Q. 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 $x$.
System of Particles and Rotational Motion
Solution:
$MI$ 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{2 M L^{2}}{3}+\left[\frac{M L^{2}}{12}+\frac{3 M L^{2}}{4}\right] $
$=\frac{2 M L^{2}}{3}+\frac{10 M L^{2}}{12}=\frac{3 M L^{2}}{3}$
$=\frac{18 M L^{2}}{12}=\frac{3}{2} M L^{2}$
Now $\frac{3}{2} M L^{2}=3 M k^{2}$
$k=\frac{L}{\sqrt{2}}$
