Moment Of Inertia

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 $M=3\text{ kg}$ and $L=2\text{ m}$ gives $I=\frac{1}{3}(3)(2{)}^2=\frac{1}{3}(3)(4)=4{\text{ kg m}}^2$. The value 1 kg m${}^2$ effectively ignores the length factor altogether. The value 2 kg m${}^2$ wrongly applies $\frac{1}{12}M{L}^2$, the centre-axis formula, with a slip in arithmetic. The value 12 kg m${}^2$ forgets to divide by three and merely computes $M{L}^2$. 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 $\frac{1}{3}M{L}^2=4$ must exceed the centre-axis value $\frac{1}{12}M{L}^2=1$, 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.