= Solution
Integrating coaxial cylindrical shells gives
$$
I=\int_0^a r^2\frac{2m r}{a^2}\,dr=\frac12ma^2.
$$
<Rolling without slipping> has spin <speed> $u/a$, so \b[$K=mu^2/2+Iu^2/(2a^2)=3mu^2/4$].
For the <rolling-cylinder loop contact threshold>, set $\ell=R-a$, the <radius> of the centre's circular path, and let $\theta$ run from the loop bottom. The centre starts $h$ above its lowest centre position. Its height gain from the bottom is $\ell(1-\cos\theta)$. <Energy> conservation gives
$$
u^2=\frac43g[h-\ell(1-\cos\theta)].
$$
At the top, contact requires $N=m(u_t^2/\ell-g)\ge0$, hence $u_t^2\ge g\ell$. Inserting the <energy> relation proves the limiting condition
$$
\boxed{h\ge\frac{11}{4}\ell=\frac{11(R-a)}4.}
$$
The same condition keeps $N\ge0$ elsewhere on the circle, since
$$
\frac{N}{mg}=\frac43(h/\ell-1)+\frac73\cos\theta
$$
is smallest at the top.
For the centre's <acceleration>, the inward normal component is $u^2/\ell$ and the tangential component is $-(2g/3)\sin\theta$. If $A=h/\ell-1\ge7/4$ and $x=\cos\theta$,
$$
|\mathbf a_C|^2=\frac{4g^2}{9}[4A^2+8Ax+3x^2+1].
$$
This convex quadratic has its largest endpoint value at $x=1$. \b[The maximum on the circular loop is at the bottom, directed vertically upward.] For the limiting release height,
$$
\boxed{a_{C,\max}=u_b^2/\ell=\frac{11g}{3}\approx36.0\,\mathrm{m\,s^{-2}}.}
$$
The approach track's curvature is unspecified, so a global bound for an arbitrary approach shape cannot be proved from the sketch alone; the result above concerns the specified circular part and the usual smooth approach interpretation.
The <finite-friction qualification of a marginal rolling loop> also matters: the contact threshold presumes the stated no-slip idealization. At equality, near the top, required tangential <friction> vanishes linearly with angular distance from the top but $N$ vanishes quadratically. Thus any finite coefficient of <static friction> requires a release height above this limiting bound to maintain rolling everywhere; no coefficient is provided. This does not change the requested ideal contact criterion.
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