Top and bottom of a swinging object

[MechaMZ]

Homework Statement

In a vertical circle with non uniform speed, why we always say that the tension at the bottom is a maximum and the tension at the top is minimum?
Is there any mathematical method to prove it? I don't understand why we always assume in this way.

The Attempt at a Solution

could it because of energy?

 

[Delphi51]

At the bottom the tension provides the centripetal force as usual but also must balance the gravitational force which is at that point entirely in the opposite direction to the tension force.

 

[MechaMZ]

so could we say the tension is zero since both of the gravitational force and tension are in the same direction?

 

[Delphi51]

At the bottom of the swing tension is up, gravity down.
At the top, it could happen that they cancel out of the speed is just right.