homework and exercises - Does the weight of fluid in a conical container act entirely on the base?

Lets assume both containers have a capacity of 300 litres.

One is a vertical tube as shown in pic 1

Other one is more or less a V shaped containers as shown in pic 2

Both are at ground level

No doubt that whole weight of 300 litres will act on the base of the tube (pic 1). But will the weight of the whole 300 litres act on the base of the container as shown in pic 2?

I.e if you keep both the containers on a weighing scale, for sure, it will show a weight of approx. 300 kg each.

But what I'm asking is whether the weight of whole 300 litres acts on the base of container 2 (since the container's walls are slanting, as opposed to the walls of the tube in pic 1).

If the whole weight will not act on the base of the container, can you state the reasoning and also the principle, so that I can further get to know about it?

Pic 1

           |         |        
           |         |
           |         |
           |         |
           |         |
           |         |
           |_________|

Pic 2

         \              /
          \            /
           \          /
            \________/

Answer

No, the entire weight will not directly rest on the base of the slanted container (although it does indirectly). There are a number of ways to approach this, but the easiest way is to observe that the total force acting on the bottom of the container is equal to the sum of the hydrostatic pressure force (the pressure at the bottom of the container multiplied by its area) and the shear force around the edge of the base (draw a free-body diagram to convince yourself of this). The hydrostatic pressure depends only on the height of the column of fluid above the given location ($P=\rho gh$), and the shear force is equal to the weight of the fluid outside of the base (supported by the walls). Since both vessels contain the same volume, but the slanted container has a wider cross section above the base, the total height of the body of fluid will be less for the container with slanted walls than the other container. Thus, the hydrostatic pressure at the base will be less. But remember, any reduction in the pressure force will be compensated by an increase in the shear force around the edge of the base. The vertical walls support no vertical force, and so they exert no shear force on the base. The slanted walls do bear some of the weight, and so they transfer this force to the base via shear. In both cases, the sum of the hydrostatic pressure force plus the shear force (if there be any) is equal to the weight of the fluid, and this is what the scale measures.