A kid of mass M stands at the edge of a platform of radius R which can be freely rotated about its axis. The moment of inertia of the platform is I . The system is at rest when a friend throws as ball of mass m and the kid catches it. If the velocity of the ball is v horizontally along the tangent to the edge of the platform when it was caught by the kid find the angular speed of the platform after the event.

Explanation

To solve the problem, we will apply the principle of conservation of angular momentum. Here are the steps to find the angular speed of the platform after the kid catches the ball:

### Step 1: Understand the System
Initially, the system consists of:
- A kid of mass \( M \) standing at the edge of a platform of radius \( R \).
- The platform has a moment of inertia \( I \).
- The system is at rest, meaning the initial angular momentum is zero.

### Step 2: Identify the Initial Angular Momentum
Before the kid catches the ball, the ball of mass \( m \) is moving with a velocity \( v \) tangentially at the edge of the platform. The angular momentum \( L_i \) of the ball about the center of the platform is given by:
\[ L_i = m \cdot v \cdot R \]
where \( R \) is the radius of the platform.

### Step 3: Identify the Final Angular Momentum
After the kid catches the ball, the system (platform + kid + ball) rotates with an angular speed \( \omega \). The total moment of inertia \( I_f \) of the system after the event is:
\[ I_f = I + M \cdot R^2 + m \cdot R^2 \]
where \( M \cdot R^2 \) is the moment of inertia of the kid and \( m \cdot R^2 \) is the moment of inertia of the ball (both at distance \( R \) from the axis).

### Step 4: Apply Conservation of Angular Momentum
According to the conservation of angular momentum:
\[ L_i = L_f \]
where \( L_f \) is the final angular momentum of the system. Thus, we have:
\[ m \cdot v \cdot R = (I + M \cdot R^2 + m \cdot R^2) \cdot \omega \]

### Step 5: Solve for Angular Speed \( \omega \)
Rearranging the equation to solve for \( \omega \):
\[ \omega = \frac{m \cdot v \cdot R}{I + M \cdot R^2 + m \cdot R^2} \]

### Final Answer
The angular speed of the platform after the event is:
\[ \omega = \frac{m \cdot v \cdot R}{I + (M + m) \cdot R^2} \]