Four point masses, each of mass $M$ are placed at the corners of a square of side $L$. The moment of inertia of the system about one of its diagonals is

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Aicharya · Accepted Answer

The given masses are shown in the figure.

Mass of each point mass $=M$ $A B=B C=C D=D A=L$ Length of diagonal $\begin{aligned} B D & =\sqrt{D A^2+A B^2} \\ & =\sqrt{L^2+L^2} \\ & =\sqrt{2 L^2}=L \sqrt{2} \\ \therefore \quad O D & =O B=\frac{B D}{2}=\frac{L \sqrt{2}}{2}=\frac{L}{\sqrt{2}} \end{aligned}$ $\therefore$ Moment of inertia of given system about the diagonal $A C$ $\begin{aligned} I_{A C} & =I_A+I_B+I_C+I_D \\ & =0+M\left(\frac{L}{\sqrt{2}}\right)^2+0+M\left(\frac{L}{\sqrt{2}}\right)^2 \\ & =\frac{M L^2}{2}+\frac{M L^2}{2}=M L^2 \end{aligned}$

Four point masses, each of mass \(M\) are placed at the corners of a square of side \(L\). The moment of…

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