# Is the vertical pressure force on the bottom of a liquid container equal the weight of the fluid above the surface area?

I had an exam question few days ago that asked about the vertical force on the bottom surface of containers shaped like the image attached. All of the containers have the same area on the bottom of them, and the same height of liquid.

Is the pressure force equal on all the containers, because F=pA , or does the last container have less pressure because the force is equal to the weight of the fluid above it?

The following sequence of diagrams show that the pressure, $$\rm P$$, at the bottom of your container $$1$$ is the same as the pressure at the bottom of your container $$2$$ (and is the same as the pressure at the bottom of your container $$3$$).

What you have neglected to consider are the walls of the container which exert forces on the liquid inside them equal to the forces that a liquid would exert if the container walls were not present.

Pressure at any given point points equally in all directions. What can we infer about the direction of the net force on the side walls of each container caused by fluid pressure?

If the fluid exerts a force with an upward component on both walls, and neither the walls nor the fluid is in motion, we know from Newton's Third Law that the walls exert a force with a downward component on the fluid, which must transmit this force as an increase in apparent weight to the bottom of the container. If the fluid exerts a force with a downward component on both walls, and neither the walls nor the fluid is in motion, we know that the walls exert a force with an upward component on the fluid, which must transmit this force as a reduction in apparent weight to the bottom of the container.

Argue from Newton's Third Law and your knowledge of the pressure at some small volume for which there is a continuous column of water directly above the sample volume to prove that the fluid pressure at all points with the same height difference from the highest point of a static fluid exposed to air must be equal.

Yes, it does, the pressure force acting on the bottoms of each container is the same.

It follows from the Navier-Stokes equation. If you write the Navier-Stokes equation in Cartesian (i.e., Descartes) coordinates and consider the projection on the vertical axis, you'll have that all the acceleration terms and viscous terms are zero and you're left with the following: $$0 = - \frac{\partial P}{\partial y} + \rho g$$ $$\partial P = \rho g \partial y$$ Integrate it and get: $$\Delta P = \rho g h$$ Here, $$h$$ is the height of the liquid which is the same for all the vessels. In order to find the force on the bottoms of the vessels, integrate the last equation over the surface of the bottoms and you get: $$F = \rho g h A$$ Here, $$A$$ is the surface of the bottom, which is the same for all the vessels. Thus, $$F$$ is the same for all the vessels.

It is a common misconception to say that hydrostatic pressure is equal to the pressure of the weight force. The weight force is equal to $$\rho g h A$$ only in the case of the cylindrical vessel. That's because $$\rho g h A = \rho g V = m g$$, where $$V$$ is the volume of fluid in the cylindrical vessel and $$mg$$ is the weight force of the liquid in the cylindrical vessel. For the rest of the vessels you have different formulas for the volume.

The only thing that's different between the vessels is the force acting on the sides. If you are welding a bucket like one of your vessels, then you have to calculate that force in order to figure out the proper welding procedure that will ensure integrity of the bucket.

P.S. If you were to integrate the hydrostatic pressure over the entire area which is in contact with the fluid - and not just over the area of the bottom - you will get different forces for each bucket. Integrating over the entire area will result in the weight force. That will be the total net force that the fluid exerts on the bucket. That's of interest when we are dealing with buoyancy force (i.e. Archimedes force).