A uniform rod of length `L` rests on a frictionless horizontal surface. The rod is pivoted about a fixed frictionless axis at one end. The rod is initially at rest. A bulled travelling parallel to the horizontal surface and perpendicular to the rod with speed `v` strikes the rod at its centre and becomes embedded in it. the mass of the bullet is one-sixth the mass of the rod.
(a). What is the final angular velocity of the rod?
(b). What is the ratio of the kinetic energy of the system after the collision to the kinetic energy of te bullet the o collision?

Explanation

To solve the problem, we will break it down into two parts as requested:

### Part (a): Finding the Final Angular Velocity of the Rod

1. **Identify the system before and after the collision:**
- Before the collision, the rod is at rest and the bullet is moving with velocity \( v \).
- After the collision, the bullet embeds into the rod, and they rotate together about the pivot.

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