Mathematics of A Freely-hanging 2),
Slinky
Spring
then 2 rings
above (continued)
and below, then 3 rings above and below. These data were noisy but the tr
lines agree strongly with the results from the cubic treatment. Using a larger baseline reduced th
Figure 5. The residuals show only a slight bias . errors inherent in recording height to the nearest half-millimetre, reducing scatter and improving
40
y = 0.0774x + 34.005
R² = 0.5298
y = 0.0854x + 33.841
R² = 0.8672
39
y = 0.0835x + 33.875
R² = 0.917
38
y = 0.0004x 2 + 0.0668x + 34.046
R² = 1
37
36
35
from quadratic
from cubic
from quartic
experimental
by difference 4
by difference 2
by difference 6
Poly. (from cubic)
Linear (experimental)
Linear (by difference 4)
Linear (by difference 2)
Linear (by difference 6)
34
Figure 6. Deviation from Hooke’s Law is small.
33
Figure 6. Deviation from Hooke’s Law is small.
32
Figure 6 shows force plotted against extension. The experimental
points were generated by the teacher, using 12 rings extending
and 25 gram masses. Half the mass of the slinky was included to
give an averaged force at an averaged extension.
0
10
20
30
40
50
60
Spacing between rings (mm)
Figure 7. Several analyses show the same changes in the spring constant.
Figure 7. Several analyses show the same changes in the
spring constant.
A spring constant was calculated by dividing the force by the
extension between rings. Figure 7 plots this spring constant
against extension, which emphasises the differences between
the models.
Despite not being replicated, it could be argued that the
properties of the data set show the data were reliable, accurate,
precise, and valid. The 52 measurements had no large outliers;
clearly the measurement technique was reliable. Half the marked
increment is not always the limit of reading. In titrations, chemistry
students are taught to record burette readings to two decimal
places by using a hand lens and reading between the lines.
Figure 5 suggests the precision in recording the raw data limited
its accuracy. Close inspection of Figure 1 suggests that using a
cell phone camera and a steel tape measure with finer increment
lines makes possible readings to 0.1 mm.
We also directly calculated the spring constant by using the force
at each measurement, and finding the extension between rings
using the raw data for the rings above and below (labelled by
difference 2), then 2 rings above and below, then 3 rings above
and below. These data were noisy but the trend lines agree
strongly with the results from the cubic treatment. Using a larger
baseline reduced the errors inherent in recording height to the
nearest half-millimetre, reducing scatter and improving R 2 .
The extra experimental work independently confirmed the
changing spring constant.
Hooke’s Law describes springs subjected to small stresses.
For very small forces, the slinky rings were in physical contact.
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SCIENCE EDUCATIONAL NEWS VOL 68 NO 3