Question

Three identical rods, each of length $l$, are joined to from 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. $\frac{l}{2}$
• B. $\sqrt{\frac{3l}{2}}$
• C. $\frac{l}{\sqrt{2}}$
• D. $\frac{l}{\sqrt{3}}$

Answer 1

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

Let mass of each rod is $m$ and

axis of rotation is passing through point 'A'.

we know that MOI of rod AB about point 'A'

$I_{AB}=\frac{m{l}^2}{3}$

MOI of rod AC about point 'A'

$I_{AC}=\frac{m{l}^2}{3}$

MOI of rod BC about point 'P is

'

$I_{BC}^{\prime }=\frac{m{l}^2}{12}$

From the figure AP= $l\sin {60}^o=l\sqrt{3}/2$

Using Parallel axis of theorem of MOI

MOI of rod BC about point 'A'

$I_{BC}=\frac{m{l}^2}{12}+\frac{m3l^2}{4}$

By adding all the MOI

$I_{net}=I_{AB}+I_{BC}+IAC$

$I_{net}=$$\frac{m{l}^2}{3}+$$\frac{m{l}^2}{3}+$$\frac{m{l}^2}{12}+\frac{m3l^2}{4}$

$I_{net}=\frac{3m{l}^2}{2}$

let radius of gyration be 'r'

so $I_{net}=3m{r}^2$

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

$r=l\sqrt{\frac{1}{2}}$