Let's break down the solution step by step: --- ## **a. Overall Free Body Diagram (FBD), Unknowns and Equilibrium Equations** ### **Step 1: Draw Overall FBD** - **Supports:** - **A:** Pin support (can exert force in both \(x\) and \(y\) directions: \(A_x, A_y\)) - **B:** Pin support (also exerts force in both \(x\) and \(y\) directions: \(B_x, B_y\)) - **External Loads:** - Downward force \(P\) at point \(C\) - Downward force \(Q\) at some point along \(CB\) #### **FBD Sketch Description** - Draw the two bars: \(AC\) and \(CB\) - Show all external forces and support reactions: - At \(A\): \(A_x\) (right), \(A_y\) (up) - At \(B\): \(B_x\) (right), \(B_y\) (up) - At \(C\): \(P\) (down) - At point along \(CB\): \(Q\) (down) ### **Step 2: Number of Unknown Reactions** - \(A_x, A_y\) (2 unknowns at A) - \(B_x, B_y\) (2 unknowns at B) - **Total unknowns:** \(4\) ### **Step 3: Equilibrium Equations Available** - For a rigid body in 2D: \(\sum F_x = \), \(\sum F_y = \), \(\sum M = \) - **Total equations:** \(3\) --- ## **b. FBDs for Members AC and CB; Unknowns and Equilibrium Equations** ### **Member AC FBD** - **Connections:** - Pin at A (\(A_x, A_y\)) - Pin at C (internal forces: \(C_{x1}, C_{y1}\)) - **Unknowns:** - At A: \(A_x, A_y\) (2) - At C: \(C_{x1}, C_{y1}\) (2) - **Total unknowns:** \(4\) - **Equilibrium equations:** \(3\) (for 2D: \(\sum F_x = \), \(\sum F_y = \), \(\sum M = \)) ### **Member CB FBD** - **Connections:** - Pin at C (internal: \(C_{x2}, C_{y2}\)) - Pin at B (\(B_x, B_y\)) - External load \(Q\) (downward) - **Unknowns:** - At B: \(B_x, B_y\) (2) - At C: \(C_{x2}, C_{y2}\) (2) - **Total unknowns:** \(4\) - **Equilibrium equations:** \(3\) --- ## **c. Write Out All Equilibrium Equations** ### **Overall Structure:** \[ \begin{align*} \sum F_x &= : \quad A_x + B_x = \\ \sum F_y &= : \quad A_y + B_y - P - Q = \\ \sum M_A &= : \quad -P \cdot a - Q \cdot b + B_y \cdot L_{AB} = \\ \end{align*} \] *(where \(a\) and \(b\) are distances from A to where P and Q act, \(L_{AB}\) is the length from A to B)* ### **Member AC:** \[ \begin{align*} \sum F_x &= : \quad A_x + C_{x1} = \\ \sum F_y &= : \quad A_y + C_{y1} = \\ \sum M_A &= : \quad -C_{y1} \cdot L_{AC,x} + C_{x1} \cdot L_{AC,y} = \\ \end{align*} \] *(using perpendicular distances from A to C for moment calculation)* ### **Member CB:** \[ \begin{align*} \sum F_x &= : \quad B_x + C_{x2} = \\ \sum F_y &= : \quad B_y + C_{y2} - Q = \\ \sum M_B &= : \quad -Q \cdot d + C_{y2} \cdot L_{CB,x} - C_{x2} \cdot L_{CB,y} = \\ \end{align*} \] *(where \(d\) is the distance from B to where Q acts)* --- ## **Summary Table** | Part | Unknowns | Equilibrium Equations | |--------------|----------|----------------------| | Overall | 4 | 3 | | Member AC | 4 | 3 | | Member CB | 4 | 3 | --- ### **Final Answers:** **a.** - Overall FBD has **4 unknown reactions** (\(A_x, A_y, B_x, B_y\)), **3 equilibrium equations**. **b.** - FBD for **AC**: 4 unknowns, 3 equations. - FBD for **CB**: 4 unknowns, 3 equations. **c.** - All equilibrium equations are listed above for each case. --- Let me know if you need the FBDs sketched or further clarification!

## **Step-by-Step Solution to the Problem** --- ## **a. Overall Free Body Diagram (FBD), Unknowns, and Equilibrium Equations** ### **Step 1: Draw Overall FBD** - **Supports:** - **A:** Pin support (exerts forces \(A_x\) and \(A_y\)) - **B:** Pin support (exerts forces \(B_x\) and \(B_y\)) - **External Loads:** - Force \(P\) acting downward at point \(C\) - Force \(Q\) acting downward at some point along \(CB\) #### **FBD Sketch Description** - **Sketch**: Two bars \(AC\) and \(CB\), with forces indicated. - **Reactions**: - At \(A\): \(A_x\) (horizontal), \(A_y\) (vertical) - At \(B\): \(B_x\) (horizontal), \(B_y\) (vertical) - At \(C\): \(P\) (downward) - At \(C'\) (along \(CB\)): \(Q\) (downward) ### **Step 2: Number of Unknown Reactions** - **At A**: \(A_x, A_y\) (2 unknowns) - **At B**: \(B_x, B_y\) (2 unknowns) - **Total Unknowns**: \(4\) ### **Step 3: Equilibrium Equations Available** - For 2D rigid body: - \(\sum F_x = 0\) - \(\sum F_y = 0\) - \(\sum M = 0\) - **Total Equations**: \(3\) --- ## **b. Free Body Diagrams for Members AC and CB; Unknowns and Equilibrium Equations** ### **Member AC FBD** - **Connections:** - Pin at A: \(A_x, A_y\) - Pin at C: internal forces \(C_{x1}, C_{y1}\) - **Unknowns:** - At A: \(A_x, A_y\) (2) - At C: \(C_{x1}, C_{y1}\) (2) - **Total Unknowns**: \(4\) - **Equilibrium Equations**: \(3\) ### **Member CB FBD** - **Connections:** - Pin at C: internal forces \(C_{x2}, C_{y2}\) - Pin at B: \(B_x, B_y\) - External load \(Q\) (downward) - **Unknowns:** - At B: \(B_x, B_y\) (2) - At C: \(C_{x2}, C_{y2}\) (2) - **Total Unknowns**: \(4\) - **Equilibrium Equations**: \(3\) --- ## **c. Write Out All Equilibrium Equations** ### **Overall Structure:** \[ \begin{align*} \sum F_x &= : \quad A_x + B_x = 0 \\ \sum F_y &= : \quad A_y + B_y - P - Q = 0 \\ \sum M_A &= : \quad -P \cdot a - Q \cdot b + B_y \cdot L_{AB} = 0 \\ \end{align*} \] *(where \(a\) and \(b\) are distances from A to where P and Q act, and \(L_{AB}\) is the length from A to B)* ### **Member AC:** \[ \begin{align*} \sum F_x &= : \quad A_x + C_{x1} = 0 \\ \sum F_y &= : \quad A_y + C_{y1} = 0 \\ \sum M_A &= : \quad -C_{y1} \cdot L_{AC,x} + C_{x1} \cdot L_{AC,y} = 0 \\ \end{align*} \] *(using perpendicular distances from A to C for moment calculation)* ### **Member CB:** \[ \begin{align*} \sum F_x &= : \quad B_x + C_{x2} = 0 \\ \sum F_y &= : \quad B_y + C_{y2} - Q = 0 \\ \sum M_B &= : \quad -Q \cdot d + C_{y2} \cdot L_{CB,x} - C_{x2} \cdot L_{CB,y} = 0 \\ \end{align*} \] *(where \(d\) is the distance from B to where Q acts)* --- ## **Summary Table** | Part | Unknowns | Equilibrium Equations | |--------------|----------|----------------------| | Overall | 4 | 3 | | Member AC | 4 | 3 | | Member CB | 4 | 3 | --- ## **Final Answers** **a.** - Overall FBD has **4 unknown reactions** (\(A_x, A_y, B_x, B_y\)), **3 equilibrium equations**. **b.** - FBD for **AC**: 4 unknowns, 3 equations. - FBD for **CB**: 4 unknowns, 3 equations. **c.** - All equilibrium equations are listed above for each case. --- This concludes the detailed solution. If clarification or further assistance is needed, please let me know!