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Q.
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
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
$B D=\sqrt{D A^{2}+A B^{2}}$
$=\sqrt{L^{2}+L^{2}}$
$=\sqrt{2 L^{2}}=L \sqrt{2}$
$\therefore O D=O B=\frac{B D}{2}=\frac{L \sqrt{2}}{2}=\frac{L}{\sqrt{2}}$
$\therefore $ Moment of inertia of given system about the diagonal $A C$
$I_{A C}=I_{A}+I_{B}+I_{C}+I_{D}$
$=0+M\left(\frac{L}{\sqrt{2}}\right)^{2}+0+M\left(\frac{L}{\sqrt{2}}\right)^{2}$
$=\frac{M L^{2}}{2}+\frac{M L^{2}}{2}=M L^{2}$
