The normal component $a _ { \mathrm { N } } ( t )$ of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = \sin ( 2 t ) \mathbf { i } - \cos ( 2 t ) \mathbf { j } + 2 t \mathbf { k }$ is

A)4
B) $\sqrt { 8 t ^ { 2 } + 1 }$
C) $\sqrt { 56 }$
D) $\sqrt { 65 }$
E) $\frac { 8 t } { \sqrt { 8 t ^ { 2 } + 1 } }$

Question

The normal component

a _ { \mathrm { N } } ( t )

of the acceleration of a particle moving along the curve

\mathbf { r } ( t ) = \sin ( 2 t ) \mathbf { i } - \cos ( 2 t ) \mathbf { j } + 2 t \mathbf { k }

is

A)4
B) $\sqrt { 8 t ^ { 2 } + 1 }$
C) $\sqrt { 56 }$
D) $\sqrt { 65 }$
E) $\frac { 8 t } { \sqrt { 8 t ^ { 2 } + 1 } }$

Quizplus · Accepted Answer

The normal component of acceleration is given by $a _ { \mathrm { N } } ( t ) = \frac { \left\| \mathbf { r } ^ { \prime } ( t ) \times \mathbf { r } ^ { \prime \prime } ( t ) \right\| } { \left\| \mathbf { r } ^ { \prime } ( t ) \right\| }$. So, $$\begin{split} \mathbf { r } ^ { \prime } ( t ) & = 2 \cos ( 2 t ) \mathbf { i } + 2 \sin ( 2 t ) \mathbf { j } + 2 \mathbf { k } \\ \mathbf { r } ^ { \prime \prime } ( t ) & = - 4 \sin ( 2 t ) \mathbf { i } + 4 \cos ( 2 t ) \mathbf { j } + 0 \mathbf { k } \\ \mathbf { r } ^ { \prime } ( t ) \times \mathbf { r } ^ { \prime \prime } ( t ) & = \left| \begin{array} { c c c } \mathbf { i } & \mathbf { j } & \mathbf { k } \\ 2 \cos ( 2 t ) & 2 \sin ( 2 t ) & 2 \\ - 4 \sin ( 2 t ) & 4 \cos ( 2 t ) & 0 \end{array} \right| \\ & = 8 \cos ^ { 2 } ( 2 t ) \mathbf { i } + 8 \sin ^ { 2 } ( 2 t ) \mathbf { j } - 8 \sin ( 2 t ) \cos ( 2 t ) \mathbf { k } \\ & = 8 \mathbf { i } + 8 \mathbf { j } - 8 \sin ( 4 t ) \mathbf { k } \\ \left\| \mathbf { r } ^ { \prime } ( t ) \times \mathbf { r } ^ { \prime \prime } ( t ) \right\| & = \sqrt { 64 + 64 + 64 \sin ^ { 2 } ( 4 t ) } = 8 \sqrt { 2 } \\ \left\| \mathbf { r } ^ { \prime } ( t ) \right\| & = \sqrt { 4 \cos ^ { 2 } ( 2 t ) + 4 \sin ^ { 2 } ( 2 t ) + 4 } = \sqrt { 8 } \\ a _ { \mathrm { N } } ( t ) & = \frac { 8 \sqrt { 2 } } { \sqrt { 8 } } = 4 \end{split}$$

The Normal Component aN(t)a _ { \mathrm { N } } ( t )aN​(t)

The Normal Component $a _ { \mathrm { N } } ( t )$