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The container is inverted and filled with water through a hole on top and it is said the pressure on the bottom of the container is $\rho g R $ which I agree with for only the point directly bellow the hole

I suppose the normal forces from the walls of the container would have an effect but why does it have to be such that the bottom gets exactly $\rho g R $. In the whole scenario atmospheric pressure is not present.

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Water pressure increases with depth. All across the same horizontal depth the pressure is the same, ignoring the negligible curvature of the Earth. If the container is filled all of the way up to the hole as stated then the depth is equal to the radius, though the diagram shows it not quite full.

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  • $\begingroup$ but points near the bottom ends are not in the same depth since the water is more shallow,It is only R at a single point $\endgroup$ Commented May 18, 2021 at 10:53
  • $\begingroup$ Pressure is from the depth of the surface of the water, the shape of the container does not matter. Imagine a horizontal line at the water's surface, any point on the bottom will be an equal height from this line so at an equal depth. $\endgroup$ Commented May 18, 2021 at 11:01
  • $\begingroup$ @Glowingbluejuicebox, Pascal's law is also involved in this problem. See en.wikipedia.org/wiki/Pascal%27s_law $\endgroup$ Commented May 18, 2021 at 15:23
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If the pressure was lower at other points along the bottom, water wold flow horizontally until it was equal. (You do have to consider the vertical component of the reaction force from upper sections of the container.)

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