A playground merry$-$go$-$round has a radius of $\backslash ($ R$=4\mathrm{.}0\backslash$mathrm$\{$~m$\}\backslash )$ and has a moment of inertia $\backslash ($ I$\_\{\backslash$text $\{$com $\}\}=\backslash )\backslash (7\mathrm{.}0\backslash$times $10\_\{3\}\backslash$mathrm$\{$kgm$\}^\{2\}\backslash )$ about an axis passing through the center of mass. There is negligible friction about its vertical axis. Two children each of mass $\backslash ($ m$=25\backslash$mathrm$\{$~kg$\}\backslash )$ were standing on opposite sides a distance $\backslash ($ r$\_\{0\}=3\mathrm{.}0\backslash$mathrm$\{$~m$\}\backslash )$ from the central axis. The merry$-$go$-$round is initially at rest. A person on the ground applied a constant tangential force of $\backslash ($ F$=250\backslash$mathrm$\{$~N$\}\backslash )$ at the rim of merry$-$go$-$round for a time of $10$ s $.$

$($a$)$ What was the angular acceleration of the merry$-$go$-$round?

$($b$)$ What was the angular velocity of the merry$-$go$-$round when the person stopped applying the force?

$($c$)$ What average power did the person put out while pushing the merry$-$go$-$round?

$($d$)$ What was the rotational kinetic energy of the merry$-$go$-$round when the person stopped applying the force?

The two children then walked inward and stop a distance of $\backslash ($ r$\_\{1\}=1\mathrm{.}0\backslash$mathrm$\{$~m$\}\backslash )$ from the central axis of the merry$-$go$-$round.

$($e$)$ What was the angular velocity of the merry$-$go$-$round when the children reached their final position?

$($f$)$ What was the change in rotational kinetic energy of the merry$-$go$-$round when the

children reached their final position?