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 = 1/3mL^{2}.

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