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A uniform, rigid hoop of radius R and mass M is pivoted at a point A, on its perimeter. It is able to rotate (without friction) about point A in the vertical plane in which it lies. A small piece of clay of the same mass M is shot towards its center at a horizontal speed vo, sticking immediately to point B on the hoop. Point B is at the same horizontal level as O, the center of the hoop. As a result, the hoop (with the clay on it) turns around point A. (Icm =MR? for the hoop) (a) What is the moment of inertia of the hoop-clay system about point A? (b) Consider the hoop and the clay as one system. In the collision process, is the mechanical energy conserved? What about the linear momentum and the angular momentum? Give simple justification. (c) What is the angular speed of the system immediately after the clay got stuck at point B? (d) What is the minimum possible value of vo that enables the clay (stuck at point B) to reach above point A? A M. B.

A uniform, rigid hoop of radius R and mass M is pivoted at a point A, on its perimeter. It is able to rotate (without friction) about point A in the vertical plane in which it lies. A small piece of clay of the same mass M is shot towards its center at a horizontal speed vo, sticking immediately to point B on the hoop. Point B is at the same horizontal level as O, the center of the hoop. As a result, the hoop (with the clay on it) turns around point A. (Icm =MR? for the hoop) (a) What is the moment of inertia of the hoop-clay system about point A? (b) Consider the hoop and the clay as one system. In the collision process, is the mechanical energy conserved? What about the linear momentum and the angular momentum? Give simple justification. (c) What is the angular speed of the system immediately after the clay got stuck at point B? (d) What is the minimum possible value of vo that enables the clay (stuck at point B) to reach above point A? A M. B.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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### Physics Problem on Rotational Motion and Momentum #### Problem Statement: A uniform, rigid hoop of radius \( R \) and mass \( M \) is pivoted at a point \( A \) on its perimeter. It can rotate (without friction) about point \( A \) in the vertical plane in which it lies. A small piece of clay of the same mass \( M \) is shot towards its center at a horizontal speed \( v_0 \), sticking immediately to point \( B \) on the hoop. Point \( B \) is at the same horizontal level as \( O \), the center of the hoop. As a result, the hoop (with the clay on it) turns around point \( A \). \((I_{\text{cm}} = MR^2 \text{ for the hoop})\) **Diagram:** The provided diagram illustrates the scenario where: - \( A \) is the pivot point on the perimeter of a circular hoop. - \( M \) is the mass of the clay, moving towards the hoop with horizontal speed \( v_0 \). - \( B \) is the point on the hoop where the clay sticks. - The hoop is of radius \( R \) and the center of the hoop is \( O \). **Questions:** 1. What is the moment of inertia of the hoop-clay system about point \( A \)? 2. Consider the hoop and the clay as one system. In the collision process, is the mechanical energy conserved? What about the linear momentum and the angular momentum? Provide a simple justification. 3. What is the angular speed of the system immediately after the clay gets stuck at point \( B \)? 4. What is the minimum possible value of \( v_0 \) that enables the clay (stuck at point \( B \)) to reach above point \( A \)? **Detailed Diagram Description:** - **Circle Representation:** The circle represents the hoop with center \( O \). - **Points Marked:** \( A \), \( O \), and \( B \) are marked on the diagram. - **Clay Movement:** The clay mass \( M \) is shown moving horizontally towards point \( B \) on the hoop at a speed \( v_0 \). This problem engages with the principles of rotational dynamics and momentum conservation, illustrating critical concepts in physics.
Transcribed Image Text:### Physics Problem on Rotational Motion and Momentum #### Problem Statement: A uniform, rigid hoop of radius \( R \) and mass \( M \) is pivoted at a point \( A \) on its perimeter. It can rotate (without friction) about point \( A \) in the vertical plane in which it lies. A small piece of clay of the same mass \( M \) is shot towards its center at a horizontal speed \( v_0 \), sticking immediately to point \( B \) on the hoop. Point \( B \) is at the same horizontal level as \( O \), the center of the hoop. As a result, the hoop (with the clay on it) turns around point \( A \). \((I_{\text{cm}} = MR^2 \text{ for the hoop})\) **Diagram:** The provided diagram illustrates the scenario where: - \( A \) is the pivot point on the perimeter of a circular hoop. - \( M \) is the mass of the clay, moving towards the hoop with horizontal speed \( v_0 \). - \( B \) is the point on the hoop where the clay sticks. - The hoop is of radius \( R \) and the center of the hoop is \( O \). **Questions:** 1. What is the moment of inertia of the hoop-clay system about point \( A \)? 2. Consider the hoop and the clay as one system. In the collision process, is the mechanical energy conserved? What about the linear momentum and the angular momentum? Provide a simple justification. 3. What is the angular speed of the system immediately after the clay gets stuck at point \( B \)? 4. What is the minimum possible value of \( v_0 \) that enables the clay (stuck at point \( B \)) to reach above point \( A \)? **Detailed Diagram Description:** - **Circle Representation:** The circle represents the hoop with center \( O \). - **Points Marked:** \( A \), \( O \), and \( B \) are marked on the diagram. - **Clay Movement:** The clay mass \( M \) is shown moving horizontally towards point \( B \) on the hoop at a speed \( v_0 \). This problem engages with the principles of rotational dynamics and momentum conservation, illustrating critical concepts in physics.
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