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Can you provide a transcript of what the presenter was saying?
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Nothing was said, it was all about the visuals.
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1) Did you notice that the pull-string shortens during the presentation.
2) The string is being wound around a spool within the 'cart'.
3) If there is a spring-powered clock-drive, or battery-powered- motor-spool within the 'cart', this power , added to the 'system', is what's responsible for the phenomena you are experiencing, as the string continuously winds itself onto the spool.
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This is all that is inside, a magic bog roll core:
No batteries, no springs, no antenna to receive power from a Tesla coil.
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4) A parlor trick.
5) A visual distraction from the actual dynamics.
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I'd have to disagree, I think it illuminates the principle at play with surgical precision.
The diameter of the bog roll is smaller than the diameter of the wheels, so with the same torque it can pull the cart forward with greater force than the wheels pull it backwards.
Another way of looking at it is that the relative motion of the cart and string is slower that the relative motion of the cart and floor, so the same power can provide a greater force.