A stick of length L and mass M1 is in free space (no gravity) and not rotating. A point mass m2 has initial velocity v heading in a trajectory perpendicular to the stick. The mass has a perfectly inelastically collision a distance b from the center of the stick. Find the velocity of the center of mass and the final angular velocity.
A stick of length L and mass M1 is in free space (no gravity) and not rotating. A point mass m2 has initial velocity v heading in a trajectory perpendicular to the stick. The mass has a perfectly inelastically collision a distance b from the center of the stick. Find the velocity of the center of mass and the final angular velocity.
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A stick of length L and mass M1 is in free space (no gravity) and not rotating. A point mass m2 has
initial velocity v heading in a trajectory perpendicular to the stick. The mass has a perfectly inelastically
collision a distance b from the center of the stick. Find the velocity of the center of mass and the final
![### Problem 5: Inelastic Collision with a Stick
A stick of length \( L \) and mass \( M_1 \) is in free space (no gravity) and not rotating. A point mass \( m_2 \) has an initial velocity \( v \) heading in a trajectory perpendicular to the stick. The mass has a perfectly inelastic collision at a distance \( b \) from the center of the stick. Find the velocity of the center of mass and the final angular velocity.
#### Diagram Explanation:
The provided diagram is a simple graphical representation of the problem setup:
1. **Stick's Center:** The vertical stick is indicated, with its center marked along its length \( L \).
2. **Colliding Mass:** A small mass \( m_2 \) with an initial velocity \( v_i \) is shown approaching and colliding perpendicular to the stick at a distance \( b \) from the center.
#### Solution Strategy:
To solve this problem, we need to consider the following concepts:
- Conservation of Linear Momentum
- Conservation of Angular Momentum
#### Step-by-Step Solution:
1. **Conservation of Linear Momentum:**
Before the collision:
\[
\text{Initial Momentum} = m_2 \cdot v_i
\]
After the collision:
\[
\text{Final Momentum} = (M_1 + m_2) \cdot v_f
\]
By conservation of linear momentum:
\[
m_2 \cdot v_i = (M_1 + m_2) \cdot v_f \implies v_f = \frac{m_2 \cdot v_i}{M_1 + m_2}
\]
Here, \( v_f \) is the velocity of the center of mass of the stick and mass system after collision.
2. **Conservation of Angular Momentum:**
Consider the point of collision at \( b \) from the center. The initial angular momentum relative to the center of the stick is:
\[
\text{Initial Angular Momentum} = m_2 \cdot v_i \cdot b
\]
After the collision, the angular momentum is given by:
\[
\text{Final Angular Momentum} = I_{total} \cdot \omega
\]
where, \( I_{total}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0efe3c25-c766-416b-8e92-1fb25d7204be%2Fe92d6ea9-855d-4bd7-8db5-19acdd5e2a78%2F0h2hbd8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 5: Inelastic Collision with a Stick
A stick of length \( L \) and mass \( M_1 \) is in free space (no gravity) and not rotating. A point mass \( m_2 \) has an initial velocity \( v \) heading in a trajectory perpendicular to the stick. The mass has a perfectly inelastic collision at a distance \( b \) from the center of the stick. Find the velocity of the center of mass and the final angular velocity.
#### Diagram Explanation:
The provided diagram is a simple graphical representation of the problem setup:
1. **Stick's Center:** The vertical stick is indicated, with its center marked along its length \( L \).
2. **Colliding Mass:** A small mass \( m_2 \) with an initial velocity \( v_i \) is shown approaching and colliding perpendicular to the stick at a distance \( b \) from the center.
#### Solution Strategy:
To solve this problem, we need to consider the following concepts:
- Conservation of Linear Momentum
- Conservation of Angular Momentum
#### Step-by-Step Solution:
1. **Conservation of Linear Momentum:**
Before the collision:
\[
\text{Initial Momentum} = m_2 \cdot v_i
\]
After the collision:
\[
\text{Final Momentum} = (M_1 + m_2) \cdot v_f
\]
By conservation of linear momentum:
\[
m_2 \cdot v_i = (M_1 + m_2) \cdot v_f \implies v_f = \frac{m_2 \cdot v_i}{M_1 + m_2}
\]
Here, \( v_f \) is the velocity of the center of mass of the stick and mass system after collision.
2. **Conservation of Angular Momentum:**
Consider the point of collision at \( b \) from the center. The initial angular momentum relative to the center of the stick is:
\[
\text{Initial Angular Momentum} = m_2 \cdot v_i \cdot b
\]
After the collision, the angular momentum is given by:
\[
\text{Final Angular Momentum} = I_{total} \cdot \omega
\]
where, \( I_{total}
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