Physics - Rotation Question with Solution | TestHub

The L-shaped rod consists of two uniform arms, each of length $L/2$ and mass $M/2$. The axis of rotation passes through the joint of the two arms and is perpendicular to the plane of the L.

The moment of inertia of a uniform rod of mass $m^{\prime }$ and length $l^{\prime }$ about an axis passing through one of its ends and perpendicular to its length is $\frac{1}{3}{m}^{\prime }(l^{\prime }{)}^2$. For each arm, $m^{\prime }=M/2$ and $l^{\prime }=L/2$. So, the moment of inertia of one arm about the joint is $I_{arm}=\frac{1}{3}(\frac{M}{2})(\frac{L}{2}{)}^2=\frac{1}{3}\frac{M}{2}\frac{L^2}{4}=\frac{M{L}^2}{24}$.

The total moment of inertia of the L-shaped rod is the sum of the moments of inertia of the two arms, as they are both rotating about the same axis.$$I_{total}=I_{arm1}+I_{arm2}=\frac{M{L}^2}{24}+\frac{M{L}^2}{24}=\frac{2M{L}^2}{24}=\frac{M{L}^2}{12}$$