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The pressure of a point in a fluid is the sum of the external pressure and $pgh$. However, when the fluid is not placed perpendicular to the ground, then what is $h$ here? For example, a cup of water can be tilt an angle from the vertical axis.

In particular, can you explain clearly about this: a tube containing mercury and air. When the tube tilts at an angle 30○ from the vertical, then the length of the column of air will change too. But in order to calculate the change in its length, we have to know the pressure exerted by the mercury. But each point of the mercury in contact with air will exert a different amount of pressure so how can we do it?

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The variable $h$ in this equation always refers to a vertical distance from the point where you want to know the pressure up to the fluid surface in contact with the air. In the tilted cup you would still measure straight up to the fluid surface. The bit of tea in the middle of the cup doesn't "know" anything about the walls of the cup. Their orientation has no effect on the pressure inside the fluid.

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  • $\begingroup$ But I think the pressure at one point is due to the mass of the column of water above it. So how can it be h the distance from the point to the surface? $\endgroup$
    – trung hiếu lê
    Commented Aug 14, 2015 at 4:20
  • $\begingroup$ $h$ is measured straight up. "Up" means opposite to the direction gravity points, which has nothing to do with the orientation of the cup. So this always makes $h$ the vertical distance to the surface. It is actually a little more subtle than the "mass of the column of water above it". Note that pressure at any point acts inward in all directions. This tells you that there is more going on than the column of water above the point "resting" on the point of interest. $\endgroup$
    – gleedadswell
    Commented Aug 14, 2015 at 5:01
  • $\begingroup$ So you can please explain the question that I just added in my post? $\endgroup$
    – trung hiếu lê
    Commented Aug 14, 2015 at 5:37
  • $\begingroup$ I'm afraid I don't understand the new question. Why would "each point of the mercury in contact with air will exert a different amount of pressure"? The fluid surface should be horizontal even if the tube is tilted. $\endgroup$
    – gleedadswell
    Commented Aug 18, 2015 at 3:40

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