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Although angular velocity and angular acceleration can be

University Physics | 13th Edition | ISBN: 9780321675460 | Authors: Hugh D. Young, Roger A. Freedman ISBN: 9780321675460 31

Solution for problem 8DQ Chapter 9

University Physics | 13th Edition

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University Physics | 13th Edition | ISBN: 9780321675460 | Authors: Hugh D. Young, Roger A. Freedman

University Physics | 13th Edition

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Problem 8DQ

Although angular velocity and angular acceleration can be treated as vectors, the angular displacement ???, despite having a magnitude and a direction, cannot. This is because ??? does not follow the commutative law of vector addition (Eq. 1.3). Prove this to yourself in the following way: Lay your physics textbook flat on the desk in front of you with the cover side up so you can read the writing on it. Rotate it through 90° about a horizontal axis so that the farthest edge comes toward you. Call this angular displacement ???1. Then rotate it by 90° about a vertical axis so that the left edge comes toward you. Call this angular displacement ???2. The spine of the book should now face you, with the writing on it oriented so quit you can read it. Now start over again but carry out the two rotations in the reverse order. Do you get a different result? That is, does ???1 + ???2 equal ???2 + ???1? Now repeat this experiment but this line with an angle of 1° rather than 90°. Do you think that the infinitesimal displacement obeys the commutative law addition and hence qualifies as a vector? If so, how is the direction of related to the direction of ?

Step-by-Step Solution:

Solution 8DQ Step 1: Rotation in 3-dimensions is not commutative. Which means that, you rotate a book about some axis by /2, and again rotated by some different axis by /2 and end up somewhere. Then you do the reverse by rotating in the reverse direction by keeping track of the axis by which you are rotating 1st and 2nd. By intuition you should reach at the point from where you have started. But you would end up in a different place. So, rotation is not commutative in 3 or higher dimensions. Step 2: Let’s consider an infinitesimal rotation about some arbitrary axis by some small angle . We always talk about the rotation of a vector, whose magnitude does not change by a rotation. There are many such things like the dot products of two vectors which remain invariant by the rotation of the coordinates or by the rotation of the vectors itself. The rotation can be expressed by an operator or a matrix which is denoted as R. The matrix must obey certain rules as, T 1 R = R (orthogonal) det R = ± 1 Where there are significance of +1 and -1 by the group theory.

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Chapter 9, Problem 8DQ is Solved
Textbook: University Physics
Edition: 13
Author: Hugh D. Young, Roger A. Freedman
ISBN: 9780321675460

The answer to “Although angular velocity and angular acceleration can be treated as vectors, the angular displacement ???, despite having a magnitude and a direction, cannot. This is because ??? does not follow the commutative law of vector addition (Eq. 1.3). Prove this to yourself in the following way: Lay your physics textbook flat on the desk in front of you with the cover side up so you can read the writing on it. Rotate it through 90° about a horizontal axis so that the farthest edge comes toward you. Call this angular displacement ???1. Then rotate it by 90° about a vertical axis so that the left edge comes toward you. Call this angular displacement ???2. The spine of the book should now face you, with the writing on it oriented so quit you can read it. Now start over again but carry out the two rotations in the reverse order. Do you get a different result? That is, does ???1 + ???2 equal ???2 + ???1? Now repeat this experiment but this line with an angle of 1° rather than 90°. Do you think that the infinitesimal displacement obeys the commutative law addition and hence qualifies as a vector? If so, how is the direction of related to the direction of ?” is broken down into a number of easy to follow steps, and 211 words. Since the solution to 8DQ from 9 chapter was answered, more than 291 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: University Physics, edition: 13. This full solution covers the following key subjects: angular, displacement, Now, direction, Read. This expansive textbook survival guide covers 26 chapters, and 2929 solutions. The full step-by-step solution to problem: 8DQ from chapter: 9 was answered by , our top Physics solution expert on 05/06/17, 06:07PM. University Physics was written by and is associated to the ISBN: 9780321675460.

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