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I found this problem in ‘200 puzzling physics problems’

  1. An empty cylindrical beaker of mass 100 g, radius 30 mm and negligible wall thickness, has its center of gravity 100 mm above its base. To what depth should it be filled with water so as to make it as stable as possible?

Well, I solved it but want to understand the answer: why must the center of mass always lie on the water surface for maximum stability?

In the solution given in this book, it explains that the height of center of gravity would increase on filling more water after acquiring the condition of the center of mass lying on water surface but does not explain why the center of mass cannot be below it.

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  • $\begingroup$ Please note that the OP is not asking for the solution to the problem quoted at the top of the post. He is clearly asking a conceptual question: "why its center of mass always lie on the water surface for max stability". Thus this question should not be closed for the "homework-like exercise" reason. $\endgroup$ – PM 2Ring Sep 9 at 13:28
  • $\begingroup$ I agree with PM 2Ring. Should not be closed. $\endgroup$ – Bob D Sep 9 at 13:33
  • $\begingroup$ Well, is this question closed?? If yes, please tell me how to know if a question has been closed or not. $\endgroup$ – Prajwal Turkar Sep 9 at 13:34
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    $\begingroup$ It has not yet been closed, though there are two votes to close. PM 2Ring and I are arguing in favor of not closing the question. Hope that clarifies things. $\endgroup$ – Bob D Sep 9 at 13:46
  • $\begingroup$ @PM2Ring I agree that this question could be edited into a shape that's on topic. I don't think the current wording makes the bar, though. $\endgroup$ – Emilio Pisanty Sep 9 at 14:16
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It seems "max stability" is defined as follows, 1. All water added hardens like resin, so when you push the cylinder trying to tilt it, the water surface will not keep horizontal but will tilt with the cylinder; 2. The system is at its max stability if it has the maximum possible tilting angle before it falls over.

So the condition of max stability equals having the lowest center of mass, or COM. If you add a little water to the empty cylinder, it is obvious the new total COM will be lower. This is because newly added water COM is lower than the systems previous total COM. If you add a little bit more water, the total COM will be even lower, for the same reason. You keep adding water, until you reach a point when the total COM is exactly at the water surface. Before that, adding water lowers total COM. After that , adding water raises total COM. Since you need the COM to be as low as possible, this point is what you can reach. End of proof.

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  • $\begingroup$ I would look at the problem a bit differently. Assuming that the water is NOT rigid like hardened resin, the center of gravity will shift sideways in the direction in which the cylindrical container is tilted. It will not take as much torque to tip the cylinder when the water is liquid as when it is solid. However, it appears that this consideration is not intended to be part of the problem. The answer by @verdelite is good. $\endgroup$ – S. McGrew Sep 10 at 3:27
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When the water level lies below the (vertical) center of mass adding water adds mass below the center of mass. This must lower the center of mass, making the cup more stable.

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