I like the living hinge idea very much, but think it's just as useful in a static context where the wood stays in one place. I like the idea of being able to make curved wooden objects by pinning the hinge in a particular position.
Cut the pattern for both the 'sling' and the stand on a laser cutter; this design is for 1/8" plywood. Snap the pieces together by putting the tabs into the holes. It should require no glue. You may want to curve the hinge around a bottle or similar circular object.
As with all laser cut objects, be mindful of the laser kerf and focus. For the hinge, you'll probably get the best result if the scoring is parallel with the grain of the wood, although since the hinge won't move, you can get away with against the grain if you must.
If you have thicker plywood, you'll need to change the size of the tabs and receiving holes to match.