problem $7(25$ points$)$

$7.$ A physics professor stands at the center of a frictionless turntable with arms outstretched and a $5\mathrm{.}0-kg$ dumbbell in each hand. He is set rotating about the vertical axis. Assume the professor's moment of inertia with arms outstretched is $I=4\mathrm{.}0kg{m}^2$ and $I=2\mathrm{.}5kg{m}^2$ with the arms near his chest $($dumbbells not included$).$ Assume also that the dumbbells are initially at $1\mathrm{.}0$ m and move to $0\mathrm{.}2$ m at the end.

a$.$ The professor is accelerated at the rate of $=20\mathrm{.}0ra\frac{d}{s^2}$ by a student applying a constant torque for $1\mathrm{.}0$ seconds. What is the torque magnitude?

b$.$ What is the professor's angular speed after the spin$-$up$?$

c$.$ After accelerating the professor, the student moves away and the professor pulls in the dumbbells. What is his final angular speed?