Ball hitting baseball bat (with no impulse)

[SnappySeudonym]

Homework Statement

Given the setup above, what distance x should the ball be away so that there is no impulse reaction at A?

Homework Equations

Conservation of Linear and Angular momentum.

The Attempt at a Solution

Conservation of linear momentum (←+)
(considering the bat alone)
M(v_g1) + ∑ F_xdt = M(v_g2)
M(v_g2 - (v_g1)) = ∑ F_xdt
(3/2)LM(ω₂ - ω₁) = ∑ F_xdt ⇒ (1)

Conservation of angular momentum (+⊃)
I_gω₁ + ∑ M_adt = I_gω₂ +mux
I_g = (1/12)M(3L)²
I_g = (3/4)L²M
(3/4)L²Mω₁ + ∑ (F_xdt)x = (3/4)L²Mω₂ + mux
using (1)
(3/4)L²Mω₁ +((3/2)LM(ω₂ - ω₁))x = (3/4)L²Mω₂ + mux
x((3/2)LM(ω₂ - ω₁ - mux) = (3/4)L²M(ω₂ - ω₁)
x = L/(2(1-mu)

I've seen this problem before, but not with the bat/rod moving at an initial angular speed, can anyone give some insight as to where my solution is wrong, Is it because a used inertia for the COG of the bar and not the end?

 

[haruspex]

Conservation of angular momentum (+⊃)
I_g = (1/12)M(3L)²
:
∑ (F_xdt)x

The first equation above appears to be taking moments about the centre of the rod (1/12), but the second expression is a moment about the suspension point (x).