# Step-by-Step Solution: Polar Moment of Inertia of a Thin Rectangular Tube ## 1. **Understanding the Figure** - **Outer Rectangle:** 20 cm × 10 cm - **Wall Thickness:** 0.3 cm - **Shape:** Thin-walled rectangular tube (single-cell section) ## 2. **Formula for Polar Moment of Inertia (\( J \))** For a thin-walled closed section: \[ J \approx 4 \sum \left( \frac{A_i d_i^2}{t_i} \right) \] But for a thin-walled rectangle, a simpler formula is: \[ J \approx 2 t (a b^3 + b a^3) / (a + b) \] Where: - \( a \) = longer side (20 cm) - \( b \) = shorter side (10 cm) - \( t \) = thickness (0.3 cm) But for engineering use, **for thin rectangles**: \[ J \approx 4 A_m^2 / \sum \left( \frac{ds}{t} \right) \] Where: - \( A_m \) = area enclosed by the median line - \( ds \) = small length along the wall - \( t \) = thickness --- ### **Step 1: Median Line Dimensions** Median line is at a distance \( t/2 \) from the outer edge: - Median length: \( 20 - 0.3 = 19.7\, \text{cm} \) - Median width: \( 10 - 0.3 = 9.7\, \text{cm} \) --- ### **Step 2: Area Enclosed by Median Line (\(A_m\))** \[ A_m = \text{Median length} \times \text{Median width} \] \[ A_m = 19.7 \, \text{cm} \times 9.7 \, \text{cm} = 191.09\, \text{cm}^2 \] --- ### **Step 3: Perimeter Along the Median Line** \[ \text{Perimeter} = 2 \times (19.7 + 9.7) = 2 \times 29.4 = 58.8\, \text{cm} \] --- ### **Step 4: Substitute into the Formula** \[ J = \frac{4 A_m^2 t}{\text{Perimeter}} \] Where: - \( t = 0.3 \) cm - \( A_m = 191.09 \) cm² - Perimeter = 58.8 cm --- ### **Step 5: Calculate \(J\)** First, \(A_m^2\): \[ A_m^2 = (191.09)^2 = 36515.68 \] Now plug into the formula: \[ J = \frac{4 \times 36515.68 \times 0.3}{58.8} \] \[ J = \frac{43818.816}{58.8} \] \[ J \approx 745.45 \, \text{cm}^4 \] --- ## **Final Answer** ### **The polar moment of inertia \( J \) of the given thin-walled rectangular section is:** \[ \boxed{J \approx 745.5 \, \text{cm}^4} \] --- ### **Summary Table** | Symbol | Value | Unit | |--------|---------------|-----------| | \( a \) | 20 | cm | | \( b \) | 10 | cm | | \( t \) | 0.3 | cm | | \( A_m \) | 191.09 | cm² | | Perimeter | 58.8 | cm | | \( J \) | 745.5 | cm⁴ | If you have any step you want further explained, let me know!