The bird, which at first sight hangs in an unnatural way on its bill, the swinging parrot, the acrobat bear, and the cyclist – are toys that show the question of the barycentre. Hanging on a blade they don’t fall but swing lazily.
In all these toys the natural barycentre has been moved by adding some weight in a way to make the real barycentre lay exactly under the point of support. If we put them in an oscillatory motion they swing slowly because the barycentre is not far from the rotation point, while the moment of inertia is big.
Where is your barycentre when you are in equilibrium over a rope? Do we need to bring some heavy luggage in order not to fall down?