Why isn't the tension force's module equal to the gravity force

[teaspoon46]

http://www.physicsclassroom.com/Class/waves/u10l0c2.gif Why isn't the tension force's module equal to the gravity force? If the tension force is larger then shouldn't the body be moving up?

[swansont]

The body does move up. And down.

edit to add: the velocity vector changes, so there is an acceleration.