A woman of mass m$=$67 kg is standing on the edge of a horizontal disk of radius R$=2$ m and mass M$=120$ kg$.$ The disk is at rest and is free to rotate about a frictionless vertical axis through its center. She starts walking clockwise along the edge with a constant velocity of $1\mathrm{.}5$ m$/$s with respect to the Earth $($the moment of inertia about the disk's center of mass is I$=(1/2)$MR$^2).$

$($a$)$ In what direction and with what angular speed does the turntable rotate?

$($b$)$ How much chemical energy does the woman$’$s body convert into mechanical energy of the woman–turnable system as the woman sets herself and the turntable into motion?