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When changing the volume of mercury in the container, its mass may increase or decrease, so the gravitational force acting on the inside bottom of the container may increase or decrease, and therefore the norm force of the inside bottom of the container may increase or decrease, won't that affect the height of $\rm Hg$ in the tube?
Also, if the area of the mercury exposed to air is increased or decreased (even if the mass of mercury itself did not change), the force of the gases acting on it will increase or decrease, and therefore, the norm force of the inside bottom of the container may increase or decrease, won't that also affect the height of $\rm Hg$ in the tube?

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The important quantity is not the volume or mass of the mercury in the container. Also, the diameter of the tube is not important. Rather, it is the height difference between the mercury in the tube and the mercury in the container that is important. This is confirmed by Torricelli's experiment.

The net force pushing upward on the mercury in the tube is equal to air pressure (at the height of mercury in the container) times the cross sectional area of the tube; it is not at all dependent on the area of the surface of the mercury in the container.

The force of gravity on the mercury in the tube is equal to the weight of the mercury, which is equal to the mass density of the mercury times the height of the mercury in the tube above the mercury in the container, times the cross-sectional area of the tube.

If you write all that out in an equation that relates the height of the mercury in the tube to the air pressure when the upward force balances the weight of the mercury in the tube, you will find that the cross-sectional area of the tube cancels out, leaving only air pressure, mass density, and height difference in the equation.

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  • $\begingroup$ But how the force that acts on the surface is converted into a force that acts on the mercury column? $\endgroup$ – Mason Jan 8 '19 at 6:36
  • $\begingroup$ Pressure is transmitted uniformly in all directions. Hg in the container pushes up against the air; the air pushes down against the Hg. Hg at the same level pushes up from the container into the tube, and Hg in the tube pushes down onto the Hg in the container. It all balances. $\endgroup$ – S. McGrew Jan 8 '19 at 13:24

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