The $12-$ lb sphere and $9-$ lb block $($shown in section$)$ are secured to the arm of negligible mass which rotates in the vertical plane about a horizontal axis at $\backslash ($ O $\backslash ).$ The $5-$lb plug is released from rest at $\backslash ($ A $\backslash )$ and falls into the recess in the block when the arm has reached the horizontal position. An instant before engagement, the arm has an angular velocity $\backslash (\backslash$omega$\_\{0\}=1\mathrm{.}5\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Determine the angular velocity $\backslash (\backslash$omega $\backslash )($positive if counterclockwise, negative if clockwise$)$ of the arm immediately after the plug has wedged itself in the block.