Columns: Determine the slenderness ratio for a prismatic vertical column of square cross-section 2.5 cm × 2.5 cm and length 300 cm (assume pin-ended and use the least radius of gyration).

Introduction / Context:Slenderness ratio is a key parameter in column design and stability checks. It relates the effective length to the least radius of gyration and indicates the tendency to buckle under axial compression.

Given Data / Assumptions:

• Square section: side b = 2.5 cm.
• Length L = 300 cm; assume pin-ended, so effective length Le = L.
• Use least radius of gyration r for the cross-section.

Concept / Approach:Slenderness ratio λ = Le / r. For a square of side b, the radius of gyration about centroidal axis is r = b / sqrt(12). The least r equals this value for a square (same about both principal centroidal axes).

Step-by-Step Solution:Compute r = b / sqrt(12) = 2.5 / sqrt(12) cm.sqrt(12) ≈ 3.464; r ≈ 2.5 / 3.464 ≈ 0.722 cm.Slenderness λ = Le / r = 300 / 0.722 ≈ 415.8.Rounded to nearest option: 416.

Verification / Alternative check:Check units: Le (cm) divided by r (cm) yields a pure number, matching a dimensionless slenderness ratio.

Why Other Options Are Wrong:200, 240, 360, and 500 do not match the computed ratio given the provided dimensions and assumptions.

Common Pitfalls:Using radius of gyration formula for a rectangle incorrectly; forgetting to use the least radius; mixing length units (cm vs m).

Final Answer:416