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The spring paradox: burn a string and the weight settles 103 mm higher. What the reel shows: a 0.30 kg weight hangs on two springs (11.8 N/m, 0.10 m long) joined by a 0.06 m string. Burn the string and it drops onto a pad. Add two cords hanging loose beside the springs, each with 0.02 m of slack between the two paths, burn the string again: the cords snap straight, the weight leaps 187 mm above its start, then settles 103 mm higher. A rig with 3 springs and 4 cords does it again: leap 288 mm, settled 155 mm. Add two loose cords, burn the string between two springs and the weight goes up, not down. Why: springs in series each carry the whole load F, so each stretches F/k. Cut the link and the load splits over two parallel paths, each carrying half, so each spring is shorter by F/2k, and the weight rises by that much less the cords' slack. In general, rise = (1 − 1/n) F/k − δ: with F/k = 249 mm and δ = 20 mm, 104.7 mm for 2 springs and 146.3 mm for 3 (the screen's 105 and 146). Simplified: that law has massless springs. The reel's 103 and 155 mm keep each spring's mass as a lump, the lean of the spring eyes and 0.15 N of friction on the guide rods (a judgment), so they sit a few millimetres off the law. Left out: each steel tie arm's mass (about 1 g, which would shift the 3-spring run about 1 mm settled and 3 mm at the leap) and a real spring's initial tension. The tie arms are drawn 3 mm thick, 1.5 mm in fact, and the spring wire 3.2 mm, 1.1 mm in fact, to be seen. Sources: J. E. Cohen and P. Horowitz, «Paradoxical behaviour of mechanical and electrical networks», Nature 352, 699 (1991); Y.-Y. Chen, «On a Mechanical Analogue of the Braess Paradox», Chinese Journal of Physics 43, 1004 (2005). The paradox is the mechanical cousin of Braess's paradox, where adding a road can slow traffic. No voice, no music: every sound is a cause on screen, the torch's hiss, the string's burn, the cords snapping taut, the coils shutting. #physics #springs #braessparadox

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