$A$ submarine experiences a pressure of $5.05 \times 10^6 \, Pa$ at a depth of $d_1$ in a sea. When it goes further to a depth of $d_2,$ it experiences a pressure of $8.08 \times 10^6 \, Pa.$ Then $d_2 - d_1$ is approximately ........ $m$ (density of water $= 10^3 \, kg/m^3$ and acceleration due to gravity $= 10 \, m/s^2$).

Why are two holes made to empty an oil tin?

$(a)$ 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.

From the adjacent figure,the correct observation is:

$A$ manometer reads the pressure of a gas in an enclosure as shown in Figure $(a)$. When a pump removes some of the gas,the manometer reads as in Figure $(b)$. The liquid used in the manometers is mercury and the atmospheric pressure is $76 \; cm$ of mercury.
$(a)$ Give the absolute and gauge pressure of the gas in the enclosure for cases $(a)$ and $(b)$,in units of $cm$ of mercury.
$(b)$ How would the levels change in case $(b)$ if $13.6 \; cm$ of water (immiscible with mercury) are poured into the right limb of the manometer? (Ignore the small change in the volume of the gas).

The pressure at half the depth of a lake is $2/3$ of the pressure at its bottom. What is the depth of the lake in $m$?