A large turntable $($a merry$-$go$-$round$)$ is able to rotate with friction about a fixed vertical axis through its center. The moment of inertia of the turntable about this axis is $1220$kg$$m$2$kg$$m$2.$ A daring child, with a mass of $45\mathrm{.}0$kg$$kg $,$ is riding the merry$-$go$-$round while wearing roller skates. She is initially standing at the center of the turntable $($and spinning with the turntable$)$; at this time, the turntable is making one revolution every $6\mathrm{.}40$seconds$$seconds $.$ The adventurous child finds that as soon as she is slightly off$-$center she naturally moves toward the rim of the merry$-$go$-$round. Playfully, she manages to move outward along a radius at a very slow constant speed by grabbing one of the radial support rails and applying just the right amount grip pressure to allow herself to slip slowly outwards. Assume that you can treat the child as a point mass.

What is the spin rate of the turntable when the child is a distance of $2\mathrm{.}50$m$$m from the center?