Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.5 Motion in 3-Space - Preliminary Questions - Page 744: 6



Work Step by Step

We have $$ \mathbf{a}(t)=a_{\mathbf{T}}(t) \mathbf{T}(t)+a_{\mathbf{N}}(t) \mathbf{N}(t) $$ and moreover, $v(t)=4$. Then $v'(t)=0$. Also, the circle of raduis $2$ has the curvature $\kappa(t)=\frac{1}{2}$, and $a_N=\kappa(t) v(t)^2$. Now, we have $$ \mathbf{a}(t)=0+\frac{1}{2} 4^2 \mathbf{N}(t)=8\mathbf{N}(t) $$ and hence $$\|\mathbf{a}(t)\|=8.$$
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