Chapter 11 - Vectors and Vector-Valued Functions - 11.9 Curvature and Normal Vectors - 11.9 Exercises - Page 852: 40

$${a_N} = 20{\text{ and }}{a_T} = 0$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {20\cos t,20\sin t,30t} \right\rangle \cr & {\text{Calculate }}{\bf{v}}\left( t \right){\text{, }}\left| {{\bf{v}}\left( t \right)} \right|{\text{ and }}{\bf{a}}\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \left\langle { - 20\sin t,20\cos t,30} \right\rangle \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {400{{\sin }^2}t + 400{{\cos }^2}t + 900} = \sqrt {1300} \cr & \left| {{\bf{v}}\left( t \right)} \right| = 10\sqrt {13} \cr & \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \left\langle { - 20\cos t, - 20\sin t,0} \right\rangle \cr & \cr & {\text{Find the components of acceleration: }}{a_N}{\bf{N}} + {a_T}{\bf{T}}, \cr & {\text{Where }}{a_N} = \kappa {\left| {\bf{v}} \right|^2} = \frac{{\left| {{\bf{v}} \times {\bf{a}}} \right|}}{{\left| {\bf{v}} \right|}}{\text{ and }}{a_T} = \frac{{{d^2}s}}{{d{t^2}}} = \frac{{{\bf{v}} \cdot {\bf{a}}}}{{\left| {\bf{v}} \right|}} \cr & {\text{Then}}{\text{,}} \cr} $$ \[{\mathbf{v}} \times {\mathbf{a}} = \left| {\begin{array}{*{20}{c}} {\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\ { - 20\sin t}&{20\cos t}&{30} \\ { - 20\cos t}&{ - 20\sin t}&0 \end{array}} \right|\] $$\eqalign{ & {\bf{v}} \times {\bf{a}} = 600\sin t{\bf{j}}\, - 600\cos t + \left( {400{{\sin }^2}t + 400{{\cos }^2}t} \right){\bf{k}} \cr & {\bf{v}} \times {\bf{a}} = 600\sin t{\bf{j}}\, - 600\cos t + 400{\bf{k}} \cr & \left| {{\bf{v}} \times {\bf{a}}} \right| = \sqrt {{{600}^2}{{\sin }^2}t + {{600}^2}{{\cos }^2}t + {{400}^2}} \cr & \left| {{\bf{v}} \times {\bf{a}}} \right| = \sqrt {520000} = 200\sqrt {13} \cr & \cr & {a_N} = \frac{{\left| {{\bf{v}} \times {\bf{a}}} \right|}}{{\left| {\bf{v}} \right|}} = \frac{{200\sqrt {13} }}{{10\sqrt {13} }} \cr & {a_N} = 20 \cr & \cr & and \cr & \cr & {\bf{v}} \cdot {\bf{a}} = \left\langle { - 20\sin t,20\cos t,30} \right\rangle \cdot \left\langle { - 20\cos t, - 20\sin t,0} \right\rangle \cr & {\bf{v}} \cdot {\bf{a}} = 400\sin t\cos t - 400\sin t\cos t + 0 \cr & {\bf{v}} \cdot {\bf{a}} = 0 \cr & {a_T} = \frac{{{\bf{v}} \cdot {\bf{a}}}}{{\left| {\bf{v}} \right|}} = 0 \cr & {a_T} = 0 \cr & \cr & {a_N} = 20{\text{ and }}{a_T} = 0 \cr} $$