Let’s break down the problem step by step: ### **Given:** - Four-bar linkage (see figure) - Link 2 rotates from \(\theta_2 = 30^\circ\) to \(\theta_2' = 120^\circ\) - Point Q moves from \(A\) to \(B\) along link 3 ### **Objective:** Find the absolute displacement of \(Q\). --- ### **Step 1: Position Vectors** Let: - \(O_2\) and \(O_4\) be the fixed pivots - Link 2: \(O_2A\) - Link 3: \(AB\) - Link 4: \(O_4B\) Let: - Length of \(O_2A = r_2\) - Length of \(AB = r_3\) - Length of \(O_4B = r_4\) - Length of \(O_2O_4 = r_1\) Assume \(r_2, r_3, r_4, r_1\) are known or arbitrary (since not specified). --- ### **Step 2: Initial and Final Position of Q** #### **At \(\theta_2 = 30^\circ\):** - Position of \(A\): \[ \vec{O_2A} = r_2 (\cos 30^\circ\,\mathbf{i} + \sin 30^\circ\,\mathbf{j}) \] - Position of \(B\): \[ \vec{AB} = r_3 (\cos \theta_3\,\mathbf{i} + \sin \theta_3\,\mathbf{j}) \] (where \(\theta_3\) is the angle of link 3 with respect to \(x\)-axis; to be determined from geometry) #### **At \(\theta_2' = 120^\circ\):** - Position of \(A'\): \[ \vec{O_2A'} = r_2 (\cos 120^\circ\,\mathbf{i} + \sin 120^\circ\,\mathbf{j}) \] - Position of \(B'\): \[ \vec{A'B'} = r_3 (\cos \theta_3'\,\mathbf{i} + \sin \theta_3'\,\mathbf{j}) \] (where \(\theta_3'\) is the new angle of link 3) --- ### **Step 3: Absolute Displacement of Q** If Q moves from \(A\) to \(B\) as the linkage moves from position 1 to position 2, the absolute displacement vector is: \[ \Delta \vec{Q} = \vec{OB'} - \vec{OA} \] But since \(Q\) moves from \(A\) to \(B\), the displacement is: \[ \Delta \vec{Q} = \vec{O_4B'} - \vec{O_2A} \] But if you need the **absolute displacement** (vector from starting point to final point in the global frame): - **Starting point**: Position of \(A\) at \(\theta_2 = 30^\circ\) - **Ending point**: Position of \(B'\) at \(\theta_2' = 120^\circ\) So: \[ \Delta \vec{Q} = \vec{O_4B'} - \vec{O_2A} \] --- ### **Step 4: Substitute Values** If you are given the lengths, substitute them. If not, leave the answer **in terms of \(r_1, r_2, r_3, r_4\)** and the angles. --- ### **Step 5: Final Formula** \[ \boxed{ \Delta \vec{Q} = [r_1 \mathbf{i}] + [r_4 (\cos\theta_4' \mathbf{i} + \sin\theta_4' \mathbf{j})] - [r_2 (\cos 30^\circ \mathbf{i} + \sin 30^\circ \mathbf{j})] } \] Where \(\theta_4'\) is the angle of link 4 at the new position, found from the four-bar linkage equations (loop closure equations). --- ### **Summary** - Write the position vectors for the initial and final positions. - Displacement is the difference between the final position (of \(B'\)) and initial position (of \(A\)). - For exact values, you would need the link lengths and use the loop-closure equation to solve for the unknown angles. Let me know if you have the lengths and want to compute a numerical answer!