How to make a moving object "stick" to a stationary object in box2D

I have been experimenting with the box2D sample project within cocos2D for the iPhone and am wondering if box2D is the appropriate engine to use to make a moving object "stick" to a stationary object when the moving object is finished moving in a certain direction.

Here is a simplification of what I am trying to achieve: I have MovingObject, a dynamic rigid body, that moves vertically against gravity when enough force is applied to it. As MovingObject moves, it may overlap a static object, StationaryObject. When gravity diminishes MovingObject's velocity to zero such that it is no longer moving, I would like to have MovingObject remain where it is ONLY if it overlaps StationaryObject. If the object's do not overlap, MovingObject should start to move back down towards the ground per the force of gravity. During that descent, if MovingObject at any time overlaps StationaryObject, it should stop its descent and remain in that location as if it is stuck on StationaryObject.

I am able to get MovingObject to move per the forces I am applying to it, but not really sure how to make it stop and stay there once it reaches the top of its ascent, assuming it is overlapping StationaryObject.

Currently, I am experimenting with simple square/box objects but eventually both MovingObject StationaryObject will be defined as very different complex polygon shapes.

Thanks in advance for any insights and/or suggestions for achieving this.

Sounds like you'll want to change the type of fixture used for "MovingObject" while it "ascending" and then change it when it is "descending" so that it reacts differently (to overlaps).

by "overlap" it sounds like you want to achieve something similar to "one sided platforms" in a platform game (ie; Mario Bros.) - I would recommend looking into the solutions for one-sided platforms for starters.