Only $16\mathrm{.}107!!!!!!$ A $300-$mm$-$radius cylinder of mass $8$ kg rests on a $3-kg$ carriage. The system is at rest when a force $P$ of magnitude $20$ N is applied. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine $($a$)$ the acceleration of the carriage, $($b$)$ the acceleration of point $A,($c$)$ the distance the cylinder has rolled with respect to the carriage after $0\mathrm{.}5$ s $.$

$106$

Fig. P$16\mathrm{.}107$