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A thin rod of mass M and length L is rotating counter-clockwise, with angular speed wo, about a fixed pivot point as seen in the figure below. Before m vo M, L Pivot Wo After 200 m y Pivot M,L X As the rod rotates into the position seen in the figure, it is struck by a right-traveling object of mass m moving with speed vo. (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 200. What is the final angular speed of the rod?

A thin rod of mass M and length L is rotating counter-clockwise, with angular speed wo, about a fixed pivot point as seen in the figure below. Before m vo M, L Pivot Wo After 200 m y Pivot M,L X As the rod rotates into the position seen in the figure, it is struck by a right-traveling object of mass m moving with speed vo. (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 200. What is the final angular speed of the rod?

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### 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
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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