$(50$ pts $)$ The sliding door shown below has a mass of $50$ kg and a center of mass at point G $.$ The rollers at A and $B$ are massless and frictionless. Starting from rest, a motor applies a constant force F $,$ causing the door to move $2\mathrm{.}5$ m to the right in $3$ s $.$

a$.$ Determine the magnitude of $F$ and the vertical reaction at A and $B$ in the case above.

b$.$ Suppose both rollers get stuck and drag along the ground with a coefficient of kinetic friction of $0\mathrm{.}34.$ What is the magnitude of $F$ and the vertical reaction at A and $B$ in this case?