1. **Problem Statement:** Given a column with an external diameter of 200 mm and a maximum allowable stress of 480 N/mm², determine the internal diameter of the column if the factor of safety is 4. 2. **Understanding the problem:** The factor of safety (FOS) is the ratio of the material's strength to the allowable stress. The allowable stress is the maximum stress the column can safely withstand. 3. **Formula:** The actual stress must be less than or equal to the allowable stress divided by the factor of safety: $$\sigma_{actual} \leq \frac{\sigma_{max}}{FOS}$$ 4. **Calculating allowable stress:** $$\sigma_{allowable} = \frac{480}{4} = 120 \text{ N/mm}^2$$ 5. **Stress and cross-sectional area relationship:** Stress is force divided by area. Assuming the column is hollow with external diameter $d_o = 200$ mm and internal diameter $d_i$, the cross-sectional area $A$ is: $$A = \frac{\pi}{4} (d_o^2 - d_i^2)$$ 6. **Since the problem does not provide load or force, we assume the maximum stress is related to the cross-sectional area. To find $d_i$, we rearrange the formula for stress:** Given the maximum stress is 480 N/mm², and allowable stress is 120 N/mm², the ratio of areas is: $$\frac{A_{actual}}{A_{max}} = \frac{\sigma_{max}}{\sigma_{allowable}} = \frac{480}{120} = 4$$ 7. **Relate areas:** $$\frac{\pi}{4} (d_o^2 - d_i^2) = \frac{1}{4} \times \frac{\pi}{4} d_o^2$$ Simplify: $$d_o^2 - d_i^2 = \frac{d_o^2}{4}$$ 8. **Solve for $d_i^2$:** $$d_i^2 = d_o^2 - \frac{d_o^2}{4} = \frac{3}{4} d_o^2$$ 9. **Calculate $d_i$:** $$d_i = d_o \sqrt{\frac{3}{4}} = 200 \times \frac{\sqrt{3}}{2} = 200 \times 0.866 = 173.2 \text{ mm}$$ **Final answer:** The internal diameter of the column is approximately **173.2 mm**.