Moment Of Inertia

Question

A uniform rod of mass 3 kg and length 2 m is set rotating about an axis perpendicular to the rod and passing through one of its ends. What is its moment of inertia about that axis?

RankGuru · Accepted Answer

Moment of inertia measures how a body's mass is distributed relative to a rotation axis and depends strongly on that distance. For a thin uniform rod rotating about a perpendicular axis through one end, the standard result obtained by integrating $\int x^2\,dm$ along the rod is $I = \frac{1}{3} M L^2$. Substituting $M = 3	ext{ kg}$ and $L = 2	ext{ m}$ gives $I = \frac{1}{3}(3)(2)^2 = \frac{1}{3}(3)(4) = 4	ext{ kg m}^2$. The value 1 kg m$^2$ effectively ignores the length factor altogether. The value 2 kg m$^2$ wrongly applies $\frac{1}{12}ML^2$, the centre-axis formula, with a slip in arithmetic. The value 12 kg m$^2$ forgets to divide by three and merely computes $ML^2$. This end-axis result follows directly from the NCERT derivation of the moment of inertia of a rod. As a plausibility check, the end-axis value $\frac{1}{3}ML^2 = 4$ must exceed the centre-axis value $\frac{1}{12}ML^2 = 1$, since mass lies farther from an end axis, and our answer respects that ordering. Physically, the farther a rod's mass is shifted from the chosen axis, the greater the torque required to produce a given angular acceleration, which is exactly why a door is hinged at its edge rather than through its middle.

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