Question

Four point size dense bodies of same mass are attached at four corners of a light square frame. Identify the decreasing order of their moments of inertia about following axes:
I) passing through any side
II) passing through opposite corners
III) perpendicular bisector of any side
IV) perpendicular to the plane and passing through any corner

• A. III, IV, I, II
• B. IV, III, I, II
• C. III, II, IV, I
• D. IV, III, II, I

Answer 1

Answer: (B)

IV, III, I, II

Let the masses be $M$ and side of cube be $L$.

For first case, the MI is $M(0{)}^2+M(0{)}^2+M(L{)}^2+M(L{)}^2=2M{L}^2$

For second case, the MI is $M(0{)}^2+M(0{)}^2+M(L/sqrt2{)}^2+M(L/sqrt2{)}^2=M{L}^2$

For third case, the MI is $M(L/2{)}^2+M(L/2{)}^2+M(\sqrt{5}L/2{)}^2+M(\sqrt{5}L/2{)}^2=3M{L}^2$

For Fourth case, the MI is $M(0{)}^2+M(L{)}^2+M(L{)}^2+M(L\sqrt{2}{)}^2=4M{L}^2$

Arranging in decreasing order, IV > III > I > II