A solid disc of radius $0\mathrm{.}5$ m is angularly accelerated at a constant rate from a frequency of $100$ Hz to $800$ Hz over an interval of $10$ s $.$

$(1)$ What are the disc's initial and final angular velocities?

$(2)$ What is the final speed of a point at the disc's edge?

$(3)$ What is final speed of an interior point at $r=0\mathrm{.}2m?$

$(4)$ What is the angular acceleration from $i$ to $f?$

$(5)$ What is the tangential acceleration of a point at the edge over the interval?

$(6)$ What are the initial and final centripetal accelerations of a point at the edge?

$(7)$ From $i$ to $f$ how many radians does each point on the disc rotate through?

$(8)$ What is this change in angle expressed in revolutions?