Let's solve **Question 2** step by step, as requested, and include a graph for clarity. --- ## **Question 2:** > Determine the smallest allowable diameter of the solid shaft ACDB, knowing that: > a) Shearing stress = 60 MPa, > b) The shaft is made of carbon steel AISI 104 annealed, > c) The safety factor is 2. --- ## **Step 1: Calculate Maximum Shear Stress** Given: - Maximum load at point C = 6 kN = 600 N - Safety factor (SF) = 2 - Allowable Shear Stress = 60 MPa ### **Shear Stress in Shaft:** \[ \tau_{max} = \frac{T_{max} \cdot r}{J} \] Where: - \(T_{max}\) = Maximum torque - \(r\) = outer radius of shaft - \(J\) = polar moment of inertia = \(\frac{\pi d^4}{32}\) for a solid shaft ### **Maximum Torque (T) on the Shaft:** The maximum moment will be at point C (where the 6 kN force acts). Let's calculate the maximum moment: - Distance from A to C: \(175\, \text{mm} + 150\, \text{mm} = 325\, \text{mm} = .325\, \text{m}\) - Torque at C: \(T = F \times L = 600\, \text{N} \times .325\, \text{m} = 195\, \text{Nm}\) ### **Step 2: Find Minimum Diameter (d)** \[ \tau_{max} = \frac{T_{max} \cdot r}{J} \] \[ \tau_{max} = \frac{T_{max} \cdot \frac{d}{2}}{\frac{\pi d^4}{32}} = \frac{16 T_{max}}{\pi d^3} \] Include safety factor: \[ \tau_{allowable} = \frac{\tau_{max}}{SF} = \frac{60\, \text{MPa}}{2} = 30\, \text{MPa} \] Set \(\tau_{allowable}\) equal to calculated shear stress: \[ 30 \times 10^6 = \frac{16 \times 195}{\pi d^3} \] Solve for \(d\): \[ d^3 = \frac{16 \times 195}{\pi \times 30 \times 10^6} \] \[ d^3 = \frac{31200}{94,247,779.6} \] \[ d^3 = .000331 \] \[ d = \sqrt[3]{.000331} = .069\, \text{m} = 69\, \text{mm} \] --- ## **Graph: Shear Stress vs. Shaft Diameter** ```plaintext Shear Stress (MPa) | |\ | \ | \ | \ | \ | \ | \ | \ | \ | \ |----------\-------------------> Diameter (mm) 69 ``` *As diameter increases, allowable shear stress decreases. At 69 mm diameter, the shaft is safe (with SF = 2).* --- ## **Final Answer** - **Smallest allowable diameter:** \(\boxed{69\, \text{mm}}\) --- ### **Summary Table** | Parameter | Value | |---------------------|----------------| | Shear Stress | 60 MPa | | Safety Factor | 2 | | Allowable Stress | 30 MPa | | Max Moment (Torque) | 195 Nm | | **Min Diameter** | **69 mm** | --- **Example:** If the safety factor was 1, the diameter would be smaller, but for higher reliability, 69 mm is required. Let me know if you want the solution for a different question or more detailed diagrams!