How to perfect the bottle flip?

A lot of skill goes into whether you can achieve a bottle flip. But with a bit of physics, you can maximize your chances by optimizing the volume of water in the bottle. Over- or under-filling the water bottle yields a high center of gravity, which can cause the bottle to tip over on landing, even if the throw is otherwise perfect. To stabilize the bottle's landing, the center of gravity should be as low as possible.

Treat an empty water bottle as a cylinder with mass \(\SI{12}{\gram}\) evenly distributed along its length. What is the water level \(h\) in centimeters such that the bottle's center of gravity is lowest?

Note: The density of water is about \( \SI[per-mode=symbol]{1}{\gram \per \centi \meter \cubed}.\)

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