A stone is attached to one end of a string and is whirled in a...

At the highest point of the vertical circle, the forces acting on the stone are its weight $mg$ and the tension $T$ in the string, both acting downwards towards the center of the circle.

The equation for centripetal force is given by $T + mg = \dfrac{mv^2}{R}$.

For the stone to just complete the circle, the minimum speed at the highest point corresponds to the condition where the tension in the string becomes zero ($T = 0$).

Substituting $T = 0$ in the equation, we get $mg = \dfrac{mv^2}{R}$.

Solving for $v$, we get $v^2 = Rg \Rightarrow v = \sqrt{Rg}$.

Thus, the minimum speed the stone can have at the highest point is $\sqrt{Rg}$.