## Step-by-Step Solution --- ### Step 1: Find the Smallest Moment of Inertia The **moment of inertia** for a rectangle about its centroidal axis (bending about the horizontal axis) is: \[ I = \frac{1}{12} b h^3 \] where: - \( b \) = base width (horizontal) - \( h \) = height (vertical) Let's compare each: #### Cross-section A: - \( b = 125 \) mm, \( h = 60 \) mm - \( I_A = \frac{1}{12} \times 125 \times (60)^3 \) #### Cross-section B: - \( b = 60 \) mm, \( h = 125 \) mm - \( I_B = \frac{1}{12} \times 60 \times (125)^3 \) #### Cross-section C: - \( b = 125 \) mm, \( h = 60 \) mm - \( I_C = \frac{1}{12} \times 125 \times (60)^3 \) #### Cross-section D: - \( b = 60 \) mm, \( h = 125 \) mm - \( I_D = \frac{1}{12} \times 60 \times (125)^3 \) Between A and C, and between B and D, the dimensions are the same, so only two unique values. Since the moment of inertia is proportional to \( h^3 \), the smallest height gives the smallest \( I \). - For A and C: \( h = 60 \) mm — **smaller** than for B and D (\( h = 125 \) mm). - **A and C have the smallest moment of inertia.** #### **Explanation:** The moment of inertia increases dramatically with increased height, so the cross-sections with the smallest height (A and C) have the smallest moment of inertia. --- ### Step 2: Which Axis is the Bending Axis? Given the load is along the **y-axis** and the beam is fixed along the **x-axis**, the beam will bend about the **z-axis**. #### **Explanation:** For a beam loaded vertically (along y), it bends about the horizontal axis perpendicular to the load (here, the z-axis). --- ## Final Answers a) **Which cross-section has the smallest moment of inertia?** **A** or **C** b) **Which axis is the bending axis?** **z-axis**