The yo$-$yo in Fig. $5$ is pulled by a string with constant magnitude tension force $T$ applied at an angle $.$ The string is secured to the inner radius $r=\frac{1}{2}R$ where $R$ is the outer radius. If the yo$-$yo has mass $m$ and radius of gyration $k_G$

$($a$)$ Draw a free body diagram of the yo$-$yo

$($b$)$ Derive an equation for the angular acceleration $$ of the yo$-$yo as a function of $.($Your answer should include the known constants $T,R,m,$ and $k_G)$

$($c$)$ At what critical angle will the yo$-$yo roll with constant horizontal velocity? Describe the motion just greater and just less than this critical angle.

$($d$)$ If $=0,$ what is the minimum coefficient of static friction needed to prevent slipping?