question_answer

Three thin metal rods, each of mass M and length L, are welded to form an equilateral triangle. The moment of inertia of the composite structure about an axis passing through the centre of mass of the structure and perpendicular to its plane is

A)  \[\frac{1}{2}M{{L}^{2}}\]  

B)  \[\frac{1}{3}M{{L}^{2}}\]

C)  \[\frac{2}{3}M{{L}^{2}}\]                            

D)  \[\frac{1}{4}M{{L}^{2}}\]

Correct Answer: A

Solution :

                 Moment of inertia of rod BC about an axis perpendicular to plane of triangle ABC and passing through the mid-point of rod BC (i.e., D) is                 \[{{l}_{1}}=\frac{{{m}^{2}}}{12}\] From theorem of parallel axes, moment of inertia of this rod about the asked axis is                                 \[{{l}_{2}}={{l}_{1}}+M{{r}^{2}}=\frac{{{M}^{2}}}{12}+M\left( \frac{{}}{2\sqrt{3}} \right)=\frac{{{M}^{2}}}{6}\] \[\therefore \] Moment of inertia of all three rod is                 \[l=3{{l}_{2}}=3\left( \frac{{{M}^{2}}}{6} \right)=\frac{{{M}^{2}}}{2}\]