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**

\(1)r\ll R\)

\(2)M=2kg\)

\(3)m=3Kg\)

\(4)R=2 m\)

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

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