A spring's force law is fitted to loaded-spring data

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 law

F(x) = k(x-x_0)

is fitted to loaded-spring data, with

k

\mathrm{N\,m^{-1}}

and

x_0

\mathrm{mm}

. The curvature (Hessian) matrix of

\chi^2

, built by finite differences at the minimum, is

H = \begin{pmatrix} 5.0 & 3.0 \\ 3.0 & 4.0 \end{pmatrix}

in units such that

2H^{-1}

gives the covariance matrix directly in

\mathrm{(N\,m^{-1})^2}

and

\mathrm{mm^2}

. (a) Compute the covariance matrix

2H^{-1}

. (b) Read off the standard uncertainty in

k

and in

x_0

. (c) Compute the correlation coefficient between

k

and

x_0

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