Gena Gorin Block III Physics SL 18 January 12

Lab #5: Hookes Law

Research The goal of this experiment is to ascertain via a controlled setup the deviation from Hookes Law of five metal springs, of various sizes and spring constants, due to a constant force being exerted on all possible pairs of springs in series. It is presumed that the equation governing this situation is equivalent to that governing a number of capacitors in series; that is, . Sais

assumption will be evaluated via percent error between k values derived from directly testing two springs and mathematically deriving the k value from individual constants. This setup specifically explores the quantitative difference between using several springs and mathematically deriving the equivalent spring constant, with a consideration for the uncertainties and variables unconsidered.

Variables The independent variable is the spring or combination of springs. The dependent variable is the vertical change in the spring length, allowing one to ascertain the spring constant. The controlled variables include the instruments, the temperature, and the mass.

Materials y y y y Spring stand. Five springs, marked as to be differentiable. 200-gram mass. Ruler.

Procedure y y Set up spring stand; attach spring 1 to one end. Measure the initial position of the bottom of the spring.

y y y y

Attach the 200 g mass to the bottom of the spring. Measure the final position of the bottom of the spring. Repeat with springs 2-5. Repeat with all ten possible combinations of springs.

Data For all trials, m = .2 kg, g = 9.81 ms-2, and F = 1.962 N. The uncertainties for these quantities are unknown and are thus ignored as negligible. Significant digits are taken into account in the last table to avoid significant numeric losses prior.

Table 1: Individual Spring Constants (Nm-1)Spring Spring 1 Spring 2 Spring 3 Spring 4 Spring 5 x1 (cm).1 22 22.7 23.3 21.9 25.8 x2 (cm).1 14.8 16 15.8 14.5 19.2 x (cm).2 7.2 6.7 7.5 7.4 6.6 x (m).002 0.072 0.067 0.075 0.074 0.066 % unc 2.777778 2.985075 2.666667 2.702703 3.030303 k (Nm ) 27.25 29.28358 26.16 26.51351 29.72727-1

k unc (Nm ) 0.756944 0.874137 0.6976 0.716581 0.900826

-1

Springs x1 (cm).1 S12 28 S13 28.3 S14 28.4 S15 24.2 S23 27.7 S24 28 S25 24.2 S34 28.6 S35 24.5 S45 25.2

Table 2: Series Spring Constants (Nm-1) x2 (cm).1 x (cm).2 x (m).002 % unc 13.7 14.3 0.143 1.398601 13.2 15.1 0.151 1.324503 13.1 15.3 0.153 1.30719 10.3 13.9 0.139 1.438849 13.3 14.4 0.144 1.388889 13.5 14.5 0.145 1.37931 10.2 14 0.14 1.428571 13 15.6 0.156 1.282051 10.1 14.4 0.144 1.388889 10.5 14.7 0.147 1.360544

k (Nm-1) k unc (Nm-1) 13.72028 0.191892 12.99338 0.172098 12.82353 0.167628 14.11511 0.203095 13.625 0.189236 13.53103 0.186635 14.01429 0.200204 12.57692 0.161243 13.625 0.189236 13.34694 0.181591

Table 3: Theoretical Spring Constants (Nm-1) Spring 2 Spring 3 Spring 4 Spring 5 Spring 1 14.11511 13.34694 13.43836 14.21739 Spring 2 13.8169 13.91489 14.75188 Spring 3 13.16779 13.91489 Spring 4 14.01429

Table 4: Theoretical v. actual spring constantsSpring S12 S13 S14 S15 S23 S24 S25 S34 S35 S45 kt (Nm-1) kt unc (Nm-1) 14.11511 0.813433 13.34694 0.726667 13.43836 0.736486 14.21739 0.825758 13.8169 0.780896 13.91489 0.791448 14.75188 0.887381 13.16779 0.707027 13.91489 0.792727 14.01429 0.80344 [theoretical] ka (Nm-1) ka unc (Nm-1) 13.72028 0.191892 12.99338 0.172098 12.82353 0.167628 14.11511 0.203095 13.625 0.189236 13.53103 0.186635 14.01429 0.200204 12.57692 0.161243 13.625 0.189236 13.34694 0.181591 [actual] % error % error unc 2.797203 0.20032 2.649007 0.17931 4.575163 0.310547 0.719424 0.052136 1.388889 0.097787 2.758621 0.194954 5 0.372197 4.487179 0.298461 2.083333 0.147622 4.761905 0.337788 [per cent error] D 0.226713 0.201008 -0.04597 0.520379 0.399758 0.220954 -0.05042 -0.04508 0.313598 -0.0455

Analysis All spring constant calculations are done via F = -kx (or, for theoretical constants, ). The results, as seen in tables above and to the right, show quite clearly the spring constants for springs in series.S S12 S13 S14 S15 S23 S24 S25 S34 S35 S45

Conclusion

Table 5: Table 4 values, rounded kt kt ka ka %err 14.12 0.81 13.72 0.19 2.8 13.35 0.73 12.99 0.17 2.65 13.44 0.74 12.82 0.17 4.58 14.22 0.83 14.12 0.2 0.72 13.82 0.78 13.63 0.19 1.39 13.91 0.79 13.53 0.19 2.76 14.75 0.89 14.01 0.2 5 13.17 0.71 12.58 0.16 4.49 13.91 0.79 13.63 0.19 2.08 14.01 0.8 13.35 0.18 4.76

%err 0.2 0.18 0.31 0.05 0.1 0.19 0.37 0.3 0.15 0.34

The per cent error varies from 0.72% to 5% (relatively small values), thereby indicating the validity of the initial assumption, viz., that electromagnetic capacitors and springs, as energy storage components utilizing respectively electric fields and elasticity, are governed by essentially the same law of constant equivalence for series placement. This is, as a conclusion, reasonable.

Evaluation The procedure is reasonably precise; most of the deviations from ideally predicted behavior fall within experimental error: [ka - kaunc] [kt - ktunc] describes the difference between the lowest possible actual result and lowest possible theoretical result given that all theoretical values are higher than and have greater uncertainties than experimental results, if the number is positive, the experimental result, including uncertainty, is wholly contained within the domain of the

theoretical result. The quantitative data are shown in the D column of Table 4; six of ten actual results can be said to be fully accounted for in theoretical results. The fact that the theoretical values are greater than the experimental ones implies that, as the k constant is inversely proportional to the change in length, the theoretical change would be smaller than the actual one. This is reasonable, as actual springs have more points of elasticity than could be accounted for by (a single application of) Hookes Law. Furthermore, linear, algebraic Hookes law does not account for the springs mass.

Improvements The key improvement in this lab would be to use a greater range of springs with a wider variance in k constants. If the springs are larger, the change in length would be greater, leading to more negligible uncertainty values.