another newtons cradle question...

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another newtons cradle question...
« on: 22/11/2007 08:28:19 »
A recent post by Andrew (sophiecentar) got me thinking, suppose the 5 balls are made of different metals. lead, brass, copper, steel and aluminium. if you pick the first ball and let it swing, does the last ball have a higher arc, than normal?

what happens if two or three balls are swung?
is there a noticeable difference to the effect of a cradle where all the balls are made from the same material?


Offline daveshorts

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another newtons cradle question...
« Reply #1 on: 22/11/2007 09:36:13 »
there are a couple of effects that changing material will have

Changing the density will mean that the balls will no longer all have the same mass so they won't do the neat stopping dead in the collision thing.
This is because a collision is symmetric about the centre of mass, so in the conventional newtons cradle has one ball hitting another identical mass ball
which is best looked at in the centre of mass frame which is moving at v/2
So in this frame the collision is symmetric, which means in the not moving frame the second ball ends up moving at v and the first is stopped.

If the masses are not the same the symmetry isn't there so the first ball will not stop if it is lighter it will bounce off, if it is heavier it will carry on.

The ball that is hit will bounce off faster (at a max of 2v) if it is lighter than the first ball.

The other effect is that materials such as copper and aluminium are not going to behave elastically so energy will be lost and things will slow down.


Offline syhprum

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another newtons cradle question...
« Reply #2 on: 22/11/2007 09:53:02 »
When the first ball is dropped energy is injected into the system that moves as a wave of compression thru the line of balls at the speed of sound in the relevant materiel.
This can be compared to an electrical delay line consisting of inductors and capacitors with an element of resistance that leads to the eventual dissipation of the energy (the ratio of stored energy to dissipated energy is refereed to as the "Q" factor that we will consider to be high for the moment ) .
When the balls are all of a similar materiel there is a complete transfer of energy at each ball to ball junction but as in the case of the delay line where the ratio of inductance to capacitance varies between sections some of the energy is reflected back to the input and causes a premature movement of the first ball and a secondary pulse of energy to move thru the system .
If the system is terminated by a ball dissimilar to the others the same energy will cause it to rise higher or lower as the case my be.
Balls of softer materials will have lower "Q" factors and cause a more rapid dissipation of energy but also cause lower amplitude reflections when their sound speed mismatches.
« Last Edit: 22/11/2007 17:24:10 by syhprum »