Groovy groove

We have a disk of mass \(M\) and radius \(R\) placed on a horizontal plane. A cylindrical groove of radius \(r\) is made on a diameter.Now a sphere of mass \(m\) and radius \(r\) is placed in the groove at the circumference of the disk.

At \(t=0\) the whole system is rotated with an angular velocity \(1 rad/s\) as shown in the figure.Also at the same time the sphere was projected with a velocity \(v\) towards the centre of the disk.Find the minimum velocity \(v\) of the sphere such that it is able to reach the other end of the groove.

Details and Assumptions

\(1)r\ll R\)



\(4)R=2 m\)

\(5)\)The disk is fixed about it's centre that is it can only rotate


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