Let the center of mass of the rod have downward acceleration a. Then
W-F=ma ---------(1)
The angular momentum equation implies
WL/2 = Iω, ----------(2)
where I=moment of inertia about an end = 1/3mL2.
One has ω=a/(L/2),
a relation that is true for small ω, i.e short times.
From (1) and (2),
F=W/4