[Physics] Experimental data for asymmetric Newton cradle

classical-mechanicsexperimental-physicsnewtonian-mechanics

Using a "successive impact model" (as if each ball were seperated from the other ones), I produced the following animations:

You can see any combination of balls with masses of 1 or 2 (left) or 1 and 4 (right).

Unfortunately, I do not have any Newton cradle to make some experiments, and I'd like to compare my results with observations. I contacted the author of this website, which is the most instructive one I found about asymmetric Newton cradle. In particular, he writes that the "OoO" configuration behaves as the "ooo" one, which is not what I simulate. But the author is not sure whether who is wrong because he did the experiments a long time ago.

I am especially interested in asymmetric Newton cradle with simultaneous impacts: left and right balls collinding at the same time.

Please let me know if you have any idea on where to find such experimental data.

Edit Just of few general remarks to avoid extending the comments.

Energy and momentum are conserved. However, they are sufficient to ensure uniqueness of the post-impact velocities only when there are 2 balls. The successive impact model precisely consists in propagating the impact ball after ball, and therefore leads to a unique solution (which conserves momentum and energy globally).
From the waves point of view, changing the mass of a ball can be done by increasing the diameter, or the density (or a combination of both). This simple model cannot account for such subtleties of course, but maybe it does not matter when the balls are "small enough".
Even if I think it's quite basic, following WetSavannaAnimalakaRodVance's comment, I am ready to hand out my Mathematica code to anybody who wants it. I'll check for good ways to share it.

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Experimental results can deviate from ideal. Outcomes depend sensitively on small differences in mass and alignment, and the extent to which kinetic energy is conserved. Alignment of non-identical balls is much more difficult. Considerable effort and expense may be required to achieve reliable results.

Although Simanek displays his apparatus, his webpage mostly contains animations. The only video clip (ooO <--> Ooo) does agree reasonably well with your simulation (#2 and #5). Perhaps his other videos were less convincing and left out for that reason.

Simanek's observation that the OoO collision is symmetrical might have been correct for the size of balls he used. (Judging by his video clip, I suspect that his apparatus did not actually show perfect symmetry.) In your simulation, although the 2:1 mass collision is not symmetrical, the 4:1 mass collision is much closer. Probably the n:1 collision gets more symmetrical as as n --> infinity.

Your simulation implements conservation of kinetic energy as well as momentum perfectly, so it ought to be more reliable than experimental results, which cannot guarantee perfectly elastic collisions and perfect alignment. However, investigating the differences with experiment may unearth interesting physics, particularly when the "successive impact model" does not apply.

The following Wolfram simulation includes the transfer of compression waves along the chain of balls :

http://demonstrations.wolfram.com/PhenomenologicalApproximationToNewtonsCradle/#

There are a number of very detailed and informative answers on the operation of and ideal conditions for Newton's Cradle already on Physics SE, such as

Newton's cradle
Newton's Cradle: why does it stay symmetric?

The group at California Institute of Technology (cited in the Wolfram demo) seems to have done the most recent published research (2008). They report different results when the balls are initially touching vs a small gap between them. You could try contacting them about their results :

https://www.researchgate.net/publication/253981162_Newton's_cradle_undone_Experiments_and_collision_models_for_the_normal_collision_of_three_solid_spheres

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Experimental Physics – How to Fit Power to Experimental Data

Being a mathematician, I feel that I should point out a common misconception here. Don't feel too bad, plenty of mathematicians (including myself) have fallen into this trap. Basically, if you want to fit data to power law using least squares methods, then you should NOT fit a straight line in log log space. You should fit a power law using non-linear least squares to the original data in linear space without any kind of transformation.

The basic intuition behind this is this. Least squares method assume that the error distribution is normal meaning that there is one true (unknown) value which you are trying to measure and the measurements you are taking are above/below the true value staying close to the true value so the distribution is gaussian. The problem is that log is not a linear function so when you take log log of the data, numbers bigger than one are pushed together and numbers less than one are spread apart and namely, the error distribution is not normal any more in log log space....meaning that least squares is not guaranteed to converge to the true values any more. In fact it has been proven that fitting in log-log space consistently gives you biased results.

Fitting a straight line in log-log was fashionable back in the day before GHz processors. Now whatever you are using to do the computation, most likely has the ability to do non-linear least squares power law fit to the original data so that is the one you should do. Since power-law is so prevalent in science, there are many packages and techniques for doing them efficiently, correctly, and fast.

I refer you to these published results and a rant here. This is the my question which sparked this discussion.

Addendum:

So by OP's request here is an actual example. The actual function is $y=f(x)=2x^{-4}=ax^b$. I take a bunch of points from $x\in[1,2]$ and $x=10$. Note that $f(10)=0.0002$ and I change this $y$-value to $y=0.00002$ while keeping all others the same and then fitting them using least squares to estimate $a$ and $b$.

Using non-linear least squares fit to the functional form $y=ax^b$ gives us

\begin{eqnarray} a&=&2\\ b&=&-4\\ r^2&=&0.9999. \end{eqnarray}

Using linear least squares fit to the functional form $y=mx+n$ in log-log space gives us

\begin{eqnarray} a &=& exp(n) = 2.46\\ b &=& m = -4.57\\ r^2 &=& 0.9831. \end{eqnarray}

The $95\%$ bounds for $b$ are $(-4.689, -4.451)$ which doesn't even include the true value for $b$. Basically log pushes numbers bigger than one together and spreads apart numbers less than one. So in linear space changing $y=0.0001$ to $y=0.00001$ is inconsequential. The algorithm doesn't care too much about it. The change in the residual is very tiny. In log-log space the difference is rather huge (an entire order of magnitude) and the point has become an "outlier" from the nice linear trend. The change in residual is now large. Since the algorithm is trying to minimize the sum of the squares, the algorithm can't ignore the deviation and must try to accommodate the outlier point so the line is skewed messing up the slope and the intercept.

Here is the MATLAB code and the graphs. You can easily reproduce this and play with it yourself. Blue are the data points. Black is the power fit in linear space. Red is the fitted line from log-log least squares fitting. The top panel shows all three in linear space and the bottom panel shows all three in log-log space to emphasize the differences. The exponent is off by more than a half between the two methods. But how see how good the second fit it ($r^2=0.98$) and the confidence interval is nowhere near the true value of $b$. The algorithm is very confident that $b\neq-4$.

The same effect will work in reverse with large changes in large numbers. For example, changing a data point from 10000 to 20000 in linear space will cause a huge change but in log-log space the change is not a big deal at all so the algorithm will again give misleading results.

close all
clc
clear all

x = [1:0.01:2 10];
y = 2*x.^(-4);
y(end)
y(end) = 0.00002;

% The power law fit on the original data
ft = fittype( 'power1' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
[fitresult1, gof1] = fit(x',y', ft, opts )

% The linear fit in log-log space
logx = log(x);
logy = log(y);
[fitresult2, gof2] = fit(logx',logy',fittype( 'poly1' ))

% The plots
x1 = min(x):0.01:max(x);
y1 = fitresult1.a*x1.^fitresult1.b;
y2 = exp(fitresult2.p1*log(x1)+fitresult2.p2);
figure
subplot(2,1,1)
plot(x,y,'o',x1,y1,'k',x1,y2,'r')
xlim([min(x),max(x)+1])
subplot(2,1,2)
loglog(x,y,'o',x1,y1,'k',x1,y2,'r')
xlim([min(x),max(x)+1])
legend('Data','Power Fit','Linear Fit')

enter image description here

Second Addendum:

Does non linear least squares manage a set of points with experimental/numerical errors?

I am assuming by manage you mean if NLLSF will give us an unbiased estimate of the parameters. The answer to that is, least squares fitting works if and only if the errors are gaussian. So if the errors are random errors (with the expected value and arithmetic mean being zero) then least squares gives you an unbiased estimate of the true parameter values. If you have any type of systematic error then least squares is not guaranteed to work. We often assume/want-to-think that we only have random error unless there is good evidence to the contrary.

And also does give the uncertainties in both a and b based on those parameters?

Again assuming errors being normally distrubuted, there are standard methods for estimating the error deviation. The mean is already assumed to be zero. We estimate the standard error for each parameter and then we use it to compute the confidence intervals. So yes, if I ONLY have experimental data and I have no idea what the real values of the parameters are, then I can still compute the confidence intervals.

Now an introductory bibliographic reference other than wikipedia would be very helpful.

I don't know of a good stats/regression book I can recommend. I would say just pick your favorite intro to stats book. And there are those papers by Clauset that I linked to both available on Arxiv for free.

TL;DR

Physicist think everything is a power law. It isn't. Even if the data looks like its power law, it probably isn't.
If you do legitimately have a power law, don't fit a straight line in log-log space. Just use NLLSF to fit a power law on the original data. If there is still any doubt, I can easily make up a numerical example and show you concretely how different the two answers can be.

[Physics] Why do the experimental data for Hooke’s law not match the expected data

The coils of the spring are touching one another and the spring is initially under self-compression so it takes a finite force to move all the coils away from one another and for the spring to behave as you expected.
That initial part of the source vs stretch curve is real (you had obtained data whilst undertaking an experiment) and hence should not be ignored.

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Experimental Physics – How to Fit Power to Experimental Data

[Physics] Why do the experimental data for Hooke’s law not match the expected data

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