271 - Mechanics part 1 Questions Answers

Dear Sir kindly help provide the correct approach and solution to solve this 2016 iit jee advanced rotational question. 

Two approaches

For angular velocity about z axis there are two approaches in solutions available in books , internet , first one considering spin angular velocity and revolution angular velocity about z axis itself ; second one considering spin angular velocity and revolution angular velocity perpendicular to rod through O and then calculating angular velocity about z axis  taking cosine component of revolution angular velocity. 

May refer 2nd picture attached. 

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Dear Sir kindly help solve
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Dear Sir kindly help provide the solution 
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Find the moment of inertia of an ellipsoid about its mazor axis.

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Sir pls solve
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Sir pls solve for reaction at N
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How can the wheel and platform stay stationary since the center of mass cant move without external force?
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Can you find the ans numerically 
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