Consider a body with two small volumes, one twice the other. They are held apart by a massless and a volumeless rod. They are constrained vertically with the large volume down. When the density is ⅔ any vertical position with the large mass under water and the smaller above is in equilibrium. This is perhaps a paradoxical configuration. Skip it for now.

Consider a thin isosceles triangle constrained to float vertically. Its height is 1 and its volume is ½. It is constrained to float tip up. A table of Height of density = ρ; CG of triangle = h; moment of displaced water = mw; depth of CG of displaced water = H.

ρ | h | mw | H |

0 | ⅓ | 0 | 0 |

¾ | −1/6 | 5/48 | −5/36 |

1 | −⅔ | ⅓ | −⅔ |

integral from 0 to ½ of x(½ + x) = I x^2 + ½x = (x^3)/3 + ¼ x^2 = 1/24 + 1/16 = 5/48

m = gc*v

5/48 = H*(¾); H = 5/48 4/3 = 20/144 = 10/72 = 5/36