A 1.0 g bullet is fired into a 0.5 kg block attached to the end of a 0.6 m nonuniform rod of mass 0.5 kg. The block-rod-bullet system then rotates in the plane of the figure about a fixed axis at A (the figure shows a vertical rod labeled A at its top. A block is attached to its bottom end. The bullet flies into the block). The rotational inertia of the rod alone about the axis at A is 0.060 (kg)m^2. Treat the block as a particle. If the angular speed (ω) of the system about A just after impact is 4.5 rad/s, what is the bullet's speed just before impact.
I have the worked out solution and see that we need to use L = Iω and L = (linear momentum) x r.
The Attempt at a Solution
When I first tried this problem, I tried to do it as a linear momentum problem. I set it up as (mass of the bullet)(initial velocity of bullet) = (mass of the block, bullet and rod system)(velocity of the block, bullet, rod system). I then found the velocity of the block, bullet, rod system using V=ωR.
I understand that this is an angular momentum problem and that I should have used angular momentum equations to solve it. However, I tried to turn it into a linear momentum problem. My strategy was to use the w given to find v of the block/rod/bullet system after impact (using V=ωR). Since my strategy didn't yield the correct answer, I now see that my reasoning was flawed :).
One thing in particular that confuses me about all of this is that the actual solution requires that we take the linear momentum of the bullet and multiply it by the length of the rod...which seems like taking linear momentum and turning it into angular momentum. I therefore thought/hoped/prayed there was a way to turn angular momentum into linear momentum (by finding V from the w given)...in part because the L = r x p and L = Iω equations seem like magic whereas linear momentum equations seem to make sense.
Can someone please explain the error in my logic? While I'm sure that a completely adequate explanation requires an explanation of the cross product, I think I'm clinically incapable of understanding what it is...so if there's a conceptual way of explaining why I can't tackle this problem with linear momentum...and why the actual method of multiplying the bullet's linear momentum by R and equating it to Iω works, I'd be super grateful!