### Rotational Inertia and Point Masses**Concept Overview:**- The moment of inertia (also known as rotational inertia) of a point mass at a distance \( r \) from the axis of rotation is given by the formula \( I = mr^2 \).- For a set of point masses \( m_i \) at various distances \( r_i \) from the axis, the total moment of inertia is the sum \( I = \sum m_i r_i^2 \).**Problem:**1. **Situation:** - A baby’s toy is made up of six beads, each with mass \( M \), arranged on a ring of radius \( R \). The ring itself has negligible mass. - Determine the total moment of inertia about an axis going through the center and perpendicular to the page, treating the beads as point masses. - Repeat the process for a ring of radius \( R/2 \). - Simplify your answers.**Diagrams Explanation:**1. **Left Diagram:** - Shows a ring with a larger radius \( R \). - Six points (beads) are symmetrically placed around the circle. - The central axis is shown as a line perpendicular to the page.2. **Right Diagram:** - Shows a similar arrangement with a reduced radius \( R/2 \). - Again, six beads are symmetrically placed around the smaller circle.**Question:**- Which ring is easier to start (or stop) rotating? Explain.**Analysis:**- To determine which ring is easier to start or stop rotating, compare the moments of inertia calculated for both rings. A lower moment of inertia indicates that less torque is required to change the ring's rotational motion.**Conclusion:**- Completing the calculations as directed will lead to a comparison of moments of inertia for the two scenarios. Discuss how radius affects rotational inertia and the implications for starting and stopping the rotation of the rings.

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1. A baby's toy is made up of six beads, each of mass M on a ring of radius R. Assume the ring has negligible mass. Determine the total moment of inertia about the axis going through the center, perpendicular to the page. Treat the beads as point masses. Repeat for the ring of radius R/2. Simplify your answers. axis axis radius R/2 radius R Which ring is easier to start (or stop) rotating? Explain.

1. A baby's toy is made up of six beads, each of mass M on a ring of radius R. Assume the ring has negligible mass. Determine the total moment of inertia about the axis going through the center, perpendicular to the page. Treat the beads as point masses. Repeat for the ring of radius R/2. Simplify your answers. axis axis radius R/2 radius R Which ring is easier to start (or stop) rotating? Explain.

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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Concept explainers

Torque

Torque is the rotational equivalent of linear force in physics and mechanics. Based on the course of study, it is also known as the moment of force, rotational force, or turning effect.

Question

### Rotational Inertia and Point Masses

**Concept Overview:**
- The moment of inertia (also known as rotational inertia) of a point mass at a distance \( r \) from the axis of rotation is given by the formula \( I = mr^2 \).
- For a set of point masses \( m_i \) at various distances \( r_i \) from the axis, the total moment of inertia is the sum \( I = \sum m_i r_i^2 \).

**Problem:**
1. **Situation:**
- A baby’s toy is made up of six beads, each with mass \( M \), arranged on a ring of radius \( R \). The ring itself has negligible mass.
- Determine the total moment of inertia about an axis going through the center and perpendicular to the page, treating the beads as point masses.
- Repeat the process for a ring of radius \( R/2 \).
- Simplify your answers.

**Diagrams Explanation:**

1. **Left Diagram:**
- Shows a ring with a larger radius \( R \).
- Six points (beads) are symmetrically placed around the circle.
- The central axis is shown as a line perpendicular to the page.

2. **Right Diagram:**
- Shows a similar arrangement with a reduced radius \( R/2 \).
- Again, six beads are symmetrically placed around the smaller circle.

**Question:**
- Which ring is easier to start (or stop) rotating? Explain.

**Analysis:**
- To determine which ring is easier to start or stop rotating, compare the moments of inertia calculated for both rings. A lower moment of inertia indicates that less torque is required to change the ring's rotational motion.

**Conclusion:**
- Completing the calculations as directed will lead to a comparison of moments of inertia for the two scenarios. Discuss how radius affects rotational inertia and the implications for starting and stopping the rotation of the rings.

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