Question

A thin uniform rod of length $L$ and mass $M$ is bent at length ratio $1:2$ such that the angle between them is ${90}^{\circ }$. The moment of inertia of the bent rod about an axis passes through the bending point and perpendicular to the plane is

• A. $\frac{5M{L}^2}{81}$
• B. $\frac{M{L}^2}{9}$
• C. $\frac{8M{L}^2}{81}$
• D. $\frac{8M{L}^2}{9}$

Answer 1

The correct option is B $\frac{M{L}^2}{9}$

Since the rod is uniform, the mass of the bent rod will also be in same ratio as length. Thus the rod can be shown as

We know that,
$\text{MOI}$ of rod about an axis perpendicular to the plane and passing through one end is $\frac{M{L}^2}{3}$
$\Rightarrow I=$ $\text{MOI}$ of $OA$ about $Z{Z}^{\prime }$ $+$ $\text{MOI}$ of $OB$ about $Z{Z}^{\prime }$
$\Rightarrow I=\frac{\frac{M}{3}\times {\left(\frac{L}{3}\right)}^2}{3}+\frac{\left(\frac{2M}{3}\right){\left(\frac{2L}{3}\right)}^2}{3}$
$\Rightarrow I=\frac{M{L}^2}{9}$