ARTICLES
Mathematics of A Freely-hanging Slinky Spring (continued)
Tension was not the only force acting. Ignoring the first few loops
was a valid decision supported by later analysis of the data. Using
a new slinky allowed valid data to be collected; earlier work with
an old and damaged slinky fitted less closely to a parabola and
the data for small numbers of rings had to be discarded.
The teacher’s aims for some students to develop robust attitudes
towards measurement and data analysis were fulfilled. All
students became more experienced with data-fitting beyond
straight lines. The teacher learned more about presenting data,
imagining and quantifying errors, and more about mathematical
methods. What one cohort of students teaches him can then be
passed to later cohorts.
If repeating this experiment, we recommend replacing the
metre rule with a steel tape measure at least 2.0 metre long, to
measure more rings without disturbing the slinky. Parallax error
must be minimised. The slinky could be inverted to repeat the
measurements. A mobile phone app can be used to level the
retort ring.
Thanks to NBHS students Frank Wu and Sungyoon Kim for
assistance with recording and editing.
References:
We often use springs to measure forces from Year 7 onwards; the
linear scale is very easy to read. Sometimes the zero of a spring
balance for small forces must be adjusted if it is used horizontally
rather than vertically. This work demonstrated that the weight of
the spring and attachment hook does affect the zero.
More detailed explanations of the Legendre polynomials are
found in these sources.
Abramowitz M and Stegun I, Editors (1972) Handbook of
Mathematical Functions Dover, NY.
Olver F W J, Editor-in-Chief and Mathematics Editor (2010)
NIST Handbook of Mathematical Functions, National Institute of
Standards and Technology, US Dept. of Commerce. Cambridge
University Press
Conclusions:
To a good approximation, the height of the slinky was parabolic
as predicted by Hooke’s law. A small cubic term was detected
and multiple methods gave a consistent result for the spring
constant changing slowly with extension.
Weisstein E W. “Legendre Polynomial.” From MathWorld––A
Wolfram Web Resource.
http://mathworld.wolfram.com/LegendrePolynomial.html
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SCIENCE EDUCATIONAL NEWS VOL 68 NO 3