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How is it that always the same number of balls leave at the other end in Newton's cradle. I understand that the momentum needs to be conserved, but as momentum is defined as p=m*v couldn't you have a different number of balls move at a different speed instead of the same number of balls at the same speed?

Qmechanic
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Nickpick
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2 Answers2

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The simple answer is that not only momentum but also energy needs to be conserved.

This puts constraints on the number of balls that can be activated in the cradle.

Note that this does not always give a unique solution either. But it enforces that $ n $ balls to $ n $ balls is a "stable" solution.

Mikael Fremling
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The compression pulse that propagates through the metal spheres of Newton's cradle are not ordinary sound waves. They are approximate solitons (a nonlinear wave form that balances dispersion against nonlinearity). It is this property of soliton pulses that is responsible for the observed behavior.

More Details: Newton's cradle is a physical manifestation of the Fermi Pasta Ulam simulation conducted in the 1950s: https://en.wikipedia.org/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam_problem The expected thermalization (analogous to the expectation that more and more balls in Newton's cradle will start to move) fails to materialize because the dispersion is exactly balanced by the nonlinearity that results from the Hertz deformation law for elastic spheres. Instead, repeated occurrences of the initial conditions are observed.

In actuality the soliton solutions are only an approximate representation of the motions of the balls in Newton's cradle. As the energy get dissipated through friction, the balance between dispersion and nonlinearity breaks down and the expected thermalization finally occurs.

Lewis Miller
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  • Also, see this: https://physics.stackexchange.com/questions/231284/why-doesnt-the-9th-ball-move-in-the-break-in-the-nine-ball-pool-game/231317#231317 – Lewis Miller May 26 '16 at 22:33
  • So a Newton cradle with something else than balls would a priori not work? – anderstood Aug 23 '16 at 20:27
  • @anderstood It would probably resemble Newton's cradle for the first few bounces if the nonlinearity is especially weak. Eventually those nonlinear terns should lead to the population of other modes and the periodic behavior should be disrupted. I'm just using theory as a guide since I have never observed such a modified system. – Lewis Miller Aug 25 '16 at 02:13
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    I doubt that the people who make them are aware of how special the physics of Newton's cradle is. – Lewis Miller Aug 25 '16 at 02:17
  • Your answer implies that Newton cradle would not work with rodsinstead of balls, right? I made some rough experiments with three small rods and it seems to work exactly the same way (even with uneven contact surface !). – anderstood Mar 05 '19 at 14:50
  • Uneven surface contact would cause similar behavior. As long as the contacts are point wise the nonlinear behavior will persist – Lewis Miller Mar 05 '19 at 15:23
  • So, if I understand correctly, you are saying that the uneven surface induces a nonlinearity. But strangely, the observed behaviour seems to be exactly the solution to a 1D linear hyperbolic problem (d'Alemberts PDE). – anderstood Mar 05 '19 at 15:31
  • I'm not familiar with d'Alemberts PDE, but yes, the uneven contact does cause a nonlinear response. – Lewis Miller Mar 05 '19 at 15:37