I am not clear on the question really. Perhaps the following is the expected answer. perhaps not...
A body is whirled in a horizontal circle ABCD. The pivot P holds a string to which the stone is tied. The string is rotated by some means. A tension develops in the string to balance the weight and give the stone a centripetal force.
Let the string move on a conical surface of an angle = Ф. The string makes angle Ф with the vertical.
T Cos Ф = mg as the body does move vertically upwards or downwards
T Sin Ф = m v² / R
=> v² = R g Tan Ф
=> Kinetic energy required : 1/2 m v² = 1/2 m R g Tan Ф
=> Tension required T = mg / CosФ
Initially the body is hanging from the pivot at the end of the string. So the height to which the body is elevated while whirling = h = L (1- Cos Ф).
So the increase in PE. = m g L (1- CosФ) , where L is the length of the string.
So the amount work to be done is equal to the KE + PE as above. Once this Energy is given, the body will continue to move in the circle, as long as the tension in the string is maintained.
Energy required = mg L (1 - Cos Ф) + 1/2 m g R Tan Ф
= m g [ L(1 - CosФ) + 1/2 * R Tan Ф ]
Once, the body is in circular motion, there is no additional work, as the weight and the tension are both perpendicular to the velocity and displacement of the body.
If a body is in circular motion (uniform), horizontally, then there is no work done further to maintain it in circular motion. This is true in the gravitational field.
If a body makes a circle vertically in the gravitational field, the the net amount of work done is zero.
In any conservative field of force like magnetic, electric, gravitation , if a body makes a complete circle, then the net amount of work done is zero.