As the ring is rotating in horizontal plane tension at any instant is given by,
T = mrw^2 (constant)

Introduction:
When a ring of mass m moves in a horizontal circle of radius r about an axis passing through its center and perpendicular to its plane, it experiences a tension force. In this explanation, we will discuss the factors that affect the tension in the ring and determine its value.

Factors affecting tension:
The tension in the ring is influenced by the following factors:
1. Centripetal force: The tension in the ring provides the centripetal force required to keep the ring moving in a circular path. This force acts towards the center of the circle and is responsible for the circular motion of the ring.
2. Gravitational force: The gravitational force acting on the ring also affects the tension. As the ring moves in a horizontal circle, it experiences a gravitational force acting downwards. This force adds to the tension in the ring.
3. Angular speed: The tension in the ring is also influenced by the angular speed of the ring. The faster the ring rotates, the greater the tension required to maintain its circular motion.

Determining the tension:
To determine the tension in the ring, we can use the following steps:
1. Identify the forces acting on the ring: The two main forces acting on the ring are the tension force and the gravitational force.
2. Equate the net force with the centripetal force: In order for the ring to move in a circular path, the net force acting on the ring must be equal to the centripetal force. This can be expressed as:
Net force = Tension - Gravitational force = m * r * w^2
where Tension is the tension in the ring, m is the mass of the ring, r is the radius of the circle, and w is the angular speed.
3. Solve for the tension: Rearrange the equation and solve for the tension:
Tension = m * r * w^2 + Gravitational force

Conclusion:
The tension in the ring can be determined by equating the net force acting on the ring with the centripetal force. The tension is influenced by factors such as the centripetal force, gravitational force, and angular speed. By considering these factors and solving the equation, the value of the tension in the ring can be determined.