$A$ tank $5 \ m$ high is half-filled with water and then is filled to the top with oil of density $0.85 \ g/cm^3$. The pressure at the bottom of the tank,due to these liquids is ....... $g \ dyne/cm^2$.

Two liquids of densities $\rho_{1}$ and $\rho_{2}$ (where $\rho_{2} = 2\rho_{1}$) are filled behind a vertical wall of height $10 \; m$ and width $w$,as shown in the figure. Each liquid has a height of $h = 5 \; m$. The ratio of the force exerted by the upper liquid on the part $MN$ to the force exerted by the lower liquid on the part $NO$ is (Assume that the liquids do not mix):

Explain why:
$(a)$ The blood pressure in humans is greater at the feet than at the brain.
$(b)$ Atmospheric pressure at a height of about $6 \; km$ decreases to nearly half of its value at sea level,though the height of the atmosphere is more than $100 \; km$.
$(c)$ Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.

$A$ hollow sphere of radius $R$ is filled completely with an ideal liquid of density $\rho$. The sphere is moving horizontally with an acceleration $2g,$ where $g$ is the acceleration due to gravity. If the minimum pressure of the liquid is $P_0$,then what is the pressure at the centre of the sphere?

An open-ended $U$-tube of uniform cross-sectional area contains water (density $1.0 \text{ g/cm}^3$) standing initially $20 \text{ cm}$ from the bottom in each arm. An immiscible liquid of density $4.0 \text{ g/cm}^3$ is added to one arm until a layer $5 \text{ cm}$ high forms,as shown in the figure. What is the ratio $h_2/h_1$ of the heights of the liquid in the two arms?

$A$ liquid mixture of volume $V$ has two liquids as its ingredients with densities $\alpha$ and $\beta$. If the density of the mixture is $\sigma$,then the mass of the first liquid in the mixture is ............