Problem $5\mathrm{.}58$

The disk shown, has a weight of $20$ lb and is pinned at its center. Neglect the mass of the cord. If a vertical force of $50$ lb is applied to the cord wrapped around the disk's outer rim, the angular velocity $(\backslash (\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$sec$\}\backslash ))$ of the disk after $5$ seconds starting from rest is$.$

$($write the answer in nearest hundreds such as $8700$ or $1100$ or $400$ without comma for thousands$)$