Re: How does a Newtons Cradle work?

This is a really good question.

When I first read the question and the situation of two balls coming in from one side producing two balls coming out on the other, I thought "sure this will happen if the balls are really separated ever so slightly. Symmetry would prevail two in -> two out."

Often, in physics, engineering, and mathematics, a reasonable conjecture based on a property like symmetry, finiteness, or, linearity is a great place to start when solving a problem. Evaluating the symmetry of an equation or system can really cut down on potential errors and can help one avoid spending unnecessary effort. One of the best examples of considering symmetry before proceeding is evaluating an integral in a physics problem. An odd function integrated over an even interval is zero.

So for example if in a problem I have to itegrate the function X³ over the interval from -100 to +100, I know right away it must be zero. As another example, suppose I have to evaluate an integral involving the cosine function, I can use the idea of symmetry to give me an idea if my final answer is correct or not. If the symmetry indicates the answer should not be zero and I get a zero, I would be very suspicious.

Using symmetry may seem like a cop out on going into a gory detailed answer, but it really is applying the principles of physics/math from the beginning to avoid blind alleys and pitfalls.

I found a couple webpages on the subject.

This page seems to follow the same thought process I had.

http://www.roe.ac.uk/ajcwww/posts/newton.html

The author of the email Andrew Cooke makes some really good "thought experiment" comments. Here is a reproduction of the above

.....

        there was a thread a while back that was asking about newton's
        cradle (the balls on strings that bounce against each other).

        i can't find the thread now, but it has been bothering me for a
        while, and i've finally sorted out in my head what was
        confusing me.

        if anyone is interested (sorry if it has already been explained
        in this way) it is a neat example of how conservation laws
        come from symmetry.

        the original post wanted to know why, if two balls were swinging,
        two balls came off `the other end'.  people (engineers?)
        talked about the balls not touching perfectly, the finite
        speed of sound, etc. while physicists talked about it being a
        result of the conservation of energy and momentum.

        i believe that, while the engineers' comments may be correct,
        as an idealised physics problem, conservation of energy and
        momentum `require' the same number of balls to leave the other
        end.

        now, what was bugging me was the question - what else could happen
        anyway?  i mean, when two balls `on the left' come down, we
        would be really surprised to see nothing happen on the right!
        indeed, if more than one thing could happen - say either one ball
        moving twice as fast, or two balls moving as `normal', how could
        `nature' decide which did occur?

        taking this further, if we dropped two balls on one side, and
        only one came off on the other, then we could film this, and
        play the film backwards.  what makes the film played backwards
        different from real life?  or what makes balls at the left
        different from those on the right?

        worrying about this on the way to work i realised that we
        expect the `normal' thing to happen because of symmetry.  symmetry
        left and right in the balls.  symmetry in that we expect the
        same physics whether time runs forwards or backwards, and
        symmetry in that we expect physics to be the same at different
        places.

        i think you can see that the symmetry left and right is just
        something that is a property of newton's cradle.  that leaves
        invariance of physics with position and direction of time.

        from that, the connection to conservation of momentum and energy
        seems quite clear.

        maybe i just seem to be rambling.  for me it was quite a
        revelation.  i only understood the connection between symmetry and
        conservation laws last year, when reading this group.  this
        is the best illustration i've seen.

Tom "Even Oddity" Cull