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Asymmetrical Newton's Cradle
Name: John M.
Status: N/A
Age: 20s
Location: N/A
Country: N/A
Date: 2001-2002
Question:
When you are dealing with Newton's cradle, say you have 4 balls of
equal mass and you pull back balls 1 and 2. Balls 3 and 4 will both
go at the same velocity sticking to each other and traveling away from
the center. But, if you make the mass of ball 1 = 2x and the mass
of the rest of the balls equal x, why do balls 2, 3, and 4 each go at
a different velocity? Does it have anything to do with the larger
ball (ball 1) having only one pivot point while in the original
example, balls 1 and 2 had different pivot points? I would greatly
appreciate any input on this question. Thank you.
Replies:
John,
I can see where you would expect to get two balls going out at half the
initial speed. In a perfect situation, it would provide conservation of
momentum and kinetic energy. I can think of two things that can cause
irregularities. One is that the first ball is not EXACTLY two masses.
Being off by a few percent could cause problems. A much more likely problem
is position. If the centers of mass of all balls involved do not line up
perfectly, you can get quite unusual patterns. When all balls are
identical, lining them up is fairly easy. If one ball is twice as massive,
it is usually bigger. This makes judging a line-up much more difficult. I
cannot think of any way around the line-up barrier, if that's what is wrong.
Some would try using hollow balls for the lighter ones. The problem this
causes relates to resonance. A hollow ball can hold much more energy in its
vibrations. This can remove some kinetic energy from the collision. If you
can get two different metals, one almost twice as dense as the other, you
might have a system with fewer possible defects.
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
The simplest and most straightforward answer is that the balls move so
as to conserve kinetic energy and momentum. With two balls in and two
out, this is obviously satisfied. Now if you double the mass of the
first ball, for the three balls to move together with the same
momentum as the initial ball (2mv), they would have to have velocity
2/3*v. But to have the same kinetic energy, they would have to have
velocity square root of 2/3*v. Clearly they cannot have two different
velocities and so they move at different velocities.
You might try to calculate what other velocities the three balls could
have and still conserve energy and momentum. One obvious solution is
the two outer balls move off at the same speed as the incident ball.
Together they have the same mass as the incident ball and so with the
same velocity they must have the same energy and momentum.
But your experiment shows that the three balls move at different
speeds. With three balls, you have an infinite number of solutions.
Why your particular one is chosen by the balls then must depend on the
details of the collisions and is a difficult problem.
Incidentally, notice that putting a little putty (or chewing gum)
between two balls changes the motion drastically. That's because
kinetic energy is no longer conserved as some of the energy goes into
heating up the putty.
Dick Plano
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