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I have the following problem:

An inflatable beach ball is made of a thin, inelastic, but bendable material. When fully inflated, it has a radius $R$. In an experiment, the ball is filled half full with air, and is then continuously pushed underwater by a large, flat surface. What shape does the ball assume when fully underwater? If possible, determine the numerical values of the relevant sizes of the shape.

My initial guess to this problem was that the shape is going to be a hemisphere of radius $R$. I tried integrating the water's pressure's vertical component on the surface, and equating it with the force exerted by the flat surface on the ball. Then dividing this force by the area gives the inner pressure of the air, which turns out is always larger than or equal to the pressure of the water. This means the ball would have to exert pressure on the air, which seems to make sense, since if it's a hemisphere, it's fully streched.

However, this doesn't prove at all that the shape is indeed going to be a hemisphere, and I don't even know if that's the correct answer to the problem.

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  • $\begingroup$ Here is a related problem. What is the shape of the same beach ball held to the flat floor by gravity? $\endgroup$
    – mmesser314
    Commented Oct 16, 2021 at 16:17

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The surface area of the ball would need to be unchanged, so if the hemisphere is the answer it would need to be a different radius.

Perhaps someone can provide a more mathematical treatment, but here are some options that you could consider.

enter image description here

There is an upward pressure from the water that will force the air to move. The bottom one seems better, in the top diagram as the pressure increases with depth, there would be a stronger upward push in the middle than at the edges, so if the air was in the top shape it would change to the bottom. Once in the bottom shape it has nowhere else to be pushed to.

It is the answer that might occur if the balloon has some tension in it. You could then use volume of a torus formula to start trying the mathematical side of the problem.

If there is no tension, or the tension is negligible compared to other forces, you could also consider this third option.

enter image description here

All the best with it.

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  • $\begingroup$ I just realize this now, but in both of your drawings the top of the shape is flat, which is impossible, you cannot flatten a spherical surface. This is a problem of my solution too, though. $\endgroup$
    – AstroRP
    Commented Oct 16, 2021 at 11:06
  • $\begingroup$ @ AstroRP It's a tricky question, but try with a balloon or a tennis ball pushing e.g. against a wall, it starts to make a flat circle, but the radius presumably has a maximum value or R. $\endgroup$
    – John Hunter
    Commented Oct 16, 2021 at 11:10

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