(a) A light, rigid rod of length { = 1.00 m joins two particles, with masses m, = 4.00 kg and m, = 3.00 kg, at its ends. The combination rotates in the xy-plane about a pivot through the center of the rod (see figure below). Determine the angular momentum of the system about the origin when the speed of each particle is 4.00 m/s. (Enter the magnitude to at least two decimal places in kg - m2/s.) magnitude 7 Note that the velocity of each particle is perpendicular to the vector giving its position relative to the origin, so the cross product is particularly simple in this case. kg · m/s direction (b) What If? What would be the new angular momentum of the system (in kg - m/s) if each of the masses were instead a solid sphere 15.0 cm in diameter? (Round your answer to at least two decimal places.) 18.641 x kg - m?/s
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.
![**Angular Momentum Analysis**
**(a)** A light, rigid rod of length \( \ell = 1.0 \, \text{m} \) joins two particles with masses \( m_1 = 4.00 \, \text{kg} \) and \( m_2 = 3.00 \, \text{kg} \) at its ends. The combination rotates in the xy-plane about a pivot through the center of the rod. Determine the angular momentum of the system about the origin when the speed of each particle is \( 4.00 \, \text{m/s} \). (Enter the magnitude to at least two decimal places in kg · m²/s.)
- A diagram illustrates a rod with particles \( m_1 \) and \( m_2 \) on either end, rotating counterclockwise around the origin in the xy-plane. The vectors are perpendicular to the line joining the origin to the particles.
1. **Magnitude:** [User Input: Incorrect Example Provided: "7"]
2. **Direction:** [Choice of directions, e.g., -x]
*Note:* The velocity of each particle is perpendicular to the vector giving its position relative to the origin, so the cross product calculation is simplified.
**(b) What If?**
What would be the new angular momentum of the system (in kg · m²/s) if each of the masses were instead a solid sphere with a diameter of 15.0 cm? (Round your answer to at least two decimal places.)
- **User Input for New Angular Momentum:** [Incorrect Example Provided: "18.641"]
**Explanation:**
The rigid rod system's angular momentum depends on the mass distribution and the velocity of the particles. When analyzing the effect of changing the mass shape to a solid sphere, consider the change in moment of inertia. The input fields and corrective feedback offer insights into precise calculations involving cross products in rotational motion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F251d1f23-479f-46f1-ad40-8e7a3447a718%2F2090130a-da0d-489f-a794-c0d0df6d66b4%2Frzjxwf5_processed.png&w=3840&q=75)
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