A

To determine at what point the tension in the rope (or the force acting on the particle) equals zero while the particle completes a vertical circle, we must first understand the forces acting on the particle.

When the particle is at the highest point of the vertical circle, only the gravitational force and the centripetal force need to be considered. At this point, the gravitational force contributes to the centripetal force needed to keep the particle in circular motion.

• At the highest point, the forces acting on the particle are:
• Weight (mg) acting downwards.
• Tension (T) acting downwards

In uniform circular motion, the net force acting towards the center must equal the required centripetal force. Thus, the equation of motion at the highest point can be expressed as:
$$T + mg = \frac{mv^2}{r}$$
To find the condition where tension equals zero, we set T=0 in the equation:
$$0 + mg = \frac{mv^2}{r}$$
This simplifies to:
$$mg = \frac{mv^2}{r}$$

Therefore, this leads to:
$$v^2 = rg$$
This is the minimum velocity required at the highest point for the particle to just complete the circular motion without the need for tension in the string/rope.
Hence, the particle has zero tension at the highest point when it just manages to complete the vertical circle.
Therefore, the answer is option A.