Consider the Free Body Diagram of the system at any general time.
Now assume after time t slipping ceases and pure rolling starts, and at that time angular velocity of horizontal disc is \(
\omega
\)
and the angular velocity of vertical disc is \(
\omega'
\)
So we have
(1) \(
r\omega ' = L\omega
\)
(2) \(\omega = \omega _0 - \frac{{\mu mgL}}{{MR^2 /2}}t
\)
(3) \(
\omega ' = \frac{{L\omega }}{r} = 0 - \frac{{\mu mgr}}{{Mr^2 /2}}t
\)
from (2) and (3)
\(
\frac{L}{r}\left\{ {\omega _0 - \frac{{\mu mg2L}}{{MR^2 }}t} \right\} = \frac{{\mu mgr2}}{{mr^2 }}t
\)
\(
\Rightarrow t = \frac{{MR^2 L\omega }}{{2\mu g\left( {MR^2 + mL^2 } \right)}}
\)
