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When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. As shown in the figure below, the arclength As between the angular positions 0, and 0, is given as As = rA8. Do you agree with this equation? rotational motion (a) In the limit where Að is very small, then As can be consider as a straight line. Is this true or false? (b) It follows from statement (a) that any points along the circular path there is a tangential velocity (v) that is always perpendicular to the radius of the rotating body. Hence, if we divide both sides of the equation As = rÃo by At, justify that we will get vy = rw. This result indicates that the direction of the particle's velocity is tangential to its circular path at each point. Most importantly, for example, this result tells us the relation- ship between the angular velocity of the wheel of the car and linear velocity of the car. Reference line - A0 At (e) If the angular velocity changes by A, then the rotating object's linear speed will change by Arg. Hence, we will have An = rAw. Is this a true statement? (d) If this changes takes place in some smallAt and if we divide both sides of the equation Ar = rAw by At, justify that we will get ae = ra, where az is called the tangential acceleration and a is the rigid body angular acceleration. Most importantly, for example, this result tells us the relationship between the angular acceleration of the wheel of the car and the linear acceleration of the car.

When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. As shown in the figure below, the arclength As between the angular positions 0, and 0, is given as As = rA8. Do you agree with this equation? rotational motion (a) In the limit where Að is very small, then As can be consider as a straight line. Is this true or false? (b) It follows from statement (a) that any points along the circular path there is a tangential velocity (v) that is always perpendicular to the radius of the rotating body. Hence, if we divide both sides of the equation As = rÃo by At, justify that we will get vy = rw. This result indicates that the direction of the particle's velocity is tangential to its circular path at each point. Most importantly, for example, this result tells us the relation- ship between the angular velocity of the wheel of the car and linear velocity of the car. Reference line - A0 At (e) If the angular velocity changes by A, then the rotating object's linear speed will change by Arg. Hence, we will have An = rAw. Is this a true statement? (d) If this changes takes place in some smallAt and if we divide both sides of the equation Ar = rAw by At, justify that we will get ae = ra, where az is called the tangential acceleration and a is the rigid body angular acceleration. Most importantly, for example, this result tells us the relationship between the angular acceleration of the wheel of the car and the linear acceleration of the car.

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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**Angular and Circular Motion: Understanding Relationships** 1. **Basic Concept**: When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. The arclength \( \Delta s \) between angular positions \( \theta_1 \) and \( \theta_2 \) is given by the equation \( \Delta s = r \Delta \theta \). 2. **Assessment of Equation**: - (a) In the limit where \( \Delta \theta \) is very small, \( \Delta s \) can be considered as a straight line. True or false? - (b) Any point along the circular path has a tangential velocity (\( v_t \)) perpendicular to the radius. By dividing both sides of \( \Delta s = r \Delta \theta \) by \( \Delta t \), we get \( v_t = r \omega \), showing the relationship between angular velocity and linear velocity. 3. **Change in Angular Velocity**: - (c) If angular velocity changes by \( \Delta \omega \), linear speed changes by \( \Delta v_t \). Hence, \( \Delta v_t = r \Delta \omega \). Is this true? 4. **Angular and Tangential Acceleration**: - (d) If changes occur in some small \( \Delta t \), divide \( \Delta v_t = r \Delta \omega \) by \( \Delta t \) to get \( a_t = r \alpha \). Here, \( a_t \) is tangential acceleration, and \( \alpha \) is angular acceleration. 5. **Centripetal Acceleration**: - (e) In rotational motion, centripetal acceleration (\( a_c \)) accompanies tangential acceleration. For a child on a merry-go-round, this exists with the tangential component. The equation for centripetal acceleration is \( a_c = \frac{v^2}{r} \). The force direction is towards the center, expressed as \( F_c = m a_c \) or \( F_c = \frac{mv^2}{r} \). **Diagram**: - The diagram shows a circle illustrating rotational motion with angular displacement \( \Delta \theta \) and an arc length \( \Delta s \). Vectors indicate tangential velocity and angular velocity \( \omega \) around a reference line.
Transcribed Image Text:**Angular and Circular Motion: Understanding Relationships** 1. **Basic Concept**: When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. The arclength \( \Delta s \) between angular positions \( \theta_1 \) and \( \theta_2 \) is given by the equation \( \Delta s = r \Delta \theta \). 2. **Assessment of Equation**: - (a) In the limit where \( \Delta \theta \) is very small, \( \Delta s \) can be considered as a straight line. True or false? - (b) Any point along the circular path has a tangential velocity (\( v_t \)) perpendicular to the radius. By dividing both sides of \( \Delta s = r \Delta \theta \) by \( \Delta t \), we get \( v_t = r \omega \), showing the relationship between angular velocity and linear velocity. 3. **Change in Angular Velocity**: - (c) If angular velocity changes by \( \Delta \omega \), linear speed changes by \( \Delta v_t \). Hence, \( \Delta v_t = r \Delta \omega \). Is this true? 4. **Angular and Tangential Acceleration**: - (d) If changes occur in some small \( \Delta t \), divide \( \Delta v_t = r \Delta \omega \) by \( \Delta t \) to get \( a_t = r \alpha \). Here, \( a_t \) is tangential acceleration, and \( \alpha \) is angular acceleration. 5. **Centripetal Acceleration**: - (e) In rotational motion, centripetal acceleration (\( a_c \)) accompanies tangential acceleration. For a child on a merry-go-round, this exists with the tangential component. The equation for centripetal acceleration is \( a_c = \frac{v^2}{r} \). The force direction is towards the center, expressed as \( F_c = m a_c \) or \( F_c = \frac{mv^2}{r} \). **Diagram**: - The diagram shows a circle illustrating rotational motion with angular displacement \( \Delta \theta \) and an arc length \( \Delta s \). Vectors indicate tangential velocity and angular velocity \( \omega \) around a reference line.
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