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

To find the minimum speed with which a stone tied to a string has to be rotated in a vertical circle, we need to analyze the forces acting on the stone at the highest point of its circular path.

Let:
- $m$ be the mass of the stone,
- $v$ be the speed of the stone at the highest point,
- $r$ be the radius of the circle (length of the string, $l$), and
- $g$ be the acceleration due to gravity.

At the highest point of the vertical circle, both the tension in the string ($T$) and the gravitational force ($mg$) act downwards, towards the center of the circular path. These forces together provide the necessary centripetal force for circular motion:

$T+mg=\frac{m{v}^2}{r}$

To complete the vertical circle, the string must remain taut. This means the tension in the string at the highest point must be greater than or equal to zero ($T\ge 0$). The minimum speed ($v_{\min }$) occurs when the tension is just about to become zero ($T=0$).

Setting $T=0$ in the force equation:

$mg=\frac{m{v}_{\min }^2}{r}$

We can divide both sides of the equation by the mass of the stone ($m$):

$g=\frac{v_{\min }^2}{r}$

Solving for $v_{\min }$:

$v_{\min }^2=rg$

$v_{\min }=\sqrt{rg}$

From the derived formula, the minimum speed at the top of the circle is $\sqrt{lg}$ (since $r=l$). This expression does not contain the mass of the stone ($m$). Therefore, the minimum speed is independent of the mass of the stone.

Thus, the correct option is: Is independent of the mass of the stone.