Energy Measurements
//Archimedes Thoughtful// an oil painting by Domenico Fetti made in 1620.
Archimedes Thoughtful an oil painting by Domenico Fetti made in 1620.
Consider some laboratory measurements of $E$, the mechanical energy. This page reviews some terms often used to compare observations with theory. Let the experiment be accomplished by any combination of observation and inference whatsoever provided only that it satisfies the professional standards of experimental physicists. For example this means that instruments are painstakingly calibrated. And any new measurement techniques are carefully compared with previous methods so that any systematic variations can be evaluated. Ideally experiments are repeated and confirmed by different scientists in other laboratories. So overall; measurement is a communal activity, with ancient roots, that links specific laboratory practice to the reproducible report of some number. The twentieth century has left us with an outstanding legacy of data about nuclear particles that come from measurements like this. Some amazing displays of international teamwork give reports with more than half a dozen significant figures ! ArchimedesXlink.png is supposed to have implored, 'give me a place to stand on, and I will move the Earth'. Experimental physicists have provided some very firm ground, and WikiMechanics is built on their work. So we expect young scientists to move mountains.

Any measurement of a particle presumably involves some sort of interaction that changes its quark content. The change may be small, maybe even negligible, but nonetheless there is always a logical distinction between an observed value and the theoretical concept of the energy of an isolated particle. The customary way of assessing this is to make many observations, so consider a series of $N$ measurements with results noted by $E^{1}, \, E^{2}, \, E^{3} \ \ldots \ E^{\it{k}} \ \ldots \ E^{\it{N}}$. These observed values are related to $E$ the theoretical idea of energy by

$E = \tilde{E} \pm \delta E$

where $\tilde{E}$ is a typical or representative value called the experimental average. The other number $\delta E$ describes the variation in observed values, it is called the experimental uncertainty. For 'good' measurements $\delta E$ is small enough so that $E$ and $\tilde{E}$ are interchangeable thus reconciling theory and observation. Usually the experimental average is determined from the arithmetic meanXlink.png of the set of observations

$\tilde{E} = \frac{1}{ N} \sum_{k=1}^{N} \; E^{k}$

and the experimental uncertainty is represented by their standard deviationXlink.png

$\delta E = \sqrt{ \frac{1}{N} \sum_{k=1}^{N} \left( E ^{k} - \tilde{E} \right)^{2} \; }$

Another important number is the coefficient of variationXlink.png in the data which is defined by the ratio $\delta E / \tilde{E}$. The inverse of this quantity is known as the signal-to-noiseXlink.png ratio

$\varsigma = 10 \log{ \left(\tilde{E} / \delta E \right) }$ (dB)

expressed on a logarithmic scale in units of decibelsXlink.png.

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