In vertical circular motion while finding the minimum velocity at the bottom-most point for looping the loop, why do we take the tension at the topmost point to be zero?
To find the minimum velocity at the bottom-most point, we find the minimum velocity at the uppermost point. This minimum velocity is the one such that the centripetal force is equal to the force of gravity. Any lower and gravity will pull the object down and out of the loop, any higher and a faster velocity would be required to generate it (thus, not a minimum). Because this is the case, we set the tension to be zero because gravity can account for the entire centripetal force. Using this minimum velocity at the topmost point, we can use conservation of energy to find the minimum velocity at the bottom-most point.
A tension less than zero isn't really physical; this is the point where the string stops being taught and the object doesn't make a complete circle. Therefore, when finding the minimum velocity for an object to make it around a loop, we solve for when the tension at the top is zero as it is the minimum possible tension for the object to keep going in a circle.