Problem Statement:
A mass is attached to the end of a string of length l which is tied to a fixed point O. The mass is released from the initial horizontal position of the string. Below the point O, at what minimum distance should a peg P be fixed so that the mass turns about P and can describe a complete circle in the vertical plane? The correct answer is (3/5)l.

Solution:
To solve this problem, we need to analyze the forces acting on the mass throughout its motion. Let's break down the solution into the following steps:

Step 1: Analyzing the Initial Position:
When the mass is released from the initial horizontal position, the string will start to swing downwards due to the force of gravity. At this point, the tension in the string will be zero, and the only force acting on the mass will be its weight.

Step 2: Analyzing the Vertical Motion:
As the mass swings downwards, it will reach its lowest point when the tension in the string is at its maximum. At this point, the tension will act as the centripetal force required to keep the mass in circular motion. The tension can be calculated using the formula: T = mg + mv^2/r, where m is the mass, g is the acceleration due to gravity, v is the velocity of the mass, and r is the radius of the circular path.

Step 3: Determining the Minimum Distance:
To find the minimum distance a peg P should be fixed below point O, we need to consider the condition for the mass to describe a complete circle. For a complete circle, the tension in the string at the lowest point should be equal to zero, as the mass has to momentarily lose contact with the string.

Step 4: Applying the Condition:
To satisfy the condition, we equate the tension T to zero in the previous equation and solve for r. This will give us the radius of the circular path at the lowest point. The minimum distance a peg P should be fixed is equal to this radius.

Step 5: Calculating the Minimum Distance:
Substituting T = 0 in the formula T = mg + mv^2/r, we get mg = mv^2/r. Solving for r, we find r = v^2/g.

Step 6: Determining the Velocity:
To calculate the velocity v, we can use the conservation of energy principle. At the initial horizontal position, the mass only has potential energy, which is converted into kinetic energy at the lowest point. Therefore, mgh = 1/2 mv^2, where h is the height difference from the initial horizontal position to the lowest point.

Step 7: Substituting Values:
Substituting the expression for v into the equation for r, we get r = (2gh)/g = 2h. Since h = l - r, we can rewrite the equation as r = 2(l - r). Solving for r, we find r = 2l/3.

Therefore, the minimum distance a peg P should be fixed is 2l/3. As per the provided answer, (3/5)l, it seems there might be an error in the calculations or interpretation of the question.