I'm trying to write a very simple simulation of volume/water displacement. Every timestep a sphere gets lowered a bit further into a cylindrical container filled with water. While the physics behind this should be quite clear, I just can't wrap my head around how I would go about calculating the change in water height per timestep.

If the sphere is lowered by $y$ below the initial water level, then the water level will rise by $h$, resulting in the sphere being submerged by $y + h$, which would in turn mean it would displace more water, making the water level rise some more and so on and so on... This of course is wrong somewhere, but I fail to come up with a better idea. So, where do I err?

Assume $r_{C}$to be the radius of the cylindrical container, $h_{0}$the original water level without a sphere in it, $h_{n}$ the water level after the nth step $r_{S}$the radius of the sphere and $y_{n}$ how high the spherical cap of the submerged part (below the original water level) of the sphere is after step $n$.

From the balance of volumes before and after insertion for step 1 we get $V_{o}=V_{1}-V_{S}$ with $V_{0}$ being the original water volume, $V_{1}$ the volume after inserting part of the sphere in the first step, and $V_{S}$ the volume of the inserted spherical cap.

$$V_{0}=r_{C}^{2}\pi h_{0}\\ V_{1}=r_{C}^{2} \pi h_{1}\\ V_{S}=\frac{\pi}{3}h_{1}^{2}(3h_{1}-h_{1})$$

this is fine on its own and can be solved by solving a cubic equation, but nowhere in this does $y_{n}$ appear, and that's the part where I get hung up. How to transform these equations into something dependent on $y_{n}$, e. g. how to transform them into something dependent on how much of the sphere is below the original water level?

  • $\begingroup$ You'll need to solve a cubic equation to find the water level in terms of the sphere position. $\endgroup$ May 25, 2023 at 14:30
  • $\begingroup$ Welcome to Scicomp. Start with the math you have. What are the equations for the water body volume without the sphere? define your radii and your coordinate system etc. Sketch a plot what you'd expect the water level to look like depending on the depth of insertion. Give us something to work with:-) $\endgroup$
    – MPIchael
    May 26, 2023 at 12:44
  • $\begingroup$ @MPIchael, thank you for your answer, I edited my post, I hope this clarifies things? $\endgroup$ May 27, 2023 at 13:29


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