A spring of unknown stiffness is loaded with different masses

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 of unknown stiffness

k

is loaded with different masses

m_i

, and for each mass the period is found by timing

N_i

complete oscillations with a stopwatch whose reaction-time uncertainty is

\delta t = 0.2\,\mathrm{s}

on the total timed interval; this gives a period uncertainty

\sigma_i = \delta t/N_i

for that measurement. The data (mass, number of oscillations timed, measured period) are:

$m_i$ (kg)	0.10	0.20	0.30	0.40
$N_i$	20	20	15	10
$T_i$ (s)	0.451	0.634	0.779	0.900

The model is

T(m)=2\pi\sqrt{m/k}

, with

k

the single free parameter. (a) Write the weighted sum of squared residuals

\chi^2(k)

for this model and data set, giving the value of each

\sigma_i

and explaining why the weights differ from point to point even though the stopwatch itself is equally precise every time. (b) Evaluate

\chi^2

at the trial value

k=20\,\mathrm{N\,m^{-1}}

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