Let's solve the problem step by step: --- ### **Given Data** - Thickness of steel section, \( t = 2\, \text{mm} = 0.002\, \text{m} \) - Density of steel, \( \rho = 7850\, \text{kg/m}^3 \) - Dimensions (from the figure): - Length along \( x \): \( 100\, \text{mm} = 0.1\, \text{m} \) - Height along \( y \): \( 120\, \text{mm} = 0.12\, \text{m} \) - Depth along \( z \): \( 200\, \text{mm} = 0.2\, \text{m} \) --- ## **Step 1: Understanding the Shape** The section is a thin rectangular plate in the \( yz \)-plane, with thickness along \( x \) (2 mm). --- ## **Step 2: Volume Calculation** \[ \text{Volume} = \text{height} \times \text{width} \times \text{thickness} \] \[ V = (0.12\, \text{m}) \times (0.2\, \text{m}) \times (0.002\, \text{m}) = 0.000048\, \text{m}^3 \] --- ## **Step 3: Mass Calculation** \[ m = \rho \times V = 7850 \times 0.000048 = 0.377 \, \text{kg} \] --- ## **Step 4: Mass Products of Inertia** For a thin rectangular plate in the \( yz \)-plane (thickness in \( x \), dimensions \( a \) along \( y \), \( b \) along \( z \)), centered at the origin: \[ I_{xy} = I_{xz} = 0 \quad (\text{due to symmetry about the axes}) \] \[ I_{yz} = \int y z \, dm \] Since the plate is symmetric about both the \( y \) and \( z \) axes, the product \( I_{yz} \) will be zero if the origin is at the centroid. #### **Check: Is the origin at centroid?** If the plate is centered at the origin: - The centroid in \( y \) is at \( y = 0 \) - The centroid in \( z \) is at \( z = 0 \) So, all products of inertia are zero. --- ### **Final Answers** \[ \boxed{ I_{xy} = I_{xz} = I_{yz} = 0 } \] #### **Explanation:** - All mass products of inertia are zero because the component is a thin, flat, rectangular plate, symmetrically placed about the coordinate axes. - If the origin were not at the centroid, we would need to use the parallel axis theorem to compute the products of inertia about the origin. --- ### **If you need moments of inertia (not products), let me know!**