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Problem: Calculate the radius of gyration for a uniform solid rod of mass M and length L rotating about an axis perpendicular to its length and passing through its center. Step 1: Identify the knowns. Mass = M Length = L Axis of rotation = Center of the rod. Step 2: Recall the Moment of Inertia (I) for this specific case. For a uniform rod rotating about its center, the formula is I = (1/12) * M * L^2. Step 3: Use the Radius of Gyration formula. The relationship is I = M * K^2, where K is the radius of gyration. Rearranging for K, we get K = sqrt(I / M). Step 4: Substitute the value of I into the equation. K = sqrt( [(1/12) * M * L^2] / M ) Step 5: Simplify the expression. The mass M cancels out. K = sqrt( (1/12) * L^2 ) K = L / sqrt(12) Step 6: Final Answer. K = L / (2 * sqrt(3)) or approximately 0.289 * L. This means the mass can be considered concentrated at a distance of L/(2*sqrt(3)) from the center.