Question

Three identical vessels are filled with equal masses of three different liquids A, B, and C $\left( {{\rho _A} \gg {\rho _B} \gg {\rho _C}} \right)$. The pressure at the base will be
A) Equal in all vessels
B) Maximum in vessel A
C) Maximum in vessel B
D) Maximum in vessel C

Answer 1

For this question we have to remember the phenomenon of hydrostatic paradox. If the liquid having lesser density will have more height in the vessel due to larger volume. The pressure at the base will be the same in all vessels.

Complete answer:
As in the question mentioned three identical vessels are filled with equal masses. So, all three vessels have equal mass ‘m’.
Volume of the liquid depends on density, where the liquid having lesser density will have more height in the vessel due to larger volume. We can simply write volume of liquid in mathematically $\dfrac{m}{{{\rho _A}}}$ ,$\dfrac{m}{{{\rho _B}}}$,$\dfrac{m}{{{\rho _C}}}$.
Here we use similar vessels has same base area, then the heights of liquid in the vessel are $\dfrac{m}{{a{\rho _A}}}$ ,$\dfrac{m}{{a{\rho _B}}}$ ,$\dfrac{m}{{a{\rho _C}}}$.
Hence the pressure at the base are $\dfrac{{{\rho _A}gm}}{{a{\rho _A}}}$, $\dfrac{{{\rho _B}gm}}{{a{\rho _B}}}$ ,$\dfrac{{{\rho _C}gm}}{{a{\rho _C}}}$.
Next we cancel the similar terms, Therefore we get $\dfrac{{gm}}{a},\dfrac{{gm}}{a},\dfrac{{gm}}{a}$.
The pressure at the base will be equal in all the three vessels.

Hence the option A is the correct answer.

Additional information:
Density can be defined as mass per unit volume of a material substance. It's measured in the unit of grams.
$\rho = \dfrac{M}{V}$ this is the formula of density where M is the mass and V is the volume. So, density is inversely proportional to volume of the given substance.
In these types of questions the quantity of liquid will be variable in all three vessels, it must be dependent on density. If density is high then quantity of liquid is less. so, we can say density of given liquid is inversely proportional to density of that liquid.

Note:
The pressure acting at bottom is balanced with the weight of the liquid which is filled with vessels. Force will also be the same in all the three vessels equal to their weight. This can be also considered as a form of Newton’s third law, i.e. every action has an equal and opposite reaction.