A bucket of mass 8 kg is supported by a light rope wound around a solid wooden cylinder of mass 12 kg and radius 20 cm free to rotate about its axis. A man holding the free end of the rope, with the bucket and the cylinder at rest initially, lets go the bucket freely downwards in a well 50 m deep. Neglecting friction, obtain the speed of the bucket and the angular speed of the cylihder just before the bucket enters water.

Explanation

To solve the problem, we will use the principle of conservation of energy. The initial potential energy of the bucket will be converted into kinetic energy of the bucket and rotational kinetic energy of the cylinder as the bucket falls.

### Step-by-step Solution:

1. **Identify the given values:**
- Mass of the bucket, \( m_b = 8 \, \text{kg} \)
- Mass of the cylinder, \( m_c = 12 \, \text{kg} \)
- Radius of the cylinder, \( r = 0.2 \, \text{m} \)
...