Center of gravity

A \(3\times 3\times 3\) cube resting on the ground has a \(2\times 2\times 2\) cube cut out of its center leaving each side of the cube with thickness \(.5\) (except for the corners). It's hollow center is then filled with water so that the center of mass of the cube is as close to the ground as possible.

If the shortest distance from the ground to the center of mass of the water filled cube is \(M\) where \(M\) is in meters, find \(\lfloor 100M \rfloor\).

Important details:

The hollow part of the cube that is not filled with water is filled with helium gas. (Hint: you don't need to account for the helium gas, it changes the height of the center of mass by less than 0.0006 m)

All measurements are given in meters.

Density of the cube is \(500~\frac{Kg}{m^3}\)

Density of water is \(1000 ~ \frac{Kg}{m^3}\)

You may use Wolfram Alpha


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