Let's break down each part step by step: --- ## a. **Overall Free Body Diagram, Unknowns, and Equilibrium Equations** ### **Step 1: Draw the Overall Free Body Diagram (FBD)** - **At A**: Pin support → 2 reaction components: \(A_x\) and \(A_y\) **At B**: Pin support → 2 reaction components: \(B_x\) and \(B_y\) - **At C**: Internal pin (no external reaction, only internal member forces) - **External Forces**: P and Q are applied vertically at C - **Members**: AC and CB #### **Unknown Reactions** - \(A_x, A_y\) (at A) - \(B_x, B_y\) (at B) **Total unknowns:** 4 #### **Equilibrium Equations** - For a rigid body in 2D: 1. \(\sum F_x = \) 2. \(\sum F_y = \) 3. \(\sum M = \) **Total equations:** 3 --- ## b. **Free Body Diagrams for AC and CB** ### **Step 2: Draw FBD for Member AC** - **At A**: Pin - reactions \(A_x\), \(A_y\) - **At C**: Pin - internal forces (\(C_{AC,x}\), \(C_{AC,y}\)) - **External Force on AC**: Portion of P (if P is on AC) - **Unknowns for AC**: 4 (\(A_x, A_y, C_{AC,x}, C_{AC,y}\)) #### **Equilibrium Equations for AC** - \(\sum F_x = \) - \(\sum F_y = \) - \(\sum M_A = \) (or at any point) **Total equations:** 3 ### **Step 3: Draw FBD for Member CB** - **At B**: Pin - reactions \(B_x\), \(B_y\) - **At C**: Pin - internal forces (\(C_{CB,x}\), \(C_{CB,y}\)) - **External Force on CB**: Q (downward at C) - **Unknowns for CB**: 4 (\(B_x, B_y, C_{CB,x}, C_{CB,y}\)) #### **Equilibrium Equations for CB** - \(\sum F_x = \) - \(\sum F_y = \) - \(\sum M_B = \) (or at any point) **Total equations:** 3 --- ## c. **Write Out All Equilibrium Equations** ### **Step 4: Overall Structure Equilibrium Equations** #### **Sum of Forces in X:** \[ A_x + B_x = \] #### **Sum of Forces in Y:** \[ A_y + B_y - P - Q = \] #### **Sum of Moments about A:** \[ -B_y \cdot L_{AB} + P \cdot a + Q \cdot b = \] --- ### **Step 5: Member AC Equilibrium Equations** #### **Sum of Forces in X:** \[ A_x + C_{AC,x} = \] #### **Sum of Forces in Y:** \[ A_y + C_{AC,y} - P = \] #### **Sum of Moments about A (for AC):** \[ C_{AC,x} \cdot L_{AC} \cdot \sin(\theta_A) - C_{AC,y} \cdot L_{AC} \cdot \cos(\theta_A) + P \cdot a = \] --- ### **Step 6: Member CB Equilibrium Equations** #### **Sum of Forces in X:** \[ B_x + C_{CB,x} = \] #### **Sum of Forces in Y:** \[ B_y + C_{CB,y} - Q = \] #### **Sum of Moments about B (for CB):** \[ C_{CB,x} \cdot L_{CB} \cdot \sin(\theta_B) - C_{CB,y} \cdot L_{CB} \cdot \cos(\theta_B) + Q \cdot b = \] --- ## **Summary Table** | **Part** | **Unknowns** | **Equilibrium Equations** | |-----------------|--------------|---------------------------| | **Overall** | 4 | 3 | | **AC** | 4 | 3 | | **CB** | 4 | 3 | --- ## **Final Answers (Concise):** ### a. - **Overall FBD:** 4 unknown reactions (\(A_x, A_y, B_x, B_y\)); 3 equilibrium equations. ### b. - **AC FBD:** 4 unknowns (\(A_x, A_y, C_{AC,x}, C_{AC,y}\)); 3 equilibrium equations. - **CB FBD:** 4 unknowns (\(B_x, B_y, C_{CB,x}, C_{CB,y}\)); 3 equilibrium equations. ### c. - **Equilibrium Equations:** 1. \(\sum F_x = \) 2. \(\sum F_y = \) 3. \(\sum M = \) (about any point, such as A or B) **Write out all as above for each part.** --- **Let me know if you need the actual FBD sketches or further clarification!**

## Step-by-Step Solution --- ## a. **Overall Free Body Diagram, Unknowns, and Equilibrium Equations** ### **Step 1: Draw the Overall Free Body Diagram (FBD)** - **At A (Pin support)**: - Reaction components: \(A_x\) and \(A_y\) - **At B (Pin support)**: - Reaction components: \(B_x\) and \(B_y\) - **At C (Internal pin)**: - No external reaction, only internal member forces. - **External Forces**: - Forces \(P\) and \(Q\) acting vertically at C. - **Members**: - AC and CB. #### **Unknown Reactions** - \(A_x, A_y\) (at A) - \(B_x, B_y\) (at B) **Total unknowns:** 4 #### **Equilibrium Equations** - For a rigid body in 2D: 1. \(\sum F_x = 0\) 2. \(\sum F_y = 0\) 3. \(\sum M = 0\) **Total equations:** 3 --- ## b. **Free Body Diagrams for Members AC and CB** ### **Step 2: Draw FBD for Member AC** - **At A**: - Reactions \(A_x\), \(A_y\) - **At C**: - Internal forces \(C_{AC,x}\), \(C_{AC,y}\) - **External Force on AC**: - Component of \(P\) if applicable. #### **Unknowns for AC** - 4 (\(A_x, A_y, C_{AC,x}, C_{AC,y}\)) #### **Equilibrium Equations for AC** 1. \(\sum F_x = 0\) 2. \(\sum F_y = 0\) 3. \(\sum M_A = 0\) **Total equations:** 3 ### **Step 3: Draw FBD for Member CB** - **At B**: - Reactions \(B_x\), \(B_y\) - **At C**: - Internal forces \(C_{CB,x}\), \(C_{CB,y}\) - **External Force on CB**: - Force \(Q\) acting downward at C. #### **Unknowns for CB** - 4 (\(B_x, B_y, C_{CB,x}, C_{CB,y}\)) #### **Equilibrium Equations for CB** 1. \(\sum F_x = 0\) 2. \(\sum F_y = 0\) 3. \(\sum M_B = 0\) **Total equations:** 3 --- ## c. **Write Out All Equilibrium Equations** ### **Step 4: Overall Structure Equilibrium Equations** #### **Sum of Forces in X:** \[ A_x + B_x = 0 \] #### **Sum of Forces in Y:** \[ A_y + B_y - P - Q = 0 \] #### **Sum of Moments about A:** \[ -B_y \cdot L_{AB} + P \cdot a + Q \cdot b = 0 \] --- ### **Step 5: Member AC Equilibrium Equations** #### **Sum of Forces in X:** \[ A_x + C_{AC,x} = 0 \] #### **Sum of Forces in Y:** \[ A_y + C_{AC,y} - P = 0 \] #### **Sum of Moments about A (for AC):** \[ C_{AC,x} \cdot L_{AC} \cdot \sin(\theta_A) - C_{AC,y} \cdot L_{AC} \cdot \cos(\theta_A) + P \cdot a = 0 \] --- ### **Step 6: Member CB Equilibrium Equations** #### **Sum of Forces in X:** \[ B_x + C_{CB,x} = 0 \] #### **Sum of Forces in Y:** \[ B_y + C_{CB,y} - Q = 0 \] #### **Sum of Moments about B (for CB):** \[ C_{CB,x} \cdot L_{CB} \cdot \sin(\theta_B) - C_{CB,y} \cdot L_{CB} \cdot \cos(\theta_B) + Q \cdot b = 0 \] --- ## **Summary Table** | **Part** | **Unknowns** | **Equilibrium Equations** | |-----------------|--------------|---------------------------| | **Overall** | 4 | 3 | | **AC** | 4 | 3 | | **CB** | 4 | 3 | --- ## **Final Answers** ### a. - **Overall FBD:** 4 unknown reactions (\(A_x, A_y, B_x, B_y\)); 3 equilibrium equations. ### b. - **AC FBD:** 4 unknowns (\(A_x, A_y, C_{AC,x}, C_{AC,y}\)); 3 equilibrium equations. - **CB FBD:** 4 unknowns (\(B_x, B_y, C_{CB,x}, C_{CB,y}\)); 3 equilibrium equations. ### c. - **Equilibrium Equations:** - Overall: 1. \(\sum F_x = 0\) 2. \(\sum F_y = 0\) 3. \(\sum M = 0\) (about any point, such as A or B) **All equations written out for each member as above.**