Is the tension of the string when the stopper is at the top of the circle the same, greater than, or less than the tension of the string when the stopper is at the bottom of the circle?

I will assume that the situation is that this stopper has a string tied to it and it is being twirled at constant angular velocity in a vertical circle. If I interpreted your question wrong, well, I tried.

The stopper would fly off in a straight line if the string broke -- that is inertia. In order to continue going in a circle, the string must exert a force called centripetal force to continue changing the direction of the stopper's velocity vector. The term centripetal means that it pulls the stopper toward the center. It pulls just enough to prevent it getting farther from the center than the length of the string.

The stopper also has weight. When the stopper is at the bottom of the circle, the tension in the string must perform 2 functions.

• The tension provides the centripetal force.
• The tension provides a force equal and opposite the stopper's weight.

When the stopper is at the top of the circle, its weight helps the string provide the centripetal force. Because of that help, the tension is less at the top than at any other place in the circle.

I hope this helps,
Steve