Homework H$\mathrm{.}5.$J

Given: A homogeneous rectangular plate, having a mass of $m$ and side lengths of $b$ and $2b,$ moves within a horizontal plane. Corners C and D are constrained to move in two slots, with the two slots between perpendicular to each other. A spring is attached between corner C and a fixed point O located along the slot within which C moves. A force acts at corner D of the plate, with the force acting along the line of the slot within which D moves. The spring is known to be unstretched when $=0.$ The system is released from rest when $={}_0.$ Consider all surfaces to be smooth.

Find: Determine the angular speed of the plate when $=0.$

Use the following parameters in your analysis: ${}_0=36\mathrm{.}87,m=2kg,b=100mm,k=4\frac{N}{m}m$ and $F=50N.$