Two marks on a laboratory track are read off the same steel rule

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Uncertainty in measurement

Measurement and Method, Ch Propagating uncertainty into the quantity you actually want

Problem

Two marks on a laboratory track are read off the same steel rule, at positions

x_1 = 85.00\,\mathrm{cm}

and

x_2 = 32.00\,\mathrm{cm}

. The rule's own length may be systematically long or short by a fraction

\kappa

, with

u(\kappa) = 0.15

per cent (one standard deviation), and this scale error shifts every reading taken with the rule by the same fraction. Apart from this shared scale error, the two readings are exact. The separation of the marks is

\Delta x = x_1 - x_2

. Someone argues: "

x_1

and

x_2

share the same rule, so they are correlated, and correlation always makes an uncertainty bigger than treating the readings as independent. Treating them as independent gives

u(x_1) = 0.15\%\times85.00\,\mathrm{cm} = 0.128\,\mathrm{cm}

and

u(x_2) = 0.15\%\times32.00\,\mathrm{cm} = 0.048\,\mathrm{cm}

, so

u(\Delta x) = \sqrt{0.128^2+0.048^2}\,\mathrm{cm} = 0.14\,\mathrm{cm}

; the correlated value must be larger than this." ((a)) Write

x_1

and

x_2

in terms of the true rule readings

\ell_1

\ell_2

and the shared scale error

\kappa

, and use this to find the correct

u(\Delta x)

. ((b)) Say whether the quoted claim is right, and explain the error in the reasoning.

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