= Solution
The <parallel axis theorem> states $I_O=I_G+md^2$ for two parallel axes, one through the <centre of mass> and the other a perpendicular distance $d$ away. To see this, expand the squared distance to the translated axis in the <mass> integral; its mixed term vanishes because the centroidal first moment is zero.
For a uniform solid hemisphere, the moment about a diameter in its flat face is half the corresponding full-sphere moment. With full-sphere <mass> $2m$, it is $\tfrac12[\tfrac25(2m)a^2]=2ma^2/5$. The original PDF gives the centroidal displacement $d=3a/8$; the TeX's $3a/5$ is incorrect. Moving the diameter axis to its parallel centroidal axis therefore yields
$$
\boxed{I_G=\frac25ma^2-m\left(\frac{3a}{8}\right)^2=\frac{83}{320}ma^2}.
$$
This is the <moment of inertia of a uniform solid hemisphere>, not that of a thin hemispherical shell.
Take a cross-section perpendicular to the rotation axis. Let $O$ be the underlying sphere centre, $G$ the <centre of mass> and $P$ the contact point. During rounded-surface contact, $OP=a$ vertically and $OG=d$ points toward the curved face, with downward component $d\cos\theta$. Therefore
$$
y_G=a-d\cos\theta,\qquad b^2=GP^2=a^2+d^2-2ad\cos\theta.
$$
For <rolling without slipping>, $P$ is instantaneously at rest and the body rotates about it with angular <speed> $|\dot\theta|$. Thus \b[the centroid <speed> is $b|\dot\theta|$]. Translation and rotation contribute
$$
K=\frac12mb^2\dot\theta^2+\frac12I_G\dot\theta^2=\frac12ma^2\left(\frac75-\frac34\cos\theta\right)\dot\theta^2,\qquad V=mg\left(a-\frac{3a}{8}\cos\theta\right).
$$
The initial vertical base has $\theta=\pi/2$ and $K=0$, so conserved <energy> is $mga$. Equating the later $K+V$ to this value gives
$$
\boxed{\dot\theta^2=\frac{15g\cos\theta}{a(28-15\cos\theta)}}.
$$
On the initial roll toward the lowest centroid position, $\dot\theta<0$. Static contact friction does no work because the contacting material point is at rest; its coefficient must be large enough for the assumed no-slip motion. The formula describes the rounded-contact phase of the <rolling uniform solid hemisphere> and should not be continued through a change of contact geometry without another model.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ia/paper-4-rolling-and-orbit.png]
{title=Rolling hemisphere centroid and contact geometry, and scattering under a repulsive inverse-cube force}
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