Question

The radius of gyration of a rod of length $L$ and mass $M$ about an axis perpendicular to its length and passing through a point at a distance $L/3$ from one of its ends is

• A. $\frac{\sqrt{7}}{6}L$
• B. $\frac{L^2}{9}$
• C. $\frac{L}{3}$
• D. $\frac{\sqrt{5}}{2}L$

Answer 1

Answer: (C)

Moment of inertia of the rod about a perpendicular axis $PQ$ passing through centre of mass $C$

$I_{CM}=\frac{M{L}^2}{12}$
Let $N$ be the point which divides the length of rod $AB$ in ratio $1:3.$
This point will be at a distance $\frac{L}{6}$ from $C$.
Thus, the moment of inertia $I^{{}^{\prime }}$ about an axis parallel to $PQ$ and passing through the point $N$.
$I^{{}^{\prime }}=I_{CM}+M{\left(\frac{L}{6}\right)}^2$
$=\frac{M{L}^2}{12}+\frac{M{L}^2}{36}=\frac{M{L}^2}{9}$
If $K$ be the radius of gyration, then
$K=\sqrt{\frac{I^{{}^{\prime }}}{M}}=\sqrt{\frac{L^2}{9}}=\frac{L}{3}$