A wheel of moment of inertia `I_(1)` rotates about a vertical frictionless axle with an angular velocity `omega_(1)`. A second wheel of moment of inertia `I_2` and initially at rest drops onto the first wheel. The two wheels attain a common angular velocity of `omega_(2)` . (a) What is the value of `omega_(2)` ? (b) What is the ratio of the initial to the final rotational energy ?

Explanation

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (a): Finding the value of \( \omega_2 \)

1. **Understand the System**: We have two wheels. The first wheel has a moment of inertia \( I_1 \) and is rotating with an angular velocity \( \omega_1 \). The second wheel has a moment of inertia \( I_2 \) and is initially at rest.

2. **Apply Conservation of Angular Momentum**: Since there are no external torques acting on the system, we can apply the conservation of angular momentum. The initial angular momentum of the system is equal to the final angular momentum after the second wheel drops onto the first.

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