How do you hang a bottle off a toothpick? I came across this video: https://www.youtube.com/watch?v=mHsVxNMFWwA
How is this possible? I'm guessing the vertical toothpick is exerting an upward force on the table toothpick to balance the torque. But how? I am confus
 A: An easier way to understand why the arrangement is static is to think about energy. The arrangement will be static if it is in a minimum of energy. Addition of the two toothpicks to the rope/bottle arrangement introduces a geometric constraint that makes this arrangement a minimum of energy.
I don't know if you are familiar with the concept of virtual work, but basically, to analyze if a situation is at an energy minimum, we presume we move the assembly as allowed by its geometry by a very small distance, and see if energy is increased or decreased. If energy increases for all possible movements, then we are at an energy minimum and have static equilibrium.
First, consider the initial configuration, with only 1 toothpick, the rope and the bottle. If the bottle goes down. energy is reduced. Since the toothpick is completely free to rotate about the table, the bottle can go down, and we are not at equilibrium.
In the second situation, the two additional toothpicks add a constraint. If the "original" toothpick is rotated a bit, the bottle will go up, as the new "vertical" toothpick would force the new horizontal toothpick, in the rope, to move towards the table. Since we are pivoting about the table, this horizontal movement has to be along an arc and the horizontal movement has to be accompanied by a vertical movement in the up direction, which will raise the bottle a bit. This movement is therefore not allowed to occur spontaneously.
Note that this requires enough friction with the table. If there is no friction, a combination sliding/rotating movement is allowed and the arrangement is not stable. This is related in forces term to the fact that there must be an horizontal friction between the vertical toothpick and the original table-supported toothpick, as the "vertical" toothpick is not completely vertical.
A: Firstly, the bottle creates tension in the string which is then imparted on the tootpick. This creates a torque on the toothpick which is neutralized by the truss-like arrangement attached to both strings. The support beam (toothpick) which exerts a normal force on the vertical beam then carries over as an opposing torque on the one hung from the table. Namely, the arrangement is created to prevent the toothpick on the table to rotate and tip over by exerting this force on the section of the string from the table to the support beam through tension.
