Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 12 - Rotation of a Rigid Body - Exercises and Problems - Page 350: 37

Answer

a) $-21.2\;\hat k$ b) $24\;\hat k$

Work Step by Step

a) We can use the right-hand rule to find the direction of the resultant vector but we also can use the direction of the angle which is negative here since the angle from $\vec A$ to $\vec B$ is clockwise. $$\vec A\times \vec B=AB\sin(-45)^\circ\;\hat k$$ We assume that the two vectors are in the $x$-$y$ plane. $$\vec A\times \vec B=-AB\sin45^\circ\;\hat k$$ where $\hat k$ is the $z$-direction and since it is negative, then its direction is out of the page toward you. We can test that also by using the the right-hand rule. $$\vec A\times \vec B=-(6)(5)\sin45^\circ=\color{red}{\bf -21.2}\;\hat k$$ --- b) The angle from $\vec C$ to $\vec D$ is counterclockwise, so it is a negative direction and the resultant vector is perpendicular to the page out toward you. In other words, it is in the positive $ z$-direction. $$\vec C\times \vec D=CD\sin90^\circ\;\hat k$$ $$\vec C\times \vec D=(6)(4)\sin90^\circ=\color{red}{\bf 24}\;\hat k$$
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