A uniform rod of mass M and length L is suspended at a point (P) which is a distance L/3 from one end and is free to rotate frictionlessly about point P in a vertical plane. A small piece of clay of mass M/4 is thrown horizontally at the rod with a speed v; and it immediately sticks to the lower end of the rod. (Rod: Iem =ML?) 12 (a) What is Ip, the moment of inertia of the rod about point P? (b) Consider the rod and clay as one system. In the collision process, is the mechanical energy of the system conserved? What about the linear momentum and the angular momentum? Give simple justifications. (c) What is the angular speed of the system immediately after the collision? (d) What is the minimum speed v; of the clay that enables the rod to make complete revolutions about point P? M,L
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Physics Problem: Rotational Motion of a Rod-Pendulum System
**2. A uniform rod of mass \( M \) and length \( L \) is suspended at a point \((P)\) which is a distance \( \frac{L}{3} \) from one end and is free to rotate frictionlessly about point \( P \) in a vertical plane. A small piece of clay of mass \( \frac{M}{4} \) is thrown horizontally at the rod with a speed \( v_i \) and it immediately sticks to the lower end of the rod.**
\[
\text{(Rod: } I_{\text{cm}} = \frac{1}{12}ML^2\text{)}
\]
#### Questions:
**(a) What is \( I_P \), the moment of inertia of the rod about point \( P \)?**
**(b) Consider the rod and clay as one system. In the collision process, is the mechanical energy of the system conserved? What about the linear momentum and the angular momentum? Give simple justifications.**
**(c) What is the angular speed of the system immediately after the collision?**
**(d) What is the minimum speed \( v_i \) of the clay that enables the rod to make complete revolutions about point \( P \)?**
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### Detailed Explanation of the Diagram:
The accompanying diagram provides a visual representation of the described system:
1. **Rod and Suspension Point \( P \):**
- A vertical rod of mass \( M \) and length \( L \) is shown.
- The rod is suspended at a point \( P \), located at a distance \( \frac{L}{3} \) from one end of the rod.
2. **Clay and Initial Speed \( v_i \):**
- A small piece of clay with mass \( \frac{M}{4} \) is depicted moving horizontally towards the rod with an initial speed \( v_i \).
- The clay is illustrated as a small circle near the rod, indicating its motion towards the lower end of the rod.
3. **Post-Collision Scenario:**
- After the collision, the clay sticks to the lower end of the rod, setting the rod in rotational motion about point \( P \).
- The diagram shows an arrow in a circular path, representing the rotational movement initiated by the collision.
By analyzing this representation, students](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb397ef84-6eeb-40b9-aea4-22211e922ea0%2F945535df-c4db-41ff-834e-eea66f8316f3%2Ffvy1mds_processed.png&w=3840&q=75)
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