Problem $2$

$(10$ pts$)$

A square plate of mass $m$ has been welded into a uniform disk of radius $R$ and mass $m.$ The object is given the angular speed ${}_0$ in the direction shown and then it is placed on the incline. If $=30,$ and the coefficient of kinetic friction between the ring and the incline is ${}_k=\frac{\sqrt[\phantom{2}]{3}}{2},$ determine

$($a$)$ the time it takes for the object to start rolling without slip,

$($b$)$ the distance $d$ the mass center moves up the incline during the time found in $($a$),$

$($c$)$ the angular velocity of the object when it starts rolling without slip,

$($d$)$ the distance $d_2$ the mass center moves up the incline after it starts rolling without slipping.