ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is $$I0$$. If part ADE is removed, the moment of inertia of the remaining part about the same axis is $$NI016$$ where N is an integer. Value of N is $$__________$$ .

Question

ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is  $$I0$$. If part ADE is removed, the moment of inertia of the remaining part about the same axis is $$NI016$$ where N is an integer. Value of N is $$__________$$ .

Infinity Learn · Accepted Answer

Divide the larger plate into four equal plates as shown.If  $$I0$$ is moment of inertia of bigger plate, we may $$writeI0=$$β$$ma2$$ Where  β is a shape factor.  So each small plate will have moment of inertia about its own centroid as  $$=I=$$β $$m4a22=I016$$.So their contribution to $$I0$$  can be written as  $$I0=I016+3I016+$$ x     Using parallel axis theorem​$$I0$$−$$I016$$−$$3I016=3$$ x ⇒   $$x=I04We$$ require  Iremaining $$plate=I0$$−Ismall plate $$=I0$$−$$I016+x=I0$$−$$I016+I04=11$$ $$I016$$

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