Required information

Problem $17\mathrm{.}035-$ The angular velocities of a falling combined bar$-$gear system

The $\backslash (3\mathrm{.}500-\backslash$mathrm$\{$kg$\}\backslash )$ uniform slender bar $\backslash ($ A B $\backslash )$ is connected to the $\backslash (3-\backslash$mathrm$\{$kg$\}\backslash )$ gear $\backslash ($ B $\backslash )$ that meshes with the stationary outer gear $\backslash ($ C $\backslash ).$ The centroidal radius of gyration of gear $\backslash ($ B $\backslash )$ is $30$ mm $.$ Assume that the system is released from rest in the position shown in the figure.

Problem $17\mathrm{.}035.$a $-$ The angular velocities of a falling combined bar$-$gear system

Determine the angular velocity of the bar as it passes through the vertical position.

The angular velocity of the bar as it passes through the vertical position is