A mass is hung from a light spring and set oscillating vertically; the period of one…

PHYSICS · Multi-part · 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

A mass is hung from a light spring and set oscillating vertically; the period of one full oscillation is timed. The spring constant is obtained from

k_{\rm s} = \frac{4\pi^2 m}{T^2},

where

m

is the suspended mass and

T

is the period. Take the mass measurement and the period measurements as uncorrelated, since they are made with different instruments. The mass is read once, on an electronic balance, as

m = 0.2500\,\mathrm{kg}

. The balance's calibration certificate states an uncertainty of

\pm 0.0003\,\mathrm{kg}

, with nothing said about the shape of that uncertainty beyond the bound itself; the balance's display resolution is

0.0001\,\mathrm{kg}

. The period is timed thirty times with a photogate timer. The thirty readings have a mean

\bar T = 0.8420\,\mathrm{s}

and a sample standard deviation

s = 0.0100\,\mathrm{s}

. The timer's own display resolution is

0.0001\,\mathrm{s}

. Treat every stated bound as a rectangular distribution, and use the sensitivity coefficients that moving

m

alone and

T

alone give,

c_m = k_{\rm s}/m

and

c_T = -2k_{\rm s}/T

. ((a)) Evaluate the standard uncertainty of the mass,

u(m)

, combining the calibration bound and the display resolution, stating the distribution assumed for each. ((b)) Evaluate the standard uncertainty of the mean period,

u(\bar T)

, combining the Type A scatter of the thirty timings with the Type B contribution from the timer's resolution, stating the degrees of freedom of the Type A part. ((c)) Compute

k_{\rm s}

, then build a two-row uncertainty budget (mass, period) by moving each input by its own standard uncertainty in turn, and find the combined standard uncertainty

u_{\rm c}(k_{\rm s})

. ((d)) Report

k_{\rm s}

with an expanded uncertainty at a coverage factor of

2

. ((e)) State which row of the budget should be attacked first to reduce

u_{\rm c}(k_{\rm s})

the most, and find the new

u_{\rm c}(k_{\rm s})

if that row's contribution

c_ju(x_j)

could be halved.

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