Rotational Motion Part 2

Let the length of side of square plate is l.
The diagram of square plate having an axis passing from centre as X−X′ and an axis passing through one of the corner and perpendicular to plane of square plate Z−Z′ is shown below. The distance between the centre of square plate and the corner of square plate is
d=√(

1

2

)2+(

1

2

)2=

1

√2

We know that, moment of inertia of square plate about centre,
IXX′=

MI2

6

By parallel axis theorem, moment of inertia of square plate about axis Z−Z′ will be
  IZZ′=IXX′+M(

1

√2

)2
IZZ′=

MI2

6

+

MI2

2

=MI2[

1

6

+

1

2

]
IZZ′=

2MI2

3

Thus, the moment of inertia about an axis passing through corner of square plate and perpendicular to plane of plate is 2

MI2

3

.