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To solve this problem, let's go through each part step by step.

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### Given Data:

- Mass of the rod, \( M = 10 \, \text{kg} \)
- Length of the rod, \( L = 2.0 \, \text{m} \)
- Mass of the ball, \( m = 20 \, \text{kg} \)
- Initial angle of the rod from vertical, \( \theta = 30^\circ \)
- Moment of inertia of the rod about the end: \( I_{\text{rod, end}} = \frac{1}{3}ML^2 \)

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### (a) Angular Speed of the Rod Before it Hits the Ball

1. **Conservation of Energy**: As the rod falls, potential energy is converted into rotational kinetic energy.

   - Initial potential energy of the rod’s center of mass (located at \( \frac{L}{2} \) from the pivot):
     \[
     U = M \cdot g \cdot \frac{L}{2} \cdot (1 - \cos \theta)
     \]
     where \( g \approx 9.81 \, \text{m/s}^2 \).

   - Rotational kinetic energy at the bottom (just before collision):
     \[
     K = \frac{1}{2} I_{\text{rod, end}} \omega^2
     \]

2. **Solving for Angular Speed \( \omega \)**:
   \[
   M \cdot g \cdot \frac{L}{2} \cdot (1 - \cos \theta) = \frac{1}{2} \cdot \frac{1}{3}ML^2 \cdot \omega^2
   \]

   Simplify to find \( \omega \):
   \[
   \omega = \sqrt{\frac{3g(1 - \cos \theta)}{L}}
   \]

   Substitute \( g = 9.81 \, \text{m/s}^2 \), \( L = 2 \, \text{m} \), and \( \theta = 30^\circ \) to get \( \omega \).

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### (b) Velocity of the Ball and Angular Velocity of the Rod After the Collision

Since the collision is elastic:

1. **Conservation of Angular Momentum**: Angular momentum about the pivot just before and after the collision should be conserved.

   - Angular momentum of the rod just before the collision:
     \[
     L_{\text{initial}} = I_{\text{rod, end}} \cdot \omega
     \]

   - After the collision, let \( v \) be the velocity of the ball and \( \omega' \) the new angular velocity of the rod.

   - The angular momentum of the ball (mass \( m \), velocity \( v \)) at a distance \( L \) from the pivot:
     \[
     L_{\text{ball}} = m \cdot v \cdot L
     \]

   - Angular momentum of the rod after the collision:
     \[
     L_{\text{rod, after}} = I_{\text{rod, end}} \cdot \omega'
     \]

   Using conservation of angular momentum:
   \[
   I_{\text{rod, end}} \cdot \omega = I_{\text{rod, end}} \cdot \omega' + m \cdot v \cdot L
   \]

2. **Conservation of Energy**: Since it’s an elastic collision, kinetic energy is also conserved.

   Set up equations for conservation of energy and solve for \( v \) and \( \omega' \).

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### (c) Maximum Angle Subtended by the Rod After the Collision

1. After the collision, the rod will have rotational kinetic energy corresponding to its new angular velocity \( \omega' \).
2. The rotational kinetic energy will convert to potential energy as the rod swings upward.
3. Using energy conservation:
   \[
   \frac{1}{2} I_{\text{rod, end}} \omega'^2 = M \cdot g \cdot \frac{L}{2} \cdot (1 - \cos \alpha)
   \]
   where \( \alpha \) is the maximum angle reached by the rod.

4. Solve for \( \alpha \) in terms of \( \omega' \).

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Would you like detailed calculations and solutions for each part?

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#### Further Questions

1. How would the solution change if the rod were released from a different angle?
2. What would be the effect if the collision were inelastic instead of elastic?
3. How does the length of the rod affect the angular speed and energy outcomes?
4. What happens to the rod’s motion if the mass of the ball is significantly increased?
5. How would air resistance affect the overall motion of the rod and ball?

#### Tip
In rotational dynamics problems, conserving angular momentum and energy is often key, especially in systems involving collisions and pivots.

A thin uniform rod of mass 10 kg and length 2.0 m is released from rest at an angle of 30° from the vertical. At the bottom of its swing it hits a small dense ball of mass 20 kg which is initially at rest on a frictionless surface. The moment of inertia of the rod about an axis through the end of the rod, perpendicular to its length is given by I_rod_end = (1/3)ML^2.

(a) What is the angular speed of the rod the instant before it hits the ball? (5 points)
(b) Find the velocity of the ball and the angular velocity of the rod the instant after the collision. Assume the collision is elastic. (10 pts.)
(c) What is the maximum angle subtended by the rod after the collision? (5 pts.)

Explore the rotational dynamics of a thin uniform rod released from rest at a 30° angle, colliding elastically with a dense ball. This problem covers energy and angular momentum conservation, including calculations of angular speed, ball velocity, and maximum angle subtended after collision. Suitable for undergraduate physics students.

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