# Gas station without pumps

## 2011 November 9

### Physics Lab 4: spring constants results

Filed under: home school — gasstationwithoutpumps @ 00:11
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In Physics Lab 4: spring constants, I assigned a lab exercise to be done while I was away  in Washington, DC, measuring a dozen different springs, getting spring constants for each, and trying to relate the properties of the springs to their geometric properties.  It took 2 sessions to do all the measurements, which were reported in Physics update.

On Friday, 4 Nov, we discussed the spring data. The force vs. stretch curves looked pretty linear, except for springs F and H, so we remeasured those:

```Force (in "kg", as read from the force gauge)
stretch
0.5cm   1.0cm   1.5cm   2.0cm
F                     0.36    0.475   0.605   0.695
F                     0.365   0.48    0.62    0.72
F                     0.355   0.465   0.595   0.715
F(avg)                0.360   0.473   0.607   0.710

H                     0.97    1.36    1.83    2.26
H                     1.01    1.44    1.95    2.28
H                     1.00    1.43    1.835   2.19
H(avg)                0.993   1.410   1.872   2.243```

To look at the data, I reformatted it all into an RDB file and used gnuplot:

```# Spring constant data
# length, diameter, and wire diameter in cm
# forces in "kg" at 4 different stretches
ID	old_l	diam	old_w	cm0_5	cm1_0	cm1_5	cm2_0	turns	length	wire
S	N	N	N	N	N	N	N	N	N	N
A	1.025	0.46	0.03	0.28	0.47	0.68	0.87	20	1.025	0.05
B	1.44	0.525	0.045	0.23	0.34	0.40	0.57	27	1.415	0.05
C	0.97	0.59	0.02	0.13	0.23	0.32	0.39	18.8	0.985	0.05
D	1.27	0.78	0.045	0.04	0.06	0.10	0.12	26	1.305	0.05
E	1.53	0.78	0.03	0.04	0.06	0.08	0.11	31.5	1.585	0.05
F	2.92	0.83	0.07	0.360	0.473	0.607	0.710	37.8	2.965	0.08
G	3.58	0.705	0.045	0.16	0.26	0.36	0.47	51	3.60	0.07
H	7.005	0.36	0.045	0.993	1.410	1.872	2.243	98.5	7.03	0.07
I	3.51	0.44	0.03	0.15	0.24	0.32	0.38	65	3.45	0.05
J	2.44	0.67	0.045	0.13	0.18	0.25	0.31	39.8	2.40	0.06
K	2.89	0.80	0.045	0.05	0.10	0.14	0.19	31.2	1.935	0.06```

(The old_l and old_w were the original length and wire measurements, which I am not using.) I then used a gnuplot script to look at the spring constants:

```g = 9.80665	#standard g, in m/s^2  (or N/kg)
cm_per_m = 100.

set title "Spring constants for springs from JN Distribution"

set xlabel "Spring type"
set xtics ("A" 0, "B" 1, "C" 2, "D" 3, "E" 4, "F" 5, "G" 6, "H" 7, "I" 8, "J" 9, "K" 10)
set xrange [-0.5:10.5]

set ylabel "Spring constant (N/m)"
set logscale y
set yrange [10:3000]

set key bottom right
set key box lw 0.3 height 1 width 2

plot	'<column cm0_5 cm1_0 cm1_5 cm2_0 < spring_data.rdb| rdb2gnuplot' using 0:(cm_per_m*g*\$1/0.5) title "0.5cm", \
'<column cm0_5 cm1_0 cm1_5 cm2_0 < spring_data.rdb| rdb2gnuplot' using 0:(cm_per_m*g*\$2/1.0) title "1cm", \
'<column cm0_5 cm1_0 cm1_5 cm2_0 < spring_data.rdb| rdb2gnuplot' using 0:(cm_per_m*g*\$3/1.5) title "1.5cm", \
'<column cm0_5 cm1_0 cm1_5 cm2_0 < spring_data.rdb| rdb2gnuplot' using 0:(cm_per_m*g*\$4/2.0) title "2cm", \
'<column cm0_5 cm1_0 cm1_5 cm2_0 < spring_data.rdb| rdb2gnuplot' using 0:(cm_per_m*g*(\$1+\$2+\$3+\$4)/5.0) title "weighted avg"
```

Note that the measurements for a stretch of 0.5cm consistently give a higher estimate for the spring constant than the other measurements, which are otherwise fairly consistent.

I was not, however, able to reduce this data to a constant dependent only on the material using various dimensions of the spring.  I had expected that the stiffness of the spring would be proportional to the square of the wire diameter and inversely proportional to the length of the wire (turns*π*diam), perhaps with some correction of diameter for wire width.  But no scaling I used substantially decreased the spread of the data, though I still believe that the same material was used in all the springs.

Anyone have any thoughts on what the right model for the stiffness of a coiled spring is?

## 1 Comment »

1. […] Both the stainless-steel rod and the spring are leftovers from previous projects: Two-electrode vs. four-electrode impedance spectroscopy and Physics Lab 4: spring constants results. […]

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