Mathematics of A Freely-hanging Slinky Spring (continued)
3. Measure and record the position of each ring going up the
slinky.
4. Plot the data with the number of rings as x and the measured
distance from the bottom of the slinky as y. Fit a second-
order polynomial using the option for this built into Excel.
The program uses a Gaussian least squares method. This is
formally equivalent to using a limited subset of the Legendre
polynomials. Use the options to print the equation and
Pearson’s R2 on the graph to show how good the fit is.
5. Weigh the slinky, measure its height, and count the rings.
Calculate the mass of a single ring.
Possible inclusions:
ssible inclusions:
Results and Discussion:
The slinky had 70 rings. Diameter of slinky = 68 mm
Mass of slinky = 147.08 grams
In Figure 2, the parabola is an excellent fit to the data, as
predicted by Hooke’s Law. The first ring seen clearly separated
from the ring below was labeled as #4. 3½ rings might better
describe this first value, but the metre rule was nicely aligned for
minimal parallax error in the measurements and we chose not to
disturb it. (See Figure 1, the gap is really small.) The 70 rings of
the slinky stacked 53 mm high, so each ring was about 0.757 mm
thick. The slinky has a little bit of tension even at this spacing so
it collapses back readily to its compact shape.
Residuals should always be plotted. One outlying point revealed a
transcription error from the raw handwritten data. Figure 3 shows
the residuals from the parabolic fit. The pattern was recognized
as the third Legendre polynomial, the cubic curve orthogonal
(under uniform weighting) to the constant, linear and second
the third Legendre polynomial, the cubic curve orthogonal (under uniform weighting)
order terms. If fitting a cubic to the residuals, only this shape is
stant, linear
and second
order terms.
If fitting
a cubic
to the residuals,
only this shape i
possible.
This millimetre
variation
in data
stretching
across more
This millimetre
variation
data stretching
than variation
800 mm from
is small, real, and
than 800
mm is in small,
real, and across
shows more
a slight
Hooke’s
Law.
slight variation
from
Hooke’s Law.
{6. Measure the spring constant by repeatedly adding 25 gram
Measure the masses
spring constant
by repeatedly adding 25 gram masses to 12 rings.
to 12 rings.
7. Repeat with
the experiment
with a plastic slinky.}
Repeat the experiment
a plastic slinky.}
900
y = 0.281401x 2 - 0.029755x - 0.839135
R² = 0.999992
800
y = -0.000244x 3 + 0.302959x 2 - 0.566993x +
2.502311
R² = 0.999999
700
Plastic
(and permanent)
springs problem,
is a familiar
Plastic (and
permanent)
deformation deformation
of springs is of
a familiar
so new slinkies w
problem, so new slinkies were used to improve validity. A polymer
slinky left hanging vertically stretched obviously overnight.
improve validity. A polymer slinky left hanging vertically stretched obviously overnig
600
y = -0.0002x 3 + 0.0216x 2 - 0.5372x + 3.3414
R² = 0.85251
1.5
500
2
400
300
200
1
0.5
0
-0.5
-1
-1.5
100
-2
0
10
20
30
40
number of rings
50
Figure 2. The raw data, with cubic and parabolic fits.
Figure 2. The raw data, with cubic and parabolic fits.
sults and Discussion:
e slinky had 70 rings. Diameter of slinky = 68 mm Mass of slinky = 147.08 grams
40
60
Figure
3. Residuals from
from the
the parabola,
and and
fitted fitted
cubic. cubic.
Figure
3. Residuals
parabola,
0.9
0.75
0.6
0.45
Figure 2, the parabola is an excellent fit to the data, as predicted by Hooke’s Law. The first ring 0.3
n clearly separated from the ring below was labeled as #4. 3½ rings might better describe this 0.15
t value, but the metre rule was nicely aligned for minimal parallax error in the measurements and 0
chose not to disturb it. (See Figure 1, the gap is really small.) The 70 rings of the
24 slinky stacked -0.15
SCIENCE
EDUCATIONAL
NEWS
VOL 68
mm high, so each ring was about 0.757 mm thick. The slinky has
a little
bit of tension
even
at NO 3
spacing so it collapses back readily to its compact shape.
20
number of rings
60
0
0
-0.3
y = -0.000003x 4 + 0.000326x 3 - 0.012844x 2 +
0.189652x - 0.810305
R² = 0.127223