Please solve 31,32,33,34

31. Four particles each of mass 'm' are placed at the corners of a square of side length 'l'. The radius of gyration of the system about an axis perpendicular to the plane of square and passing through its centre is
1   l 2 2   l 2 3   l 4   2 l

32. In the above problem the moment of inertia of four bodies about an axis perpendicular to the plane of frame and passing through a corner is
1   ML 2 2   2 ML 2 3   2 2 ML 2 4   4 ML 2

33. In above problem the moment of inertia of four bodies about an axis passing through opposite corners of frame is
1   2 ML 2 2   2 ML 2 3   ML 2 4   2 2 ML 2

34. In the above problem the moment of inertia of four bodies about an axis passing through any side of frame is 
1   4 ML 2 2   2 2 ML 2 3   2 ML 2 4   2 ML 2

Dear student, 

Let   K   be   the   radius   of   gyration AC = BD AC = l 2 + l 2 = l 2 AO = CO = BO = DO AO = l 2 2 = l 2 Moment   of   inertia   of   the   given   lamina   about   an   axis   passing   through   O   and   perpendicular   to   the   lamina , I = m × l 2 2 + m × l 2 2 + m × l 2 2 + m × l 2 2 I = 4 m × l 2 2 I = 2 ml 2     . . . . . . . . . . . . . .   1 Also ,   I = mK 2     . . . . . . . . . . . . . . . . . . . . .   2 compare   1   and   2 ,   we   get , K 2 = 2 l 2 K = l 2

Regards. 

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