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Question

A square plate of edge ′a2′ is cut from a uniform square plate of edge 'a' as shown in figure. Mass of square plate of edge 'a' is M. Find out the moment of inertia of the remaining square plate about an axis passing through 'O' (center of square plate of side 'a') and perpendicular to the plane of the plate.

A
5192Ma2
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B
2764Ma2
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C
964Ma2
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D
Ma26
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Solution

The correct option is C 964Ma2
Given mass of the square plate of sidea=M
Let the mass of removed square plate =m
For uniform square plate, the surface mass density will be constant.
So, Ma2=m(a2)2m=M4 ...(i)

Moment of inertia of parent square plate about an axis passing through 'O' (center of square plate of side a) and perpendicular to the plane of the plate. =Ma26
Iparent plate =Ma26
Moment of inertia of removed square plate (Iremoved plate)
Using parallel axis theroem I=Icom+md2 Here d=a4
Iremoved plate=(m)(a2)26+(m)(a4)2

=548ma2
From equation (i)
Iremoved plate=548×M4a2 =5192Ma2 ....(ii)

Moment of inertia of the remaining square plate (Inet)
(Inet)=Iparent plate+Iremoved plate =Ma265192Ma2
=964Ma2

So, Moment of inertia of remaining square plate about an axis passing through point O and perpendicular to the plane of plate is 964Ma2.

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