A solid spool of thread has radius R$,$ mass M$,$ and moment of inertia $.$

$($a$)$ If you pull up on the thread so that the center of mass of the spool remains

suspended in the air at the same place as it unwinds, what force must you exert on

the thread? $($Variables that may be included in your final answer are M$,$ R$,$ and

known constants.$)$

$($b$)$ By the time the spool turns with angular velocity $,$ what length of thread has been

unwound from the spool? $($Variables that may be included in your final answer are

M$,$ R$,$ $,$ and known constants.$)$