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Scenario description

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

Question

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

         \              /
          \            /
           \          /
            \________/
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If you are talking of the weight of the entire body, then it will always act perpendicularly down from the center of gravity. Otherwise a fluid (in your case a liquid) always exerts equal pressure in all directions at a given depth. This pressure varies with depth and is never affected by the shape of a container. –  rahulgarg12342 Jul 15 at 7:09
    
Do both containers contain 300L AND have the same base area? Or are the base areas different??? –  Bryson S. Jul 15 at 7:36
    
@rahulgarg12342 1) Yes, initially i had 300 litres and same base area but now after seeing all your replies i have a different doubt. the question is, " So does it mean the determinants for pressure are base area and height, and not volume of liquid??" –  user52187 Jul 15 at 11:05
    
@BrysonS. " So does it mean the determinants for pressure are base area and height, and not volume of liquid??" –  user52187 Jul 15 at 11:05
    
@user52187 The determinant for pressure is depth, and depth depends on how the volume is distributed. –  Bryson S. Jul 15 at 11:58

3 Answers 3

up vote 5 down vote accepted

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.

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We have to be careful here because the force measured by holding up the base is not just the base pressure times the area. It is that force plus the vertical shear from the slanted walls around the edge of the base. In either case, the full weight of the fluid must be borne. –  Bryson S. Jul 15 at 12:04

No. the whole weight will not act on the base of the container 2. If the whole weight had acted on the base of container 2, then the pressure on the base of container 2 would be equal to that of container 1 i.e., mg/A, where mg is the whole weight of the fluid and A is area of the base. But as you know the pressure at a depth depends on the height of the fluid above it, $h\rho g$, they should be different because the heights of the fluid are different.

Another way of thinking is if you observe that when you make the slanting parts more and more slanting, you can easily see that the weight of the whole fluid is still 300kg but the base now bears very less pressure as the fluid above the base has very less height. Weight is always downwards and the whole weight doesn't act on the base. When the sides are slanting, in addition to the base area, there is a component of the area vector of the slanting side in the downward direction providing more surface area for the fluid to exert pressure downward. But for straight cylinder there is no other area vector in the downward direction other than the base so the whole weight acts on the base in this case.

But weighing is different. On weighing you get 300kg because you are measuring the whole mass multiplied by acceleration due to gravity, it doesn't matter what pressure does the base bears. It doesn't have anything to do with the base apart from the support it provides to place it. You can measure it in any position and you will get 300kg.

Consider two cylinders, one longer but small cross-sectional area another shorter but very large cross sectional area so that the latter has larger volume. If both are fully filled with a fluid, $\rho$, then the one which is longer will have more pressure at the base but that doesn't mean that the weight of the whole fluid in it is more. The weight of the second one is more.

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thanks for the reply qwartz, so if i keep the base area of second vessel as same as tube (pic 1) and increase the height of second vessel and fill it with water to make it equal with tube, then what i get is " same base area in both vessel, same height of water in both vessel, but different volume i.e more volume in second vessel say 600 litres. But the result is same,i.e same pressure on base area" Am i right? –  user52187 Jul 15 at 11:19
    
yes. You can read this faculty.wwu.edu/vawter/PhysicsNet/Topics/Pressure/… –  learner Jul 15 at 11:48
    
Heartful thanks for the link. Really useful :-) –  user52187 Jul 15 at 11:54
    
@user52187 Yes. –  Bryson S. Jul 15 at 11:59
    
I have revised my thinking on the matter. The hydrostatic pressure at the base will be reduced as the walls are slanted outwards, but a shear force around the edge of the base will arise and account for the weight of any fluid which overhangs the base. –  Bryson S. Jul 15 at 12:44

As Rahulgarg mentioned, the pressure does not depend on the shape but on the depth. However, the direction of the force caused by pressure can be approximated as being normal to the surface, hence the total force on the sides will depend on the shape. For a fluid at rest like the one I think you are assuming, the pressure at any depth will be $p=p_0+\rho g h$, where $p_0$ is the atmospheric pressure at the surface, $\rho$ is the fluid density, $g$ the acceleration of gravity and $h$ the distance from the liquid's surface to the depth you want to know $p$. In your specific example the pressure at the bottom of case 2 will be smaller than in case one because of the different heights of the vessels.

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The normal force against the floor (between the floor and the glass), as measured by a scale, will be the same in both cases, but not the pressure (force per unit area of the liquid against the glass) at the bottom of the liquid. –  julian fernandez Jul 15 at 7:28
    
...And this is because the slanted walls resist some of the pressure force, which is in part balanced by the resistance to deformation of the vessel, and in part balanced by the reaction of the floor. –  Joce Jul 15 at 8:21
    
@Joce Understood julian fernandez, Still have one doubt. doesn't pressure indirectly mean weight, i.e for example if pressure is 1.5 bar (1 + 0.5), then it means 1.5 kg/sq.cm. So if i have a base area of 200 sq.cm, with 1.5 bar pressure, does it mean that the water above is exerting WEIGHT of 300 kg on base area?? i.e 1.5 kg/sq.cm * 200 sq.cm = 300 kgs. Is it right? –  user52187 Jul 15 at 11:47

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