Experimental Uncertainty Analysis - Introduction

Introduction

Rather than providing a dry collection of equations, this article will focus on the experimental uncertainty analysis of an undergraduate physics lab experiment in which a pendulum is used to estimate the value of the local gravitational acceleration constant g. The relevant equation for an idealized simple pendulum is, approximately,

T\,=\,2\,\pi \,\sqrt {{L \over g}} \,\,\left{\mathbf{\,\,\,\,\,\,\,\,\,Eq(1)}}

where T is the period of oscillation (seconds), L is the length (meters), and θ is the initial angle. Since θ is the single time-dependent coordinate of this system, it might be better to use θ0 to denote the initial (starting) displacement angle, but it will be more convenient for notation to omit the subscript. Solving Eq(1) for the constant g,

\hat g\, = \,{{4\,\pi ^2 L} \over {T^2 }}\,\,\left^2{\mathbf{\,\,\,\,\,\,\,\,\,\,\,\,Eq(2)}}

This is the equation, or model, to be used for estimating g from observed data. There will be some slight bias introduced into the estimation of g by the fact that the term in brackets is only the first two terms of a series expansion, but in practical experiments this bias can be, and will be, ignored.

The procedure is to measure the pendulum length L and then make repeated measurements of the period T, each time starting the pendulum motion from the same initial displacement angle θ. The replicated measurements of T are averaged and then used in Eq(2) to obtain an estimate of g. Equation (2) is the means to get from the measured quantities L, T, and θ to the derived quantity g.

Note that an alternative approach would be to convert all the individual T measurements to estimates of g, using Eq(2), and then to average those g values to obtain the final result. This would not be practical without some form of mechanized computing capability (i.e., computer or calculator), since the amount of numerical calculation in evaluating Eq(2) for many T measurements would be tedious and prone to mistakes. Which of these approaches is to be preferred, in a statistical sense, will be addressed below.

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