Mass moment of inertia of a uniform thin rod about its midpoint Find the mass moment of inertia of a uniform thin rod of mass M and length l about an axis through its mid-point and perpendicular to its length. Choose the correct expression (use SI units, kg·m²).

GivenUniform thin rod, length l, mass M, axis through centroid (midpoint) and perpendicular to the rod.

Formula / ConceptThe standard mass moment of inertia of a slender rod about a centroidal axis normal to its length is: I_G = (1/12) M l^2

Derivation sketchFrom the integral definition: I = ∫ r^2 dm Let the rod lie along x-axis from −l/2 to +l/2 with uniform linear density λ = M/l and dm = λ dx. Then I = ∫_−l/2^+l/2 x^2 λ dx = λ [x^3 / 3]_−l/2^+l/2 = (M/l) (2 (l/2)^3 / 3) = (1/12) M l^2.

Unitskg·m².

Final Answer(1/12) M l^2