Introduction:
In this scenario, a particle with mass m is suspended from the ceiling through a string of length L. The particle is moving in a horizontal circle of radius r. We need to determine the speed of the particle.

Analysis:
To find the speed of the particle, we can consider the forces acting on it. There are two main forces to consider: the tension force in the string and the gravitational force.

1. Tension force:
The tension force in the string provides the centripetal force required to keep the particle moving in a circle. This force acts towards the center of the circle and has a magnitude of mv²/r, where v is the speed of the particle.

2. Gravitational force:
The gravitational force acts vertically downwards and has a magnitude of mg, where g is the acceleration due to gravity.

Equilibrium condition:
Since the particle is in equilibrium, the net force acting on it must be zero. Therefore, the tension force and the gravitational force must balance each other.

Equation:
Setting the tension force equal to the gravitational force, we have:

mv²/r = mg

Simplifying the equation, we get:

v² = rg

Taking the square root of both sides, we obtain:

v = √(rg)

Explanation:
The speed of the particle is given by the square root of the product of the radius of the circle and the acceleration due to gravity. This equation shows that the speed of the particle is directly proportional to the square root of the product of the radius and the acceleration due to gravity.

Conclusion:
In conclusion, the speed of the particle moving in a horizontal circle of radius r, suspended from the ceiling through a string of length L, is given by v = √(rg), where g is the acceleration due to gravity. This equation allows us to determine the speed of the particle based on the given parameters.

Is it √{(r²g)/[√(l²-r²)]}