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Question

An isosceles triangle piece is cut from a square plate of side l. The piece is one-fourth of the square and mass of the remaining plate is M. The moment of inertia of the plate about an axis passing through O and perpendicular to its plane is
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A
Ml26
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B
Ml212
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C
Ml224
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D
Ml23
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Solution

The correct option is A Ml26
Mass of remaining three-fourth of plate is M, thus mass of the piece cut out should be M3 because density is uniform.
Now moment of inertia for a rectangular plate about an axis perpendicular to the plane and passing through its centre is given by I=m(a2+b2)12 which becomes I=2ml212 or I=ml26, where m=M+M/3=4M/3.
This moment of inertia is the sum of moment of inertias of 4 triangular parts about their vertex at centre. Also, these 4 moments will be equal because of symmetry.
Thus, I1+I2+I3+I4=4Itriangle=(4M3)L26
or, Itriangle=ML218
Since after cutting one piece, moment of inertia becomes 3.Itriangle, we get
I=3.Itriangle=ML26


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