$A$ swimming pool has a depth of $3 \,m$. The pressure at the bottom of the pool due to water alone is: (Density of water is $1000 \,kg \,m^{-3}$, Acceleration due to gravity $= 10 \,m \,s^{-2}$)

The density of the liquid filled in the container is $900 \ kg/m^{3}$. Calculate the force exerted on the bottom of the container. (Given: $g = 10 \ m/s^{2}$) (in $N$)

$(a)$ 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.

$A$ $U$-shaped tube is partially filled with an incompressible liquid of density $1.2 \ g \ cm^{-3}$. Oil, which does not mix with the liquid, is poured into the left side of the $U$-tube until the liquid rises by $15 \ cm$ on the right side. If the density of the oil is $0.9 \ g \ cm^{-3}$, the oil level will stand higher than the liquid level on the right side of the $U$-tube by: (in $cm$)

The atmospheric pressure and height of the barometer column are $10^5 \, Pa$ and $760 \, mm$ respectively on the Earth's surface. If the barometer is taken to the Moon,then the column height will be ........ $mm$.

As the depth increases in a vessel filled with liquid, the pressure ..........