To understand why the minimum value of H required for a particle to make a complete vertical circle is 2.5 R, let's break down the concept step by step.

Understanding the Vertical Circle
In a vertical circle, a particle moves in a circular path in a vertical plane. The force acting on the particle is the gravitational force, which always acts vertically downward. The particle needs to have sufficient speed at the topmost point to prevent it from falling down due to gravity.

Minimum Speed at the Topmost Point
To determine the minimum value of H required for a complete vertical circle, we need to consider the minimum speed the particle should have at the topmost point. This minimum speed occurs when the particle just completes the vertical circle and momentarily loses contact with the circular path at the top.

Centripetal Force and Gravitational Force
At the topmost point, the centripetal force required to keep the particle moving in a circle is provided by the normal force exerted by the circular path. This normal force must be equal to the gravitational force acting on the particle.

Equating Forces
At the topmost point, we can equate the gravitational force to the centripetal force:

mg = mv²/R

Here, m is the mass of the particle, v is its speed, and R is the radius of the circular path.

Substituting Values
We can rewrite the above equation as:

v² = gR

To find the minimum value of H, we need to determine the minimum speed v at the topmost point. This minimum speed occurs when the particle just maintains contact with the circular path. This means the normal force is zero.

Using Conservation of Energy
At the topmost point, the total mechanical energy of the particle is the sum of its kinetic and potential energy. We can express this as:

mgH = 0.5mv² + mg(2R)

Here, H is the height at the topmost point, and 2R is the height of the complete vertical circle.

Simplifying the Equation
Since the particle just maintains contact at the topmost point, its speed is at the minimum value. Therefore, we can substitute v² = gR into the equation:

mgH = 0.5m(gR) + mg(2R)

Simplifying further:

H = 0.5R + 2R

H = 2.5R

Therefore, the minimum value of H required for the particle to make a complete vertical circle is 2.5 R.