Problem $4(25$ pts$)$

The end of a rope, which is wrapped around the inner radius of a spool, is being pulled with force $P$ at an angle $.$ The spool is composed of two rings that each contact the ground $($analogous to how we treat the two rear wheels on a car$).$ The spool has a moment of inertia $I_G=m(r_i^2+r_o^2)$ where $r_i=3\frac{R}{4}$ and $r_o=R.$ The spool rolls without slipping on the fixed ground. Determine:

$($a$)$ the acceleration of the mass center in terms of $P,m,$ and $,$

$($b$)$ the angular acceleration of the spool in terms of $P,m,,$ and $R,$

$($c$)$ and the friction at the contact point between the spool and ground in terms of $P$ and $.$

From your answers, determine the angle at which the acceleration of the mass center is zero.

Extra Credit $(3pts$ without slipping for any non$-$zero value of $P?$