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In the picture below, both tubes have the same pressure at the bottom. I understand that if they were weighed individually, they would show a different result since the water on the second tube is exerting an upward force on the "roof", removing "excess" force on the scale.

But what if the scale was put inside the tubes (red lines)? Also, the sides of the tubes are mounted onto an immovable wall, preventing any upward force affecting the scale from below. Now since pressure = Force/area, and area and pressure are the same in both, the scales must be subjected to the same forces as well. Will the second scale now show an incorrect result?

enter image description here

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  • $\begingroup$ The scale reads the weight of whatever is placed upon it, REGARDLESS of the shape of the container. $\endgroup$ Mar 12, 2019 at 18:23
  • $\begingroup$ So if the first tube holds 5l and the second 4l of water, then the scales will show 5kg and 4kg? This goes against the two answers so far. Could you add another answer? Or did I just interpret your comment incorrectly? $\endgroup$ Mar 12, 2019 at 19:43
  • $\begingroup$ Yes, the scales would read 5 kg and 4 kg, if the scales are outside of the containers. I maintain that both scales would read zero if they are submerged inside the containers, as your red lines indicate in the drawing. $\endgroup$ Mar 12, 2019 at 19:49
  • $\begingroup$ yes, in my scenario the scales are inside the containers. why would they read zero? isn't there a force applied to the scales? $\endgroup$ Mar 12, 2019 at 20:15
  • $\begingroup$ There is a force applied to the scale from the liquid that is above it, but there is another force applied to the bottom of the scale from the liquid that is below it. Those forces act at right angles to the surfaces that they interact with, so the upward force and the downward force are equal and opposite for a "normal" scale that isn't hermetically sealed. $\endgroup$ Mar 13, 2019 at 1:16

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Indeed, assuming your scales still work and have negligible volume (to avoid issues with buoyant forces), they'll now show the same reading. I wouldn't say that means the scale is "wrong". It's designed to measure the normal force exerted on its top part, and in both cases it's doing just that.

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  • $\begingroup$ thank you! If I poured 5 Liter into the first construction and 4,5 Liter into the second, I would be surprised to see 5kg on both scales - thus I used "incorrect". Another question: Would you agree with the statement by Wolpham jonny that my first paragraph is wrong? $\endgroup$ Mar 12, 2019 at 15:04
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    $\begingroup$ @PhilippMurry It's not wrong, there are simply two ways of thinking about the situation. You can analyze the system of the water+tube, in which case the external forces are the normal force from the scale (which determines the scale reading) and the weight, so they must be equal. Or you can analyze the tube along, in which case the external forces are the normal force from the water, the tube's own weight, and the normal force from the scale. The answer is the same: the normal force from the water pressure is equal in magnitude to the weight of the water, because the upward normal force on... $\endgroup$
    – knzhou
    Mar 12, 2019 at 15:08
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    $\begingroup$ ...the slanted part of the tube makes up for the excess on the bottom. I suppose Wolphram jonny said this is "wrong" because the analysis is certainly more involved if all you want to know is the scale reading, but it is necessary if you want to understand the forces acting on the tube alone. $\endgroup$
    – knzhou
    Mar 12, 2019 at 15:09
  • $\begingroup$ I used "incorrect" because when the scale is outside the water is not in contact with the scale, so you can only weight the entire system, including the walls of the container, which I assumed of negligible weight. $\endgroup$
    – user65081
    Mar 12, 2019 at 15:55
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I assume that by incorrect you mean the same "weight". Yes it will show the same weight, and the reason it will not show the actual weight is that the slanted part of the wall is making a force, the reaction to the water pressure force larger than if the wall were straight (because of the larger pressure with depth), into the liquid. On the otherhand your first paragraph has a mistake, if you put the balance outside the weight will be different not because of the upward force. This force is irrelevant because the water is not in contact with the scale and then that has to be considered an internal, not an external, force. In such a case the weight will be different because they have different mass in absence of external forces.

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