Chapter 16 - Planar Kinematics of a Rigid Body - Section 16.7 - Relative-Motion Analysis: Acceleration - Problems - Page 383: 103

$$ \sim V_c=3.8636 \frac{\mathrm{m}}{\mathrm{s}} \leftarrow \\ \sim a_c=17.65 \frac{\mathrm{m}}{\mathrm{s}^2} \leftarrow \\ $$

\begin{array}{l} \vec{V}B=\vec{V}_A+W{A B} \hat{k} \vec{r}_{B / A} \\ \vec{V}_B=4 \hat{k}\left(0.5 \cos 45^{\circ} \hat{\imath}+0.5 \operatorname{sen} 45^{\circ} \hat{\jmath}\right) \\ \overrightarrow{V_B}=1.4142 \hat{\jmath}-1.4142 \hat{\imath} \\ \overrightarrow{a_B}=\vec{a}A+\alpha{A B} \hat{k} \vec{r}{B / A}-w{A B}^2 r_{B / A} \\ \overrightarrow{a_B}=6 \hat{k}\left(0.5 \cos ^4 45^{\circ} \hat{\imath}+0.5 \operatorname{sen} 45^{\circ} \hat{\jmath}\right)-(4)^2\left(0.5 \cos 45^{\circ} \hat{\imath}+0.5 \operatorname{sen} 45^{\circ} \hat{\jmath}\right) \\ \overrightarrow{a_B}=2.1213 \hat{\jmath}-2.1213 \hat{\imath}-5.6568 \hat{\imath}-5.6568 \hat{\jmath} \\ \overrightarrow{a_B}=-7.778 \hat{\imath}-3.5355 \hat{\jmath} \\ \overrightarrow{V_B}=\overrightarrow{V_C}+W_{B C} \hat{k} \overrightarrow{r_B / C} \\ \vec{V}B=V_C \hat{\imath}+W{B C} \hat{k}\left(-1 \cos 60^{\circ} \hat{\imath}+1 \operatorname{sen} 60^{\circ} \hat{\jmath}\right) \\ 1.4142 \hat{\jmath}-1.4142 \hat{\imath}=V_C \hat{\imath}-0.5 W_{B C} \hat{\jmath}-0.866 W_{B C} \hat{\imath} \\ \hat{\jmath}=\hat{\jmath} \\ 1.4142=-0.5 \omega_{B C} \\ \omega_{B C}=-2.8284 \\ \sim W_{B C}=2.8284 \frac{\mathrm{rad}}{\mathrm{s}} \circlearrowright \\ \hat{\imath}=\hat{\imath} \\ -1.4142=v_c-0.866 W_{B C} \\ V_c=-1.4142+0.866(-2.8284) \\ V_c=-3.8636 \\ \sim V_c=3.8636 \frac{\mathrm{m}}{\mathrm{s}} \leftarrow \\ \overrightarrow{a_B}=\overrightarrow{a_C}+\alpha_{B C} \hat{k} \vec{r}{B / C}-W{B C}^2 \vec{r}_{B / C} \\ -7.778 \hat{\imath}-3.5355 \hat{\jmath}=a_C \hat{\imath}+\alpha_{B C} \hat{k}\left(-1 \cos 60^{\circ} \hat{\imath}+1 \operatorname{sen} 60^{\circ} \hat{\jmath}\right)-(-2.8284)^2\left(-1 \cos 60^{\circ} \hat{\imath}+1 \operatorname{sen} 60^{\circ} \hat{\jmath}\right) \\ -7.778 \hat{\imath}-3.5355 \hat{\jmath}=a_c \hat{\imath}-0.5 \alpha_{B C} \hat{\jmath}-0.866 \alpha_{B c} \hat{\imath}+4 \hat{\imath}-6.928 \hat{\jmath} \\ \hat{\jmath}=\hat{\jmath} \\ -3.5355=-0.5 \alpha_{B C}-6.928 \\ 0.5 \alpha_{B C}=-3.3925 \\ \alpha_{B C}=-6.785 \\ \left.\sim \alpha_{B C}=6.785 \frac{\mathrm{rad}}{\mathrm{s}^2}\right.\circlearrowright \\ \hat{\imath}=\hat{\imath} \\ -7.778=a_c-0.866 \alpha_{B C}+4 \\ a_c=-7.778+0.866(-6.785)-4 \\ a_c=-17.65 \\ \sim a_c=17.65 \frac{\mathrm{m}}{\mathrm{s}^2} \leftarrow \\ \end{array}