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$\begin{array}{l}{12-177 \text { . The car travels around the circular track such that }} \\ {\text { its transverse component is } \theta=\left(0.006 t^{2}\right) \text { rad, where } t \text { is in }} \\ {\text { seconds. Determine the car's radial and transverse }} \\ {\text { components of velocity and acceleration at the instant } t=4 \text { s. }}\end{array}$

$\theta=\left(0.006 t^{2}\right) r a d$

$t=4S$

$r=(400 \cos \theta) m$

$\theta=\left. 0.006 t^{2}\right|_{t=4} \Rightarrow \theta=0.096 rad=5.5^{\circ}$

$∴ \dot \theta=2*0.006t=\left.0.012 t\right|_{t=4}\Rightarrow ∴ \dot \theta=0.048 rad/s$

$\theta^{..}=0.012 \mathrm{rad} / \mathrm{s}^{2}$

$r=(400 \cos \theta) \Rightarrow ∴ \dot r=-400 \sin \theta \dot \theta \Rightarrow r^{..}=-400(\cos (\dot \theta)^2+\sin \theta \theta^{..})$

@ $\theta=0.096 \mathrm{rad}$

$∴ r=400 \mathrm{cos}(0.096)=398.158 \mathrm{m} \Rightarrow\dot r=-1.84037 \mathrm{m/s} \Rightarrow \mathrm{r^{..}}=1.377449 \mathrm{m/s}^{2}$

$v_{r}=\dot r=-1.84 \mathrm{m} / \mathrm{s}, v_{\theta}=r_ \theta=398.158(0 .048)=19.1 \mathrm{m/s}$

$∴ a_{r}=r^{..}- r(\dot \theta)^{2}=-1.377449-398.158(0.098)^{2}=-2.29 \mathrm{m} / \mathrm{s}^{2}$

$a_{\theta}=r _{\theta^{..}}+2 \dot r_{\dot \theta}=398 . 158(0.012)+2(-1.840 37)(0.048)=4.6 \mathrm{m} / \mathrm{s}^{2}$

$\begin{array}{l}{12-191 . \text { The arm of the robot moves so that } r=3 \text { ft is }} \\ {\text { constant, and its grip } A \text { moves along the path } z=(3 \sin 4 \theta) \text { ft, }} \\ {\text { where } \theta \text { is in radians. If } \theta=(0.5 t) \text { rad, where } t \text { is in seconds, }} \\ {\text { determine the magnitudes of the grip's velocity and }} \\ {\text { acceleration when } t=3 \text { s. }}\end{array}$

$r=3 f t$

$z=(3 \sin 4 \theta) f t$

$\theta=(0.5 t) rad$

@ $t=3 S$

$\theta=0.5 t\ , r=3 \ , z=3 \sin 2 t=-0.8382$

$∴\dot \theta=0.5 \quad , \dot r=0 \quad , \dot z=6 cos 2t=5.761$

$∴ \theta^{..}=0 \quad , r^{..}=0 \quad, z^{..}=-12 \sin 2 t=3.353$

${v_{r}=r^{.}=0 \quad , v_{\theta}=r_{\dot \theta}=3 * 0.5=1.5, \quad v_{z}=\dot z=5.761}$

$∴ V=\sqrt{v_{r}^{2}+v_{\theta}^{2}+v_{z}^{2}}=\sqrt{0^{2}+1.5^{2}+(5.761)^{2}}=5.95 \mathrm{ft} / \mathrm{s}$

$a_r=r^{..}-r {\dot \theta}^{2}=0-3(0.5)^{2}=-0.75, a_{\theta}=r \theta^{..}+2\dot r \dot \theta=0+0=0$

$a_ z=z^{..}=3.35{3} \Rightarrow a=\sqrt{a_{r}^{2}+a_{\theta}^{2}+a_{z}^{2}}=\sqrt{(-0.75)^{2}+0^{2}+(3.353)^{2}}$

$a=3.44 \mathrm{ft} / \mathrm{s}^ 2$