I have been trying to find the value of theta at which a string goes slack in a vertical circle, by letting the diameter of the circle be the zero potential
My question is would the gravitational potential above and below the zero potential have opposite signs?
I am well aware that potential energy is a scalar quantity however when I apply the conservation of energy $$K.E_{initial} + P.E_{initial}= K.E_{final} + P.E_{final}$$
The only way I get a satisfactory answer (one that matches the answer on the book) is when the potential energies have opposite signs.
Yes , they would have opposite signs . In this case , when you have made the potential energy of the horizontal diameter to be zero , the potential energy when the object has moved above that level is positive , and when it moves down the level the potential energy is negative . This is because potential energy is negative of work done by an internal conservative force .