In a Skew Cylinder does pressure at the bottom face depend on depth or mass of Fluid

Question

I recently learnt that pressure in a Fluid depends only on depth and not mass of a Fluid (in any orientation). In my textbook the pressure has been derived by considering an infinitesimally small cylinder with a small pressure pressure difference between the two faces. Equilibrium condition was then applied to the cylinder (Weight+Force due to liquid above=Force due to liquid below). This gives pressure as a function of depth.

Examples are also given in my book which show different orientations of the fluid in which pressure is calculated at a point at the same depth from the free surface.

The example which confuses me is the Fluid in a skew cylinder. The free surface of the Fluid is at the same height as in the other examples and the pressure is calculated (At the bottom face) by using the vertical height from the bottom surface. But for the Fluid to be at the same vertical height some more Fluid should be added to the pipe (as volume changes) right ? (As compared to a right vertical cylinder having the same height).

If some more fluid is added the the mass of the Fluid above the bottom face will change and so should the pressure.

So does pressure not depend on the mass and only depends on height ? enter image description here

See also Does the weight of fluid in a conical container act entirely on the base? — sammy gerbil, Jan 24 '20 at 11:59
If the cylinder is tilted the vertical height $h$ decreases and so does the pressure at the base. If more water is added to restore the vertical height to its original value the pressure at the base increases back to its original value also. — sammy gerbil, Jan 24 '20 at 12:23
@sammy gerbil the cylinder is not tilted. It is deformed such that the height remains constant. — Aditya Ahuja, Jan 24 '20 at 13:51
@sammy gerbil but wouldn't the fluid column also be deformed in the same way. Like a pile of coins skewed at an angle. — Aditya Ahuja, Jan 24 '20 at 14:09
Sorry yes you are correct. The cross-sectional area of the cylinder becomes narrower as the cylinder becomes closer to horizontal. The volume of the cylinder does not change and the vertical height of the water remains the same. There is no need to add more water. — sammy gerbil, Jan 24 '20 at 14:12
@sammy gerbil so why does pressure still depends on vertical height and not slant height. Is it because the mass remains constant or the pressure due to the walls does not cancel out . — Aditya Ahuja, Jan 24 '20 at 14:23
The walls vertically above the base exert a downward component of force on the water. This force makes up for the absence of water vertically above the wall, and is equal to the weight of that absent water. See this answer to Question on the hydrostatic paradox. — sammy gerbil, Jan 24 '20 at 14:33

In a Skew Cylinder does pressure at the bottom face depend on depth or mass of Fluid

0 Answers0