A horizontally oriented uniform disc of mass `M` and radius R rotates freely about a stationary vertical axis passing through its centre. The disc has a radial guide along which can slide without friction a small body of mass `m`. A light thread running down through the hollow axle of the disc is tied to the body initially the body was located at the edge of the disc and the whole system rotated with ann angular velocity `omega_(0)`. Then by means of a force `F` applied to the lower and of the thread the body was slowly pulled to the rotation axis. find:
(a). The angular velocity of the system in its final state.
(b). The work performed by the force F.

Explanation

To solve this problem, we will use the principles of conservation of angular momentum and the work-energy theorem. Let's break it down step by step.

### Step 1: Understand the system
We have a uniform disc of mass \( M \) and radius \( R \) that is rotating with an initial angular velocity \( \omega_0 \). A small body of mass \( m \) is initially at the edge of the disc and can slide towards the center along a radial guide. We need to find the final angular velocity of the system when the body reaches the center and the work done by the force \( F \).

### Step 2: Calculate the initial moment of inertia
The initial moment of inertia \( I_i \) of the system can be calculated as the sum of the moment of inertia of the disc and the moment of inertia of the mass \( m \) at the edge of the disc.

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