Consider a vessel filled with a liquid up to height $H$. The bottom of the vessel lies in the $X-Y$ plane passing through the origin. The density of the liquid varies with the $Z$-axis as $\rho(z) = \rho_0 \left[ 2 - \left( \frac{z}{H} \right)^2 \right]$. If $P_1$ and $P_2$ are the pressures at the bottom surface and top surface of the liquid respectively, the magnitude of $(P_1 - P_2)$ is:

An open $U$-tube contains mercury. When $13.6 \,cm$ of water is poured into one of the arms of the tube,then the mercury rise in the other arm from its initial level is ....... $cm$

Two cylindrical vessels $A$ and $B$ of different areas of cross-section kept on the same horizontal plane are filled with water to the same height. If the volume of water in vessel $A$ is $3$ times the volume of water in vessel $B$,then the ratio of the pressures at the bottom of the vessels $A$ and $B$ is

Mountaineers suffer from nose bleeding (epistaxis) at higher altitudes. Explain.

$A$ vertical $U$-tube of uniform cross-section contains water in both the arms. $A$ $10 \, cm$ glycerine column $(R.D. = 1.2)$ is added to one of the limbs. The level difference between the two free surfaces in the two limbs will be ...... $cm$.

$A$ cubical block of wood, of length $10 \,cm$, floats at the interface between oil of density $800 \,kg/m^3$ and water. The lower surface of the block is $1.5 \,cm$ below the interface. If the depth of water is $10 \,cm$ below the interface and oil is up to $10 \,cm$ above the interface, then the difference in pressure at the lower and the upper face of the wooden block is:
(Assume density of water, $\rho_w = 1000 \,kg/m^3$ and acceleration due to gravity, $g = 10 \,m/s^2$) (in $\,Pa$)