Question

Three identical uniform 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.

• A. $\frac{l}{\sqrt{3}}$
• B. $\frac{l}{\sqrt{2}}$
• C. $\frac{l}{\sqrt{5}}$
• D. $\frac{l}{\sqrt{7}}$

Answer 1

The correct option is B $\frac{l}{\sqrt{2}}$

Let axis of rotation is passing through point 'A'.

Moments of Inertia (MOI) of rod AB about point 'A'= $\frac{m{l}^2}{3}$,

MOI of rod AC about point 'A'= $\frac{m{l}^2}{3}$,

From the figure AP= $lsin\left({60}^o\right)=\frac{\sqrt{3}}{2}l$,

Using Parallel axis of theorem of MOI,

MOI of rod BC about point 'A'= $\frac{m{l}^2}{12}$ + $m(\frac{\sqrt{3}}{2}l{)}^2$ $=\frac{5}{6}m{l}^2$

By adding all the MOI ,

MOI = $\frac{3}{2}m{l}^2$

$\therefore$ radius of gyration 'r'

$\Rightarrow 3m{r}^2=\frac{3}{2}m{l}^2$,

$\Rightarrow r=\frac{l}{\sqrt{2}}$