1. Identify the forces in vertical circular motion

A stone of mass $m$ moves in a vertical circle with uniform speed $v$ and radius $r$.

At any point, the required centripetal force toward the center is

$\frac{mv^2}{r}.$

The forces acting on the stone are:

• tension $T$ in the string, always toward the center,
• weight $mg$ downward.

Because the direction of $mg$ relative to the center changes from point to point, the tension changes with position.

2. At the highest point

At the top of the circle, both tension and weight act toward the center.

So,

$T_{\text{top}} + mg = \frac{mv^2}{r}$

Hence,

$T_{\text{top}} = \frac{mv^2}{r} - mg.$

3. At the lowest point

At the bottom, tension acts toward the center, but weight acts away from the center.

So,

$T_{\text{bottom}} - mg = \frac{mv^2}{r}$

Hence,

$T_{\text{bottom}} = \frac{mv^2}{r} + mg.$

4. At the horizontal position

At the side point, weight is perpendicular to the radius, so it has no radial component.

Thus,

$T_{\text{horizontal}} = \frac{mv^2}{r}.$

5. Compare the three tensions

We have:

$T_{\text{top}} = \frac{mv^2}{r} - mg,$ $T_{\text{horizontal}} = \frac{mv^2}{r},$ $T_{\text{bottom}} = \frac{mv^2}{r} + mg.$

Therefore,

$T_{\text{top}} < T_{\text{horizontal}} < T_{\text{bottom}}.$

So the tension is minimum at the highest point.

6. Check options

• A: the same throughout the motion. ❌ False
• B: minimum at the highest position of the circular path. ✅ True
• C: minimum at the lowest position of the circular path. ❌ False
• D: minimum when the rope is in the horizontal position. ❌ False

7. Final answer

The correct option is:

$\boxed{\text{B}}$