Let's break down and solve both parts **step-by-step**: --- ## **Given Data** - **Rider-applied force at pedal, \( F \):** 1100 N - **Crank arm length:** 170 mm = 0.170 m - **Cross-section diameter (crank arm):** 16.0 mm = 0.016 m - **Pedal spindle diameter:** 12.0 mm = 0.012 m - **Distance from pedal spindle to pedal:** 60 mm (not needed for the crank arm calculation) --- # **Part A: Maximum Principal Stress in the Crank Arm** ### **Step 1: Find the Bending Moment** The force is applied at the end of the crank arm: \[ M = F \times L \] Where: - \( F = 1100 \) N - \( L = 170 \) mm = **0.170 m** \[ M = 1100 \times 0.170 = 187 \text{ Nm} = 187,000 \text{ Nmm} \] --- ### **Step 2: Calculate the Section Properties** The crank arm is a **circular cross-section**: - **Diameter \( d = 16.0 \) mm** **Moment of Inertia (I):** \[ I = \frac{\pi}{64} d^4 = \frac{\pi}{64} (16)^4 = \frac{\pi}{64} \times 65536 \approx 3217.0 \text{ mm}^4 \] **Maximum distance from neutral axis (\( c \)):** \[ c = \frac{d}{2} = \frac{16.0}{2} = 8.0 \text{ mm} \] --- ### **Step 3: Calculate Maximum Bending Stress (Principal Stress)** \[ \sigma_{max} = \frac{M c}{I} \] Plug in the values: \[ \sigma_{max} = \frac{187,000 \times 8}{3217.0} \] \[ \sigma_{max} = \frac{1,496,000}{3217.0} \approx 465 \text{ MPa} \] --- ### **Final Answer, Part A** \[ \boxed{465 \ \text{MPa}} \] --- # **Part B: Maximum Principal Stress in the Pedal Spindle** For the **pedal spindle**, the force is applied **perpendicular** to the spindle, and we assume the spindle is a circular cross-section loaded in bending. ## **Step 1: Find the Bending Moment** The force is applied at the end of the spindle (distance = 60 mm): \[ M = F \times L = 1100 \times 60 = 66,000 \text{ Nmm} \] --- ## **Step 2: Calculate Section Properties of Spindle** - **Diameter (\( d \)) = 12.0 mm** \[ I = \frac{\pi}{64} d^4 = \frac{\pi}{64} (12)^4 = \frac{\pi}{64} \times 20,736 \approx 1019.4 \text{ mm}^4 \] \[ c = \frac{12.0}{2} = 6.0 \text{ mm} \] --- ## **Step 3: Calculate Maximum Bending Stress** \[ \sigma_{max} = \frac{M c}{I} = \frac{66,000 \times 6}{1019.4} \] \[ \sigma_{max} = \frac{396,000}{1019.4} \approx 388.5 \text{ MPa} \] --- ## **Final Answer, Part B** \[ \boxed{389 \ \text{MPa}} \] (Rounded to three significant figures) --- # **Summary Table** | Part | Stress (MPa) | |---------|--------------| | Crank Arm (A) | 465 | | Pedal Spindle (B) | 389 | --- ## **Let me know if you'd like to see the math in more detail or have any other questions!**