$($a$)$ A $19\mathrm{.}0-$kg child is riding a playground merry$-$go$-$round that is rotating at $45\mathrm{.}0$ rpm$.$ What centripetal force must she exert to stay on if she is $2\mathrm{.}00$ m from its center?

$($b$)$ What centripetal force does she need to stay on an amusement park merry$-$go$-$round that rotates at $3\mathrm{.}00$ rpm if she is $6\mathrm{.}70$ m from its center?

$($c$)$ Compare each force with her weight.