Rolling uniform solid hemisphere (${\dot{\theta }}^2=15g\cos \theta /[a(28-15\cos \theta )]$)

Let the flat face make angle $\theta$ with horizontal while the rounded face contacts a horizontal plane. The centroid has height $a-(3a/8)\cos \theta$ and contact distance squared $b^2=a^2+(3a/8{)}^2-2a(3a/8)\cos \theta$. For rolling without slipping, its speed is $b|\dot{\theta }|$. Combining translation and rotation gives kinetic energy $\frac{1}{2}(I_G+m{b}^2){\dot{\theta }}^2$. Conservation from rest at $\theta =\pi /2$ gives the displayed result until the flat-face/contact geometry changes.