At shallow depth $h$,the pressure in the ocean is simply given by $P = P_0 + \rho gh$,in which $\rho$ is the density of water and $P_0$ is the air pressure. As we go deeper,the high pressure causes the water to compress and become denser. Which of the following sketches illustrates the correct dependence of the pressure on the depth $h$ ?

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:

Density of ice is $\rho$ and that of water is $\sigma$. What will be the decrease in volume when a mass $M$ of ice melts?

Two vessels have the same base area but different shapes. The first vessel takes twice the volume of water that the second vessel requires to fill up to a particular common height. Is the force exerted by the water on the base of the vessel the same in the two cases? If so,why do the vessels filled with water to that same height give different readings on a weighing scale?

Two immiscible liquids $A$ and $B$ are kept in an $U$-tube. If the density of liquid $A$ is smaller than the density of liquid $B$,then the equilibrium situation is

Explain why it is easier to cut an apple with a sharp knife than with a blunt knife.