquid of density $\rho$ up to a height $H$. The average pressure on the walls of the container is:

From the adjacent figure,the correct observation is: