A spring is loaded with four forces and its extension measured

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

Mathematical Methods for Physics, Ch Estimation, fitting and hypothesis testing

Problem

A spring is loaded with four forces

F

and its extension

e

measured, each reading with the stated one-sigma uncertainty

\sigma_e

F

(N):

1.0,\ 2.0,\ 3.0,\ 4.0

e

(mm):

5.1,\ 9.8,\ 15.2,\ 19.7

\sigma_e

(mm):

0.2,\ 0.2,\ 0.1,\ 0.1

Assume Hooke's law

e = a + bF

with independent Gaussian errors on

e

, and fit a weighted straight line with weights

w_j = 1/\sigma_j^2

. ((a)) Find the intercept

a

and the gradient

b

, each with its uncertainty

\sigma_a

\sigma_b

. ((b)) Compute the minimised

\chi^2

and state the number of degrees of freedom

\nu

for this fit. ((c)) Compare

\chi^2

with its expected value

\nu \pm \sqrt{2\nu}

and say whether the fit should be trusted, naming what might be wrong if not.

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