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Transcribed Image Text:### Rotation Dynamics - Problem 3
**Problem Statement:**
A thin rod of mass \( M \) and length \( L \) is rotating counter-clockwise, with angular speed \( \omega_0 \), about a fixed pivot point as seen in the figure below.
#### Diagram Description:
- **Before Collision:**
The rod is positioned vertically along the y-axis. The system is described as follows:
- Mass \( m \) is an object moving to the right with a speed \( v_0 \).
- The rotation pivot is at the origin of the coordinate system (0,0).
- The rod's angular speed is \( \omega_0 \).
- **After Collision:**
- The mass \( m \) travels to the left with a speed \( 2v_0 \) after collision.
- The rod and its specifications (mass \( M \), length \( L \)) remain the same.
#### Questions:
(a) Using the coordinate system seen in the figure, what is the angular momentum vector of the rod-object system prior to the collision?
(b) Suppose, after the collision, the object travels to the left with speed \( 2v_0 \). What is the final angular speed of the rod?
#### Solution Overview:
- **Angular Momentum Analysis before Collision:**
- Consider contributions from both the rotating rod and the moving mass \( m \).
- **Angular Momentum Conservation:**
- Apply conservation principles to ascertain the final angular speed of the rod after the collision.
### Detailed Solution:
#### (a) Angular Momentum before Collision
The angular momentum \( \vec{L}_{\text{initial}} \) of the system before the collision is the sum of the angular momentum of the rod and the angular momentum of the object \( m \).
1. **Rod (Rotating about Pivot):**
- Moment of inertia \( I_{\text{rod}} = \frac{1}{3}ML^2 \)
- Angular momentum \( \vec{L}_{\text{rod}} = I_{\text{rod}} \cdot \omega_0 = \frac{1}{3}ML^2 \omega_0 \)
2. **Object \( m \) (Moving with speed \( v_0 \)):**
- Position vector \( \vec{r} = L \hat{i} \) (since it's moving along the x-axis)
- Linear
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