0
-0.5
-1
ARTICLES
-1.5
-2
Mathematics
of A Freely-hanging Slinky Spring (continued)
0
20
40
60
number of rings
Figure 3. Residuals from the parabola, and fitted cubic.
0.9
parabola
y = -0.000003x 4 + 0.000326x 3 - 0.012844x 2 +
0.189652x - 0.810305
R² = 0.127223
0.75
0.6
0.45
0.3
0.15
quartic
x o 0.05287 0.93682 0.65015
y o (mm) -0.83992 2.23683 1.56933
slope 0 2.5E-17 0
Parabola: y = 0.2814x 2 ;
0
Cubic: y = –0.0002436x 3 + 0.3023x 2 ;
-0.15
Quartic: y = –0.000002766x 4 + 0.00007564x 3 + 0.2903x 2
-0.3
-0.45
The one, two or three constants in these equations summarise
the information that was extracted experimentally. The parabola
can give only a fixed estimate of the spring constant.
-0.6
-0.75
cubic
0
10
20
30
40
number of rings
50
60
From these equations the spacing between the rings was
evaluated as the slope, dy/dx. The force acting was calculated
as the number of rings times 2.10 gram per ring times 9.80 m/s2
for gravity.
Figure 4. The residuals from a cubic fit are consistent
trying residuals
to record data
to a the cubic
nearest fit half-millimetre.
Figure with
4. The
from
are consistent with
trying to record data to the nearest half-millimetre.
Figure 4 plots the residuals from a cubic fit. The quartic curve is
the fourth Legendre polynomial, orthogonal to all lower terms. Its
magnitude is below the precision of the recorded data. The R2 for
this term was larger than expected from pure noise. Later work
suggested this term was just noise. To accurately and reliably
measure the fourth order term would require great effort, most
likely by recording data to better than 0.1 mm by using close-up
photographs of each ring and steel tape, counting more rings,
and several repetitions to establish statistical reliability.
Figure 5 shows that recording all the data to the nearest half-
millimetre was worthwhile. Rounding error predicts a uniform
distribution of errors in the range –0.25 to +0.25 mm. This was
almost achieved. The observation errors were always less than
the thickness of the slinky ring. Towards the top of the slinky,
the slope of the metal ring was larger, which made reading the
position harder.
Figure 5. The residuals show only a slight bias .
Figure 5. The residuals show only a slight bias.
Measuring the spring constant:
When the raw data was fitted to polynomials, small linear and
constant terms were present. The slope of the curve represents
the spacing between the rings. We evaluated the position at
which this derivative became zero and evaluated our small errors
in the count of rings and the height. We did this for the best fit
polynomials of orders 2, 3 and 4, and displaced the origin. This
eliminated the constant and linear terms. The changes are small
enough to be physically plausible.
25
SCIENCE EDUCATIONAL NEWS VOL 68 NO 3