A stone tied to a string is rotated in a vertical circle. The minimum speed with which the stone has to be rotated in order to complete the circle :

$\text{Step 1: Energy Conservation [Ref. Fig.]}$

$\text{Dependence on m}$

Let $u$ be the minimum speed required at $B$.

Apply energy conservation between point $A$ and $B$ we get

              $K{E}_A+P{E}_A=K{E}_B+P{E}_B$

Taking point $A$ as zero potential level

    $\Rightarrow$            $\frac{1}{2}m{u}^2=mg(R+R)+\frac{1}{2}m{v}^2$  $....(1)$

From equation $1$, we observe that mass $m$ is cancelled on both sides.

$\Rightarrow u$ is independent of $m$.

$\text{Steo 2: Solving Equation}$

$\text{Dependence on R}$

At highest point, Tension will be zero for minimum velocity

$\Rightarrow$ gravitational pull will provide necessary centripetal force

           $\Rightarrow mg=\frac{m{v}^2}{R}$

           $\Rightarrow v^2=Rg$ .... Putting in equation $(1)$

We get, $\frac{1}{2}m{u}^2=mg(R+R)+\frac{1}{2}m(Rg)$

           $\Rightarrow u=\sqrt{5Rg}$

$\Rightarrow u$ is proportional to the square root of $R$.

Hence option $B$ is correct.