The density of a small metal block is found from its mass and its volume ,

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Uncertainty in measurement

Measurement and Method, Ch Type A, Type B and the uncertainty budget

Problem

The density of a small metal block is found from its mass

m

and its volume

V

\rho = m/V

. A correction

\delta_{\rm buoy}

for the buoyancy of the air acting on the block during weighing has a best estimate of zero but a non-zero uncertainty, so the measurement model is

\rho = (m + \delta_{\rm buoy})/V

. Ten independent weighings of the block give a mean

\bar m = 89.34\,\mathrm{g}

with sample standard deviation

s = 0.05\,\mathrm{g}

. The balance's calibration certificate states the reading is correct to within

\pm 0.03\,\mathrm{g}

at coverage factor

k = 2

. The buoyancy correction

\delta_{\rm buoy}

is judged, from the density of air and the block, to lie within

\pm 0.02\,\mathrm{g}

of zero on the mass reading, with nothing favouring any part of that range. The block's volume, found separately from calliper measurements of its three sides and their propagated uncertainty, is

V = 1.000\times10^{-5}\,\mathrm{m^3}

with

u(V) = 6\times10^{-9}\,\mathrm{m^3}

. (a) Write the measurement model with every input classified (measured, correction with zero best estimate, or exact). (b) Evaluate the standard uncertainty of each mass-related row and combine them into a single

u(m)

. (c) Combine

u(m)

and

u(V)

into

u_{\rm c}(\rho)

and quote

\rho

with its standard uncertainty in

\mathrm{kg\,m^{-3}}

. (d) State which row of the budget dominates and what change to the experiment would reduce it first.

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