ENGR216 2024 Dynamics Q A2 (ENGR5004)

ENGR216 2024 Dynamics Q A2Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2024 Dynamics Q A2[VALID]

Figure A2 shows a lightweight lever-type steel pushing machine. The crank $AB$ is rotating with $\omega_{OA} = 0.5$ rad/s, connecting the crank $AB$ to make the rod $O_1B$ rotate around the axis $O_1$ . Rod $DC$ is connected to slider block $C$ through a hinge, and the slider block $C$ can slide along rod $O_1B$ . $OA = r$ , $AB = \sqrt{3}\,r$ , $O_1B = \frac{2}{3}l$ , $r = 0.2$ m, $l = 1$ m. At this instant, $BC = \frac{4}{3}l$ , $AB$ and $DC$ are both horizontal, and the induced angle between $CB$ and the horizontal line is $60^\circ$ . Neglect the tangential acceleration of point $A$ .

Figure A2: crank OA, horizontal link AB, rod O1B pivoted at O1, slider C on O1B pushing horizontal rod DC.

Formulas you may need

Rotation about a fixed axis: $\mathbf{v} = \boldsymbol\omega\times\mathbf{r}$ , $\mathbf{a} = \boldsymbol\alpha\times\mathbf{r} - \omega^2\mathbf{r}$ , $a_t = \alpha r$ , $a_n = \omega^2 r$ (learn this)
Relative velocity on a rigid body $\mathbf{v}_B = \mathbf{v}_A + \boldsymbol\omega\times\mathbf{r}_{B/A}$ (learn this)
Relative acceleration on a rigid body $\mathbf{a}_B = \mathbf{a}_A + \boldsymbol\alpha\times\mathbf{r}_{B/A} - \omega^2\mathbf{r}_{B/A}$ (learn this)
Point sliding on a rotating body (rotating axes): $\mathbf{v}_C = \mathbf{v}_{C'} + \mathbf{v}_{rel}$ , $\mathbf{a}_C = \mathbf{a}_{C'} + \mathbf{a}_{rel} + 2\boldsymbol\omega\times\mathbf{v}_{rel}$ (learn this)
Cross products in the plane: $\mathbf{k}\times\mathbf{i} = \mathbf{j}$ , $\mathbf{k}\times\mathbf{j} = -\mathbf{i}$ (learn this)

(a)
Show that the velocity of point $B$ with respect to $A$ , $v_{BA}$ , is $\frac{\sqrt{3}}{30}\,\mathbf{j}$ m/s.
v_B/A (j component) =m/s
omega_O1B (CCW +) =rad/s
omega_AB (CCW +) =rad/s
[5]2:2
(b)
Determine the tangential acceleration and angular acceleration of point $B$ with respect to $O_1$ .
(a_B)_t (along v_B +) =m/s^2
alpha_O1B (CCW +) =rad/s^2
[10]
(c)
Determine the velocity of the slider block $C$ .
v_C (right +) =m/s
[5]2:2
(d)
Determine the acceleration of the slider block $C$ .
a_C (right +) =m/s^2
[5]First