A heavy stick of length L = 3.3 m and mass M = 20 kg hangs from a low-friction axle. A bullet of mass m = 0.014 kg traveling at v = 117 m/s strikes near the bottom of the stick and quickly buries itself in the stick. Just after the impact, what is the angular speed ω of the stick (with the bullet embedded in it)? (Note that the center of mass of the stick has a speed ωL/2. The moment of inertia of a uniform rod about one end is (1/3)ML^2.)
Here's a picture for reference
2. Some background information
Okay so this was a homework problem we got two weeks ago (with different numerical values) and we were made to re-do it because the website we use to do homework on (webassign) was apparently marking this question wrong. I never did it the first time. I have a dilemma because I worked out this question and I got an answer of 0.0743rad/s BUT when I use this method for the homework problem from two weeks ago, I get the same answer as in the answer key. The answer key even gives the same formula I got when I worked out this question. The problem here is that, apparently this is wrong! Im not sure whats going on here, if anyone can confirm I've used the right method to solve this problem, that would be greatly appreciated!
3. Attempt at solution
Im sure I'm correct but to confirm it to myself I used three different methods to solve this problem, all which gave the same answer.
M=mass of rod
m=mass of bullet
(1/12)ML2 = Moment of inertia for rod about center
(1/3)ML2 = Moment of inertia for rod about one end
A=point about which rod rotates
There is no doubt in my mind that, initially, only the bullet has angular momentum about A
[Li]=initial angular momentum=Lmv
Now I've modeled the final state in three ways, first as the bullet and and center of mass of the rod having translational angular momentum about A, and the rod having rotational angular momentum about its center of mass
[Lf]=Lm(ωL)+(L/2)M(ωL/2)+(1/12)ML2ω <---This was the equation in the answer key
Next, I modeled it as the bullet and center of mass of rod connected to A by massless rods so that I can say the bullet and center of mass of rod have rotational angular momentum about A. Of course the rod must also have rotational angular momentum about its own center of mass as well. This equation turns out to be the same as the above equation.
Finally, I modeled it as the bullet having translational angular momentum about A and the rod having rotational angular momentum about A.
Solving for ω in any of these equations gives the EXACT same answer (to the last digit in my calculator) of 0.0743rad/s. I don't get how this can be wrong even though it apparently is, someone please help!