Relative Velocity & Acceleration - Statics and Dynamics

1. Using the velocity loop equation $\mathbf{v}_B = \mathbf{v}_A + \boldsymbol{\omega}_{AB}\times\mathbf{r}_{B/A}$, combined with the constraint that B moves perpendicular to O₂B (i.e., B's velocity is horizontal at this instant), find $\omega_{AB}$ (angular velocity of the coupler AB).
   Answer: $\omega_{AB} = -\frac{4}{\sqrt{3}}\,\text{rad/s} \approx -2.31\,\text{rad/s}$ (CW), obtained by enforcing the constraint that $v_{By} = 0$ and solving the $\hat{j}$-component of the relative velocity equation. When analyzing four-bar linkage velocities, your strategy should be to write the vector loop equation, express each cross product in component form, then apply kinematic constraints to isolate unknowns — one equation per unknown. Here, the coupler AB makes 30° above horizontal, so $\mathbf{r}_{B/A} = 0.5\cos30°\,\hat{i} + 0.5\sin30°\,\hat{j} = \frac{\sqrt{3}}{4}\hat{i} + \frac{1}{4}\hat{j}$ m. With $\boldsymbol{\omega}_{AB} = \omega_{AB}\hat{k}$, the cross product gives $\boldsymbol{\omega}_{AB}\times\mathbf{r}_{B/A} = \omega_{AB}(-\frac{1}{4}\hat{i} + \frac{\sqrt{3}}{4}\hat{j})$. Since O₂B is vertical, B can only move horizontally, meaning $v_{By} = 0$. Applying the $\hat{j}$-component of $\mathbf{v}_B = \mathbf{v}_A + \boldsymbol{\omega}_{AB}\times\mathbf{r}_{B/A}$: $0 = 2 + \omega_{AB}\cdot\frac{\sqrt{3}}{4}$, which yields $\omega_{AB} = -\frac{8}{\sqrt{3}} \cdot \frac{1}{2} = -\frac{4}{\sqrt{3}} \approx -2.31\,\text{rad/s}$. The negative sign confirms clockwise rotation. This is answer A, the correct choice. Answer B reaches the same magnitude but positive, reflecting a sign error — typically from reversing $\mathbf{r}_{B/A}$ to point from B to A instead of A to B, which flips the cross product components incorrectly. Answer C uses only the $\hat{i}$-component without enforcing $v_{By} = 0$, leaving one equation with two unknowns ($\omega_{AB}$ and $v_{Bx}$) and guessing a value — a fundamental setup error. Answer D simply divides $|\mathbf{v}_A|$ by the coupler length, ignoring both the angle of the coupler and the kinematic constraint on B's motion entirely. Your key takeaway: always apply the geometric constraint on B's velocity direction before solving — it's what makes the system determinate.

Start this quiz