how can you prove the water always come to rest at a same level
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It is OK if you haven’t done any grand experiments and rely simply on your own observations. In all of your observations, you have seen that water does something and it does it consistently and you call that “seeking level.” I totally agree with you.
However, you have never done an experiment (I have) that distinguishes between these two* rules:
1) Water always seeks its own level, and that level is necessarily planar.**
2) Water always seeks its own level, and that level is necessarily perpendicular at each point on the surface of the water to a vector from the water to the center of gravity of the Earth.***
You know for certain the first part - and most everyone who has considered the issue will agree with you. Water does seek its own level, or in more pedestrian terms, water flows downhill if it can. But that doesn’t tell you whether the rule is #1 or #2.
If you had done an experiment that could distinguish between #1 and #2, you would find that #2 is the rule and #1 is not. Alternatively, you could have relied on observations or experiments from people you trust and those would also tell you that #2 is the rule and not #1.
For a glass of water, you can’t distinguish between #1 and #2. For a big swimming pool, without very complicated equipment and ideal conditions, you also cannot distinguish.
* There is a third rule: 3) Water always seeks its own level, but if it is not in an inertial frame, the water is not going to be level. But, I’ll assume you’ve done an experiment to determine what water does in a non-inertial frame, such as a whirlpools or a fast rotating bucket. The rules #1 and #2 above can be limited to inertial frames.
However, you have never done an experiment (I have) that distinguishes between these two* rules:
1) Water always seeks its own level, and that level is necessarily planar.**
2) Water always seeks its own level, and that level is necessarily perpendicular at each point on the surface of the water to a vector from the water to the center of gravity of the Earth.***
You know for certain the first part - and most everyone who has considered the issue will agree with you. Water does seek its own level, or in more pedestrian terms, water flows downhill if it can. But that doesn’t tell you whether the rule is #1 or #2.
If you had done an experiment that could distinguish between #1 and #2, you would find that #2 is the rule and #1 is not. Alternatively, you could have relied on observations or experiments from people you trust and those would also tell you that #2 is the rule and not #1.
For a glass of water, you can’t distinguish between #1 and #2. For a big swimming pool, without very complicated equipment and ideal conditions, you also cannot distinguish.
* There is a third rule: 3) Water always seeks its own level, but if it is not in an inertial frame, the water is not going to be level. But, I’ll assume you’ve done an experiment to determine what water does in a non-inertial frame, such as a whirlpools or a fast rotating bucket. The rules #1 and #2 above can be limited to inertial frames.
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