= Solution
Interpret $h$ as the vertical drop of the cylinder's <centre of mass>, and initially take the suspended rod to be vertical. The <moment of inertia of a uniform solid cylinder> is $I=mr^2/2$. Before impact, <rolling without slipping> gives $\omega=u/r$, so <conservation of mechanical energy> gives
$$
mgh=\frac12mu^2+\frac12I\frac{u^2}{r^2}=\frac34mu^2,
\qquad u=\sqrt{\frac{4gh}{3}}.
$$
Use the usual instantaneous smooth <elastic collision> model: the horizontal impact <impulse> passes through the cylinder's <centre of mass>, and the floor supplies negligible tangential <impulse> during impact. The cylinder's <angular velocity> is then unchanged during impact; its <rotational kinetic energy> does not transfer to the bob. The rod supplies no horizontal <impulse> when vertical. If $v$ and $V$ are the signed cylinder and bob <velocities> immediately afterwards, <conservation of momentum> and the unit <coefficient of restitution> give
$$
mu=mv+2mV,\qquad V-v=u,
\qquad v=-\frac u3,\quad V=\frac{2u}{3}.
$$
During the subsequent <simple pendulum> swing, <conservation of mechanical energy> gives
$$
\frac12(2m)V^2=2mgh(1-\cos\theta_{\max}),
\qquad
\boxed{\cos\theta_{\max}=\frac{19}{27},\quad
\theta_{\max}=\cos^{-1}\frac{19}{27}\simeq45.28^\circ.}
$$
The cylinder generally slips immediately after this impact, which is consistent with imposing <rolling without slipping> only on the descent.
\b[Impact convention matters.] The <floor-impulse dependence of a rolling-cylinder collision> gives a different answer if a different constraint is imposed. If instead the floor enforces <rolling without slipping> throughout the impulsive <collision>, it supplies a nonzero horizontal <impulse>. The cylinder's effective translational <mass> is then $m+I/r^2=3m/2$. Writing $J$ for the <impulse> delivered to the bob gives $(3m/2)(v-u)=-J$, $2mV=J$ and $V-v=u$. Hence $V=6u/7$ and $\cos\theta_{\max}=25/49$, giving $59.32^\circ$. The original wording specifies the descent constraint but does not specify an impulsive floor-friction law; the first boxed value uses the smooth-impact convention.
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