= Rolling uniform solid hemisphere
{title2=$\dot\theta^2=15g\cos\theta/[a(28-15\cos\theta)]$}
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 $\tfrac12(I_G+mb^2)\dot\theta^2$. Conservation from rest at $\theta=\pi/2$ gives the displayed result until the flat-face/contact geometry changes.
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