Question

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

• A. $2M{L}^2$
• B. $M{L}^2$
• C. $4M{L}^2$
• D. $6M{L}^2$

Answer 1

Answer: (B)

The given masses are shown in the figure.

Mass of each point mass $=M$
$AB=BC=CD=DA=L$
Length of diagonal
$BD=\sqrt{D{A}^2+A{B}^2}$
$=\sqrt{L^2+L^2}$
$=\sqrt{2L^2}=L\sqrt{2}$
$\therefore OD=OB=\frac{BD}{2}=\frac{L\sqrt{2}}{2}=\frac{L}{\sqrt{2}}$
$\therefore$ Moment of inertia of given system about the diagonal $AC$
$I_{AC}=I_A+I_B+I_C+I_D$
$=0+M{\left(\frac{L}{\sqrt{2}}\right)}^2+0+M{\left(\frac{L}{\sqrt{2}}\right)}^2$
$=\frac{M{L}^2}{2}+\frac{M{L}^2}{2}=M{L}^2$