This Free Response Question includes the following concepts: Center of Mass of a System of Particles, Translational and Rotational Kinetic Energy, Conservation of Linear and Angular Momentum, Angular Momentum of a System of Particles and Rigid Objects and Moment of Inertia.

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Enjoy an EDpuzzle Lesson for this video.

Link: AP Physics C 1998 Released Exam
The problem is on pages 156 - 158 of this pdf.
The solution is on pages 182 - 184 of this pdf.
Please enjoy my solutions as well.

Content Times:
0:22 Describing the Problem
0:57 Part (a i)
3:58 Part (a ii)
5:01 Part (b)
5:48 Part (b i)
7:00 Part (b ii)
7:41 Part (b iii)
9:57 Part (b iv) Describing this part of the Problem
11:48 Part (b iv) Why is Angular Momentum Conserved?
13:16 Part (b iv) Initial Angular Momentum of the Rod System
13:34 Part (b iv) Initial Angular Momentum of the Clay
16:26 Part (b iv) Final Angular Momentum of the Whole System
17:41 Part (b iv) Final Moment of Inertia of the Whole System
19:07 Part (b iv) Back to Conservation of Angular Momentum
20:09 Part (b v)

Here is a Veritasium YouTube video that explains quite well the differences between parts (a) and (b) of this problem.  In other words, how the clay lump that hits off center can have a smaller change in kinetic energy than when it hits at the center of mass.