$7\mathrm{.}3\mathrm{.}31.[$Level $2]$ A uniform cylinder of mass $\backslash ($ m $\backslash ),$ radius $\backslash ($ r $\backslash ),$ and moment of inertia $\backslash (\backslash$frac$\{$m r$^\{2\}\}\{2\}\backslash )$ rolls down an inclined surface. The surface is simultaneously accelerated upward at a constant rate, $\backslash (\backslash$ddot$\{$z$\}=$a$\_\{$z$\}\backslash ).$

E$7\mathrm{.}3\mathrm{.}31$

a$.$ Find the acceleration of the cylinder's center of mass under no$-$slip conditions.

b$.$ What is the minimum coefficient of static friction for which no$-$slip conditions occur?