A spring's force constant is measured by hanging fifteen different loads and…

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory

Computational Physics, Ch Finding the best parameters: minimisation and fitting

Problem

A spring's force constant

k

is measured by hanging fifteen different loads and recording the extension

x_i

for each applied force, with the model

F_i = k x_i

(Hooke's law, taken to hold exactly over the range used). Each force is read with the same uncertainty

\sigma_F = 0.050\,\mathrm{N}

, and the fifteen extensions satisfy

\sum_{i=1}^{15} x_i^2 = 0.980\,\mathrm{m^2}

. Because the model is linear in

k

, the objective

\chi^2(k) = \sum_{i=1}^{15} \frac{(F_i-kx_i)^2}{\sigma_F^2}

is exactly quadratic in

k

, so the curvature-based uncertainty

\sigma_k = \frac{\sigma_F}{\sqrt{\sum_i x_i^2}}

is exact rather than approximate. (a) Compute

\sigma_k

from the formula above. (b) A thousand synthetic replicas of the fifteen

F_i

are generated at seed

2024

, each point scattered about the fitted line with the stated

\sigma_F=0.050\,\mathrm{N}

, and each replica is fitted for

k

. Predict the standard deviation you expect across the thousand fitted values of

k

, and say whether you expect it to agree with

\sigma_k

closely or only roughly, giving a reason tied to the model being linear. (c) A second batch of a thousand replicas is generated with the same seed rule but with the actual scatter of

F_i

doubled to

0.10\,\mathrm{N}

, while

\sigma_k

is still computed from the formula above using the stated

\sigma_F=0.050\,\mathrm{N}

. Predict the standard deviation of the thousand fitted

k

values in this second batch, and compare it with the unchanged

\sigma_k

of part (a).

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