What is the difference between moment of inertia of a rod about an axis passing through its one end and through its centre perpendicular to its length ? a) M L^23 b) M L^24 c) M L^26 d) M L^212

The moment of inertia of a uniform rod depends on the axis of rotation. For a rod of mass $M$ and length $L$:

• About an axis through its centre (perpendicular to length): $$I_{\text{centre}}=\frac{M{L}^2}{12}$$

• About an axis through one end (perpendicular to length): By the parallel axis theorem, $I_{\text{end}}=I_{\text{centre}}+M{d}^2$, where $d=\frac{L}{2}$: $$I_{\text{end}}=\frac{M{L}^2}{12}+M{\left(\frac{L}{2}\right)}^2=\frac{M{L}^2}{12}+\frac{M{L}^2}{4}=\frac{M{L}^2}{3}$$

Difference: $$I_{\text{end}}-I_{\text{centre}}=\frac{M{L}^2}{3}-\frac{M{L}^2}{12}=\frac{M{L}^2}{4}$$

Thus, the correct difference is option (b).