Distance moved till rolling.

A horizontal plank having mass \( m \) lies on a smooth horizontal surface. A sphere of same mass and radius \( R \) is spined to an angular frequency \( \omega \) and gently placed on the plank. If coefficient of friction between the plank and the sphere is \( \mu \), find the distance moved by the plank till the sphere starts pure rolling on the plank. The plank is long enough. If answer is of the form \( \frac { a }{ b } \frac { { { \omega }^{ 2 }R }^{ 2 } }{ \mu g } \), find \( a+b \) where \( a \) and \( b \) are two co-primes integers.

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