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

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.

An electron originates at a point A lying on the axis of a straight solenoid and moves with a velocity v at an angle α to the axis. The magnetic induction of the field is B. Find the distance r from the axis to the point on the screen into which the electron strikes. The screen is oriented at right angles to the axis and is located at a distance l from the point A.

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Dear Sir kindly help provide the solution along with right approach to solve such questions

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On a horizontal floor, two identical cylindrical vessels A and B connected by a thin tube near their bottoms contain some water. When an ice block of volume 100cm^3 is gently put inside the vessel A, it gets half submerged in the water. How much mass of water will flow through the connecting tube during the process of melting of the ice cube? (Density of water=1000kg/m^3 and Density of ice=900kg/m^3)