am not sure I'm following completely. I've tried this experiment now with several different model cradles (all with 1st ball mass 2m and remaining balls mass m) and get the same result each time: the last ball flies out and the remaining balls barely move. The balls clearly do *not* swing out in a fan pattern predicted by pairwise collisions. then the balls come together and interact again. after this *second* set of collisions the the pairwise model seems accurate (because the balls started off slightly apart)— the balls fan out in the predicted pattern. On Mon, Dec 13, 2010 at 10:20 AM, Bill Thurston <wpt4@cornell.edu> wrote:
Thanks for filling in the numbers, and confirming the intuition that thinking of it as a chain of 2-dimensional collisions is a good model.
To summarize: the reason 2 balls can't fly out at the other end when a double-mass ball hits is that the intermediate balls can't individually transmit the given momentum with the given amount of energy. An object of half the mass requires twice as much energy to carry a given amount of momentum, hence the initial ball must retain some of the momentum to be transmitted later, creating a more complicated outcome. It's quite different than when two balls are dropped, because in that case instead of one indigestible large dose of momentum, two quick (to us) doses of half the strength are administered.
As I said in my earlier message, I think the Newton's cradle system with the initial mass twice the weight is continous at the time of collision near the particular trajectory we've been discussing, where all but the heavy ball are initially stationary. Trajectories for collisions of equal masses in one dimension are continuous near the multiple collision loci. The only way there could be a discontinuity (when the order of collisions changes) would be if the heavy ball and two (or more) others were all converging relative to each other toward a simultaneous collision. This can't happen near the time of the first collision. Later on, when the motion is more chaotic, there could be discontinuities.
Question: is there some sequence of masses for Newton's cradle that would exhibit visible discontinuity when the ball at one end is dropped against the others? Perhaps sometimes the initial ball flies out energetically, and sometimes not.
Question: what about the break shot in billiards, where initially the balls are racked so they are in contact in a triangular pattern? Is the outcome of the collision discontinuous in this situation, or do the trajectories just vary quickly but continuously depending on the initial impact?
Bill Thurston On Dec 13, 2010, at 12:08 PM, Veit Elser wrote:
While I don't want to discourage the mathematical tangents this thread has generated, since it started as a physics question I think you deserve a physics answer.
The role of physics is not just to provide general principles, such as conservation laws. Sometimes it just comes down to numbers, plain and simple. The rules that one should apply to swinging steel balls are very different than what is appropriate to, say a couple of krypton atoms adsorbed on a graphite surface and struck by a third krypton atom. In that case one needs to consider the force of every atom on every other atom as the collision unfolds. By contrast, steel balls are simpler.
So here are some numbers. The speed of sound in steel is about 5000 m/s. A ball dropped from 5 cm reaches a speed of about 1 m/s. This tells us that the stresses imparted to the steel balls can be treated as quasi-static to a very good approximation. There are no shock waves.
The Young's modulus of steel is E = 200 Gpa. If we uniaxially compress a cylinder with cross sectional area A and length L by x, then the stored elastic energy is E A x^2/(2L). Equating this with the energy of the incident ball, say corresponding to m = 1 gm dropped an equivalent vertical rise of h = 10 cm or mgh = .01 J, I get x = 1 micron when A and L are of order 1 cm^2 and 1 cm. One micron is .01 the diameter of a human hair. Using the Young's modulus again, we find that the pressure at maximum compression, E x/L, is about 10 MPa. By contrast, the pressure supporting the weight of a ball placed on top of another is only about mg/A = 1 kPa -- four orders of magnitude smaller. So even when two balls in Newton's cradle are "in contact" because the strings supporting them make a slight angle with respect to the vertical, they may as well be miles apart with regard to the scale of the forces.
To help you digest these numbers, think of a pair of steel balls as two very stiff springs. The springs are in a collision course. The springs have a natural oscillation period but the collision is very slow on this scale. Start playing Smetana's Die Moldau (or Ravel's Bolero, your choice) right when the springs first touch. At about the time the Moldau crescendos into the Elbe the kinetic energy is completely and adiabatically (without the generation of entropy) converted into elastic energy. The springs will have compressed by a fraction of a millimeter. If you picture this in the center-of-mass frame, all motion ceases at an instant that marks the half-way point of the collision; thereafter the rest of the collision unfolds as the time reversal of the first half (and so even anharmonicity is not an issue).
I hope these numbers have convinced you that the pairwise collision model is the appropriate approximation to use for Newton's cradle, billiard ball dynamics, etc.
Here's a fun application of the pairwise model that you can try at home. Hold a basketball while resting a tennis ball on top of it. Now drop the basketball (the tennis ball should be co-moving and nearly in contact). You'll find that after all the collisions (pairwise!) have finished the tennis ball is shot up to a surprising height.
Veit
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