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A spool of thread of mass m and moment o...

A spool of thread of mass `m` and moment of inertia `I` is attached to the block of mass `M`. Along a horizontal line passing through the centre of the spool, a constant force `F` is applied as shown. The surface under the block is smooth while beneath the spool is rough.
Show that if `M=(I)/(r(R-r))`, no frictional force will act on the spool. Now suppose `M=(2I)/(r(R-r))`, find in which direction will the frictional force act? Also find in this case the rate of increase of length `'ab'` of the thread at time `t` after the application of force. (No slipping occurs.)

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