A cup of coffee cools according to Newton's law of cooling, T(t) = T_ room + (T_0 -…

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 cup of coffee cools according to Newton's law of cooling,

T(t) = T_{\rm room} + (T_0 - T_{\rm room})e^{-kt},

where

T_{\rm room}=20.0\text{°C}

is the known, fixed room temperature,

T_0

is the (unknown) initial temperature of the coffee and

k

(in

\mathrm{min^{-1}}

) is the unknown cooling rate. Five readings are taken with a thermometer whose stated resolution is

\pm 0.5\text{°C}

at every single reading:

$t$ (min)	0	5	10	15	20
$T$ (°C)	95.0	82.0	71.0	63.0	57.0

(a) Write the weighted sum of squared residuals

\chi^2(T_0,k)

for this model and this data set, stating the weight attached to each point and why it takes that value. (b) Evaluate

\chi^2

at the trial parameters

T_0=95.0\text{°C}

k=0.0300\,\mathrm{min^{-1}}

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