Moment of thin Rod at pivot when acted upon by a Force

[Narwhalest]

This is of my own interest/ practice.

Homework Statement

A thin rod (of width zero, but not uniform) is pivoted freely at one end about the horizontal z axis , being free to swing in the xy plane (x horizontal, y vertically down). Its mass is m and its CM is a distance a from the pivot.
The rod is struck with a horizontal force F which delivers an impulse F Δt = ξ a distance b below the pivot.
Find the impulse η delivered to the pivot.


2. Homework Equations
F Δt = ξ @b
L = T × Δt = r⋅J
p = mv_CM
T = r × F
v = ωr

The Attempt at a Solution

So first I found the angular momentum in terms of impulse. L = Iω = b ξ (j×i = -k).
Using the relation, I solve for angular velocity: ω = bξ/I and find the linear momentum.

Now here's my trouble, my friend is telling me I should add impulse and momentum for the total impulse delivered to the pivot. But I don't understand why that would be true.

 

[TSny]

Now here's my trouble, my friend is telling me I should add impulse and momentum for the total impulse delivered to the pivot. But I don't understand why that would be true.

I don't think that's quite right.

Consider any system and suppose it has zero total linear momentum intitially. Then suppose a net external impulse J_net acts on the system. Let P_f be the final linear momentum of the system just after the impulse acts. What is the relation between J_net and P_f?

For your problem, how would you write J_net in terms of the impulses ξ and η? Remember, η is the impulse delivered to the pivot.