pressure at the bottom of container

Consider a point under water which is at a depth $h$ from the water surface. Consider the small horizontal surface $\delta A$ at that point. The external force in addition to that due to the atmospheric pressure on $\delta A$ is the weight of the column of water above $\delta A$ that is:

$\begin{aligned} \delta F & = \delta mg & \small \color{#3D99F6} \text{where }\delta m \text{ is the mass of the column of water.} \\ & = \rho h \delta A g & \small \color{#3D99F6} \text{where }\rho \text{ is the density of water.} \\\frac {\delta F}{\delta A} & = \rho gh & \small \color{#3D99F6} \text{As } \delta A \to dA \\ \implies P & = \dfrac {dF}{dA} = \rho gh & \small \color{#3D99F6} \text{where }\rho \text{ pressure at the point }h \text{ below the water pressure.} \end{aligned}$

Note that since $\rho$ and $g$ are constants, $P$ is only dependent of $h$ , the height. Since water is filled to the same height in the four containers, the pressure at the bottom of the four containers are the same.

Note also that when $\delta A \to dA$ , $dA$ can be facing any direction, the pressure is the same. Therefore, pressure at a point in a liquid is equal in all directions.

The total pressure at any depth of an open container is equal to the atmospheric pressure plus the unit weight of liquid multiplied by the height from the free surface. Since the height is the same and the kind of liquid is the same, the pressures at the bottom of the containers are also the same.

Note: The difference in pressure between any two points in a homogeneous fluid (constant unit weight) at rest varies directly as the difference in depth or elevation of the two points.

Since the pressure of the water is vertical, and the four containers are filled to the same height, then the pressure at the bottom of the four containers are the same.

there is a concept easy to be misunderstood.

what does the "pressure" mean here?

if it means "density of pressure"(which is written with the letter P), the answer should be the last one.

but if it means "a kind of force"(written with the letter F), the answer should be B and C, for their area of bottom is the smallest. the equation F=PS tells us when P is same, force F decreases with the decreasing of area S.

This effect us known as the hydrostatic paradox by virtue of which irrespective of the area of the cross section of the container in which the liquid is enclosed, the pressure exerted remains the same provided the volume of the liquid in each container remains the same

The pressure in bottom of container is depend on height.

It maintains this formula,

$\large pressure\propto h\quad \boxed{\text{when g and density of water is constant.}}$

From this formula we can see, pressure is not depend on the size of container besides height.

Because of being same height, the answer must be $\boxed{\color{#20A900}\text{The pressures at the bottom of the four containers are the same}}$

Pressure is defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed. In other words, we can think of pressure in this situation as the force applied by a column of water that extends over each square centimetre on the base of the container. If we think of it this way, it becomes clear that the pressure exerted over the base of each container is equal to the pressure being exerted by each "column" of water, and has nothing to do with the surface area of the base. Since the amount of pressure exerted by each column depends only on its height, and each container is filled to the same height with the same liquid, we can say that the pressures at the bottom of the four containers are the same.

Since pressure depends on three quantities height,density of liquid acceleration due to gravity in the formula P=hdg. H is constant in all four containers so P at the bottom will be same in all the cases.

Hydrostatic pressure depends only on the depth; thus, theoretically speaking, it should be it for gases too, due to the nature of bonds between molecules. In our study of thermodynamics though we neglect the difference of pressure in an ideal gas between two different heights. This hypotesis makes sense considering the very low density of gases.

Simply Pascal law says that pressure acting at all molecules in an horizontal plane is equal and ass all are in a same plane so pressure experienced by all of them in bottom will be same

P = u.g.H, u - density, g - gravity, H - liquid height All the liquids are the same, so the density are equal in each one of them. Gravity its equal too The liquid height it's the same.

So, the pressure in the bottom of them are equal.

Pressure is given by pgh. Since the height and density of the liquid column is same the pressure will be same in all the three cases.

Its called hydrostatic paradox in pascal law Where under ideal condition pressure in every of different shape of vessel is same as it is shown above.

We know, P=h rho g i.e., Pressure is the multiplication of height and density of the liquid, times acceleration due to gravity on that place. So, pressure does not depend on the area of the bottom, results the answer.

In Engineering, the pressure for a fluid at a specific height is calculated as $\rho$ gh. This means that the only variables that do play a role in the pressure at the bottom of the tank is density, gravitational acceleration and height . The shape will therefore not influence the pressure at the bottom of the tank.

The equation for pressure is the density of the liquid * gravity * height. They all have the same liquid and gravity and height, therefore, they have the same pressure