Question

Three rods each of mass $m$ and length $\ell$ are joined together in the form of an equilateral triangle as shown in the figure. What is the moment of inertia about an axis passing through centre of mass of the system $\left(G\right)$ and perpendicular to the plane of paper?

• A. $2m{\ell }^2$
• B. $\frac{m{\ell }^2}{2}$
• C. $\frac{m{\ell }^2}{3}$
• D. $\frac{m{\ell }^2}{6}$

Answer 1

The correct option is B $\frac{m{\ell }^2}{2}$

The centre of mass of equilateral triangle is the centroid of the equilateral triangle.

As shown in figure, $OP=\frac{L}{2}tan\left({30}^o\right)=\frac{L}{2\sqrt{3}}$

Moment of inertia(MOI) of rod about centre of mass, $I_{cm}=\frac{1}{12}m{L}^2$

Using parallel axis theorem of MOI, $I=I_{cm}+m{x}^2$, Where $I_{cm}$ is MOI about centre of mass and $x$ is distance between centre of mass and axis of rotation.

MOI of the system $=3(\frac{1}{12}m{L}^2+m(\frac{L}{2\sqrt{3}}{)}^2)=3\times \frac{1}{6}m{L}^2=\frac{m{L}^2}{2}$