Problem — Stevin's necklace

Problem

Stevin’s necklace. Simon Stevin (1548–1620), a century before Newton, wanted to show that perpetual motion is impossible, and in doing so found a method for analysing inclined planes. He imagined a chain of identical balls, draped over a prism formed by two inclined planes of different slope, letting the lower part hang symmetrically like a garland. Stevin’s question: does the chain set itself in motion? If not, what is the ratio between the weights on the two planes? Argue using the principle of conservation of energy (what would happen to the total energy if the chain rotated by one bead?) and derive from this the law of the inclined plane. Adapted from (Povey 2015, §7.1).

Solution

Stevin’s argument (by contradiction). Suppose the chain sets itself in motion, sliding by one bead around the prism. After the shift the configuration of the chain is identical to the starting one: each ball has simply taken the place of the next. So the height of the centre of mass has not changed, and the total potential energy is the same as before.

But if the chain kept turning, we would have perpetual motion that neither consumes nor returns energy: the chain would gain kinetic energy from nothing, violating conservation of energy. That is impossible. Conclusion: the chain does not set itself in motion, it stays in equilibrium.

The lower garland. The part hanging below the prism is symmetric: equal weights on the left and right balance each other and can be removed without altering the equilibrium. Only the balls resting on the two inclined planes remain, and these alone must be in equilibrium.

Law of the inclined plane. On a plane of length $\ell$ and inclination $\theta$ the number of balls is proportional to the length, $n \propto \ell$. The component of each ball’s weight along the plane is $m\,g\,\sin\theta$. The total push along the plane is therefore $F_\parallel \propto n\cdot m\,g\,\sin\theta \propto \ell\,\sin\theta$ Since the two planes share the same height $H$ of the prism, for each side $\ell\,\sin\theta = H$ (the same for both). Hence the two pushes along the planes are equal and balance: equilibrium confirmed.

This gives the law of the inclined plane: the force that holds a weight on a smooth plane is proportional to $\sin\theta$, and the number of balls (i.e. the weights resting) is in the ratio of the lengths of the planes: $\ev{\dfrac{W_1}{W_2} = \dfrac{\ell_1}{\ell_2} = \dfrac{\sin\theta_2}{\sin\theta_1}}$

Collegamenti

Argomenti: Lavoro ed energia Concetti: Conservazione dell’energia meccanica Competenze: Analisi di casi limite e fantafisica Oggetti: Piano inclinato