Mass Moment — Thin Rod About Its Centre For a thin, uniform rod of mass m and length l, what is the mass moment of inertia about an axis through its centre and perpendicular to its length?

Introduction / Context: Standard mass moment formulas for slender members are used in vibration and rotational dynamics. The rod-about-centre result is one of the most cited identities in mechanical design and analysis.

Given Data / Assumptions:

• Uniform rod, length l, mass m.
• Axis through the midpoint, perpendicular to the rod’s length.

Concept / Approach: For a slender rod, distributing mass along a line, the integral I = ∫ r^2 dm with symmetric limits gives I_center = (1/12) m l^2. About an end, it is (1/3) m l^2; the two are connected by the parallel-axis theorem.

Step-by-Step Solution (outline):

Verification / Alternative check: Using parallel-axis: I_end = I_center + m (l/2)^2 = (1/12) m l^2 + (1/4) m l^2 = (1/3) m l^2, a known correct endpoint value.

Why Other Options Are Wrong: m l^2 / 4 and m l^2 / 6 overpredict; m l^2 / 8 is not a standard value for any principal rod axis.

Common Pitfalls: Confusing area versus mass moments; mixing the end-axis (1/3 m l^2) with center-axis (1/12 m l^2).

Final Answer: m l^2 / 12.