Introduction:
In this problem, we are given a body moving in a vertical circle of radius r. We need to find the speed of the particle when the string is horizontal. To solve this problem, we will use the concept of centripetal acceleration and the conservation of mechanical energy.

Given:
- Radius of the vertical circle = r

Approach:
To find the speed of the particle when the string is horizontal, we need to consider the forces acting on the body at that point. When the string is horizontal, the gravitational force acting on the body will provide the necessary centripetal force to keep it moving in a circle.

Step 1: Centripetal acceleration:
When the string is horizontal, the gravitational force acting on the body provides the necessary centripetal force. The centripetal acceleration can be calculated using the formula:

a = v^2 / r

Where:
a = centripetal acceleration
v = velocity of the particle
r = radius of the circle

Step 2: Conservation of mechanical energy:
At the highest point, the body has maximum potential energy and zero kinetic energy. When the string is horizontal, the body has maximum kinetic energy and zero potential energy. According to the conservation of mechanical energy:

Initial mechanical energy = Final mechanical energy

mgh = 1/2 mv^2

Where:
m = mass of the particle
g = acceleration due to gravity
h = height of the highest point

Step 3: Solving for velocity:
Now, we can substitute the value of h in terms of r to simplify the equation:

mg(2r) = 1/2 mv^2

Simplifying further:

2gr = v^2

Taking the square root of both sides:

v = √(2gr)

Step 4: Final answer:
Since we are looking for the speed of the particle when the string is horizontal, we only need to consider the magnitude of the velocity. Therefore, the final answer is:

Speed of the particle when the string is horizontal = √(2gr)

Conclusion:
The speed of the particle when the string is horizontal is given by the equation √(2gr). This result can be derived using the concept of centripetal acceleration and the conservation of mechanical energy.

Use energy conservation at the bottom most pt