Three identical smooth cylinders, each of mass m and radius r are resting in equilibrium within a fixed smooth cylinder of radius R (only a part of this cylinder has been shown in the figure). Find the largest value of R in terms of r, for the small cylinders to remain in equilibrium. If R is expressed as $R=r\left(1+2\sqrt{K}\right)$  , what is value of  $K$?

$O=$Centre of larger cylinder. 
DO will pass through centre of lower cylinder (B)
$N_1=$Contact force between A and B (and A and C)
$N_2=$Contact force between B and the large cylinder 
$N_3=$Contact force between B and C.
For the vertical equilibrium of A
$2N_1\cos {30}^o=mg\Rightarrow \sqrt{3}{N}_1=mg$………….(1)
For horizontal equilibrium of B
$N_1\sin {30}^o=N_2\sin \theta$
$\frac{mg}{2\sqrt{3}}=N_2\sin \theta$ ………..(2)
For vertical equilibrium of whole system
$$\begin{gathered} 2N_2\cos \theta =3mg \\ \therefore N_2\cos \theta =\frac{3}{2}mg \\ \tan \theta =\frac{1}{3\sqrt{3}}\Rightarrow \frac{r}{\sqrt{{(R-r)}^2-r^2}}=\frac{1}{3\sqrt{3}} \end{gathered}$$

$$\begin{gathered} \Rightarrow 27r^2={(R-r)}^2-r^2 \\ \Rightarrow {(R-r)}^2=28r^2 \\ R-r=2\sqrt{7}r \\ \therefore R=r(1+2\sqrt{7}) \end{gathered}$$