rotational kinematics, gravity :

$1.($a$)$ Calculate the Earth's mass given that g at the North Pole is $\backslash (9\mathrm{.}830\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ and the radius of the Earth is $6371$ km from pole to pole.

$($b$)$ Compare your value with the accepted value of $\backslash (5\mathrm{.}979\backslash$times $10^\{24\}\backslash$mathrm$\{$~kg$\}\backslash ).$

$2.$ Two children are balanced on a seesaw $($teeter$-$toter$)$ of negligible mass. The first child has a mass of $22\mathrm{.}0$ kg and sits $1\mathrm{.}50$ m from the pivot. $($a$)$ If the second child has a mass of $35\mathrm{.}0$ kg $,$ how far is she from the pivot? $($b$)$ What is $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$n$\}\}\backslash ),$ the supporting force exerted by the $|$pivot$?$

$3.$ Consider a father pushing a playground merry$-$go$-$round with a force of $200$ N at the edge of the $\backslash (50\mathrm{.}0-\backslash$mathrm$\{$kg$\}\backslash )$ merry$-$go$-$round, which has a $2\mathrm{.}00$ m radius. Calculate the angular acceleration produced when an $\backslash (18\mathrm{.}0-\backslash$mathrm$\{$kg$\}\backslash )$ child sits $1\mathrm{.}25$ m away from the center. Consider the merry$-$go$-$round as a uniform disk with negligible friction.