Fig.B-15 The hydrostatic paradoxon: the hydrostatic force
acting on the flat plate bottom is the same in all three cases in spite of the fact that weight of liquid poured into each vessel is different and in the case C the force actually acts in the opposite, upwards direction.










Fig.B-16 Below: two-arm scales for demonstration
of the hydrostatic paradoxon: With equal bottom piston
areas and equal height of liquid surfaces the scales will be
balanced in spite of the fact that weights (= gravity
forces) in the both vessels differ substantially. History of
this device goes back to Simon Stevin (1548-1620), military
and civil engineer under
Maurice of Nassau
= =
interpretation, however, is valid only for vesels of prismatic shape - i.e. having same cross-sectional area over their whole height. In nonprismatic vessels the inconsistence of the weight of liquid poured in and the generated force may be surprising - when compared with vessel A, the vessel B in Fig.B-15 contains less liquid, the weight of which is therefore smaller. An unprepared observer might be led to expectation that also the force acting on the bottom in the case B will be smaller than in the case A, where the mass of fluid inside the vessel is larger. This apparent discrepancy, hydrostatic paradoxon is, of course easily explained: the acting force is pressure force, dependent upon bottom area and hydrostatic pressure (which is a function of the depth of the bottom under liquid surface) and has nothing to do with mass of liquid in the vessel. In fact, an equal force in the case C is acting upwards, opposite to gravity.

2) The case of an inclined plane wall
- inclined at an angle relative to the free surface. Note that this case includes even the vertical walls, where equals 90 degrees. In this case, the height difference is different from one position to another and the integration (Fig. B-13) performed to obtain the force acting on a flat panel (let us say on a lid or cover covering an opening in the vessel wall, Fig.B-17) is no more a trivial one. Let us specifiy the position of a surface element of the panel by the distance measured from the intersection of panel plane with the liquid surface plane. The height difference is then . An insertion of this expression into the integral Fig.B-13results (with constant extracted in front of the integral) in
Fig.B-17

- where is the static moment, which may be re-written using the well-known definition of the position of centre of gravity:

By inserting the expression for centre of gravity depth

the final result is

This is certainly no surprising result as it closely corresponds to what we have obtained earlier in Fig.B-14


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This is page Nr. B06 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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