# Gear System Let's solve gear system problem stated in your. ### **Problem Statement (Parapased):** Given: \( n_{13} = 650 \) rpm (ccw, as viewed from left) \( n_{16} = 750 \) rpm (ccw, as viewed from left) (Take ccw from left as positive) - **Find:** \( n_{12} \) and \( n_{15} \) --- ## **Step 1: Draw the Gear Train Graph** Below is a simplified diagram of the gear train for reference (based on the image): ``` T2 - 26 T5 - 20 | | | | T3 - 20 T4 - 40 | | | | T6 - 24 T1 - 100 ``` - Each gear in mesh rotates in the **opposite direction** to its neighbor. --- ## **Step 2: Solve for Unknown Speeds** ### **Gear Ratios and Direction** For two meshed gears \( A \) and \( B \): \[ \frac{n_A}{n_B} = -\frac{T_B}{T_A} \] (The negative sign indicates **opposite rotation**.) ### **Label the Gears** From the diagram: - Gear 1 (T1 = 100) - Gear 2 (T2 = 26) - Gear 3 (T3 = 20) - Gear 4 (T4 = 40) - Gear 5 (T5 = 20) - Gear 6 (T6 = 24) Let: - \( n_{12} \) = Speed of shaft with gears 1 & 2 - \( n_{13} \) = Speed of shaft with gears 1 & 3 (given) - \( n_{14} \) = Speed of shaft with gears 1 & 4 - \( n_{15} \) = Speed of shaft with gears 1 & 5 - \( n_{16} \) = Speed of shaft with gears 1 & 6 (given) #### **Path from Gear 3 to Gear 6** Let's follow the path and calculate step by step. #### **From n13 to n12 (through gears 3 & 2):** - \( n_{13} \to n_{12} \) via gears 3 and 2: \[ \frac{n_{13}}{n_{12}} = -\frac{T_2}{T_3} \] \[ n_{12} = -n_{13} \cdot \frac{T_3}{T_2} \] Plug in values: \[ n_{12} = -650 \cdot \frac{20}{26} \] \[ n_{12} = -650 \cdot .7692 = -500 \text{ rpm} \] - **Interpretation:** The negative sign means **clockwise (cw)** as viewed from the left. --- #### **From n16 to n15 (through gears 6 & 5):** \[ \frac{n_{16}}{n_{15}} = -\frac{T_5}{T_6} \] \[ n_{15} = -n_{16} \cdot \frac{T_6}{T_5} \] Plug in values: \[ n_{15} = -750 \cdot \frac{24}{20} \] \[ n_{15} = -750 \cdot 1.2 = -900 \text{ rpm} \] - **Interpretation:** The negative sign means **clockwise (cw)** as viewed from the left. --- ## **Final Answers** **Step 1 Example:** - \( n_{13} = 650 \) rpm (ccw), \( T_3 = 20 \), \( T_2 = 26 \) - Find \( n_{12} \): \[ n_{12} = -650 \cdot \frac{20}{26} = -500 \text{ rpm} \] **Step 2 Example:** - \( n_{16} = 750 \) rpm (ccw), \( T_6 = 24 \), \( T_5 = 20 \) - Find \( n_{15} \): \[ n_{15} = -750 \cdot \frac{24}{20} = -900 \text{ rpm} \] --- ### **Summary Table** | Gear | Speed (rpm) | Direction | |------|-------------|-----------| | n12 | -500 | cw | | n15 | -900 | cw | --- ### **Graph Reference**  *Alt text: Simplified gear train for problem reference.* --- ## **Final Answers** - \( \boxed{n_{12} = -500 \text{ rpm (cw from left)}} \) - \( \boxed{n_{15} = -900 \text{ rpm (cw from left)}} \) Let me know if you need further clarifications or a step-by-step with all intermediate gears!