Find the acceleration of the block of mass $2m$ as shown in the figure. (Strings are massless and inextensible)

The correct option is B $\frac{g}{3}$
From constrained condition and using method of intercept we have,
${\stackrel{..}{L}}_1+{\stackrel{..}{L}}_2+{\stackrel{..}{L}}_3=0$
$\Rightarrow -a_1+a_2+a_2=0\Rightarrow a_1=2a_2\dots \dots \left(1\right)$

From FBD of masses $m$ and $2m$ we have force equations as

$T=m{a}_1$...........(i)
$2mg-2T=2m{a}_2$............(ii)
From eq (1), we get

$\frac{T}{m}=2[g-\frac{T}{m}]$
$\frac{3T}{m}=2g\Rightarrow T=\frac{2mg}{3}$
$\therefore$ $a_2=[g-\frac{T}{m}]=[g-\frac{2g}{3}]$
$\Rightarrow a_2=\frac{g}{3}$
Thus, the acceleration of $2m$ mass is $\frac{g}{3}$