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Yes, I think the solution is rather involved as you would need to know how much force is required to accelerate a single pin to a suitable height, I expect it could exceed the 2kg of the pin itself. The greater the height, the greater the effective increase in the "apparent" weight of the juggler for that instant. Once the pin has left the hand of the juggler, the weight of that pin and the reactionary force generated by the throw is no longer an influence on the bridge. Any pin that is in the air is of no consequence to the equation at all until it is caught again as each one is in free fall.
Myth Busters did a similar experiment with a remote controlled helicopter in an enclosed room and found that the total mass of the room did not change when the helicopter was flying as the force expended in keeping it airborne created a downward pressure onto the floor of the room equal to the weight of the helicopter. However, if the object were in free fall, as the pins are, they do not present a force onto the ground (or bridge in this case) and so are only part of the equation whilst in contact with the juggler.
Do we have a juggler in our midst who has access to a good pair of scales? This would provide the best way to get some real experimental results.
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