Let's break down the **gear forces problem step by step**: --- ## **Given Data** - **Normal diametral pitch** (\(P_n\)): 5 teeth/in - **Pressure angle** (\(\phi\)): 20° - **Helix angle** (\(\psi\)): 30° - **Transmitted load** (\(F_t\)): 800 lbf (tangential force) - **Gear 2 rotates clockwise about the y-axis** - **Gear 3 is an idler** We are to **find the forces exerted by gears 2 and 3 on their shafts**. --- ## **Step 1: Identify Forces on Helical Gears** Helical gears transmit three main forces: 1. **Tangential force (\(F_t\))**: Drives rotation. 2. **Radial force (\(F_r\))**: Points toward the center of the gear. 3. **Axial force (\(F_a\))**: Along the axis (due to helix angle). ### **Formulas** \[ F_r = F_t \tan\phi / \cos\psi \] \[ F_a = F_t \tan\psi \] Where: - \(\phi\) = pressure angle - \(\psi\) = helix angle --- ## **Step 2: Calculate the Forces** ### **Given:** - \(F_t = 800~\text{lbf}\) - \(\phi = 20^\circ\) - \(\psi = 30^\circ\) ### **Radial Force (\(F_r\)):** \[ F_r = F_t \frac{\tan\phi}{\cos\psi} \] \[ \tan 20^\circ = 0.3640 \] \[ \cos 30^\circ = 0.8660 \] \[ F_r = 800 \times \frac{0.3640}{0.8660} \] \[ F_r = 800 \times 0.4203 = \boxed{336.2~\text{lbf}} \] --- ### **Axial Force (\(F_a\)):** \[ F_a = F_t \tan\psi \] \[ \tan 30^\circ = 0.5774 \] \[ F_a = 800 \times 0.5774 = \boxed{461.9~\text{lbf}} \] --- ## **Step 3: Force Directions** - **Tangential (\(F_t\))**: In the direction of rotation (clockwise for gear 2). - **Radial (\(F_r\))**: Points to the center of the gear. - **Axial (\(F_a\))**: Along the shaft (direction depends on helix and rotation). --- ## **Step 4: Forces on Gears 2 and 3** Since **gear 3 is an idler**, it transmits the same load but reverses direction for tangential and axial forces. ### **Gear 2 (Driving Gear):** - **Tangential (\(F_t\))**: 800 lbf (clockwise) - **Radial (\(F_r\))**: 336.2 lbf (toward gear center) - **Axial (\(F_a\))**: 461.9 lbf (along shaft, direction depends on gear hand and rotation) ### **Gear 3 (Idler):** - **Tangential (\(F_t\))**: 800 lbf (opposite to gear 2) - **Radial (\(F_r\))**: 336.2 lbf (toward gear center) - **Axial (\(F_a\))**: 461.9 lbf (opposite direction to gear 2) --- ## **Summary Table** | Gear | \(F_t\) (Tangential) | \(F_r\) (Radial) | \(F_a\) (Axial) | |------|----------------------|------------------|-----------------| | 2 | 800 lbf | 336.2 lbf | 461.9 lbf | | 3 | 800 lbf (opposite) | 336.2 lbf | 461.9 lbf (opp) | - **Directions**: For precise directions, reference the gear rotation, helix hand, and shaft orientation. --- ## **Final Answers** \[ \boxed{ \begin{aligned} \text{For gear 2:} \quad & F_t = 800~\text{lbf}, \quad F_r = 336.2~\text{lbf}, \quad F_a = 461.9~\text{lbf} \\ \text{For gear 3:} \quad & F_t = 800~\text{lbf (opposite)}, \quad F_r = 336.2~\text{lbf}, \quad F_a = 461.9~\text{lbf (opposite)} \\ \end{aligned} } \] Let me know if you need help with showing the vector directions or a free-body diagram!