Two children are on a merry$-$go$-$round. Child $A$ is located $1$ meter from the center of the merry$-$go$-$round and child $B$ is located at the edge of the merry$-$go$-$round $2\mathrm{.}5$ meters from the center. Child A has a mass of $35$ Kg $,$ Child $B$ has a mass of $60$ Kg $,$ and the merry$-$go$-$round can be approximated as a disk with a mass of $100$ kg $.$ The merry$-$go$-$round is initially traveling with an angular velocity of $0\mathrm{.}50ra\frac{d}{s}.$ Calculate the following:

a$.$ The moment of inertia for the entire system.

b$.$ The center of mass for the system.

c$.$ If child $B,$ jumps off the merry$-$go$-$round, what is the new angular velocity?