Question

Three identical rods, each of length $l$, are joined to form a rigid equilateral triangle. Its radius of gyration about an axis passing through a corner and perpendicular to the plane of the triangle is $\frac{l}{\sqrt{x}}.$ Find $x$.

Answer 1

Answer: 2

$MI$ of the system w.r.t an axis $\perp$ to plane \& passing through one corner
$=\frac{M{L}^2}{3}+\frac{M{L}^2}{3}+\left[\frac{M{L}^2}{12}+M{\left(\frac{\sqrt{3}L}{2}\right)}^2\right]$
$=\frac{2M{L}^2}{3}+\left[\frac{M{L}^2}{12}+\frac{3M{L}^2}{4}\right]$
$=\frac{2M{L}^2}{3}+\frac{10M{L}^2}{12}=\frac{3M{L}^2}{3}$
$=\frac{18M{L}^2}{12}=\frac{3}{2}M{L}^2$
Now $\frac{3}{2}M{L}^2=3M{k}^2$
$k=\frac{L}{\sqrt{2}}$