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Mechanical Energy

Mechanical energy was assessed by Paul Dirac pictured here at age 31.
Paul Dirac in 1933.

Consider a particle P that is described by its rest mass  m and momentum  p. And please notice that these numbers have been defined by a methodical description of sensation. The mechanical energy of P is defined by

E \equiv \sqrt{ c^{2}p^{2} + m^{2}c^{4} \rule{0px}{11px}\; }

where  c is a constant. This statement comes from Paul Dirac . As a special case for material particles, we can divide by  mc^{2} to get

E = mc^{2} \sqrt{ \; 1 + \left( p/mc  \right)^{2} \; }

The square root may be expanded in a binomial series as

1 + \dfrac{p^{2}}{2m^{2}c^{2}}  -  \dfrac{p^{4}}{8m^{4}c^{4}} + \dfrac{3p^{6}}{48m^{6}c^{6}} + \ldots

And if p \ll mc we can ignore the smaller terms to approximate the mechanical energy with the expression

E \simeq mc^{2} \left(      1 + \dfrac{p^{2}}{2m^{2}c^{2}} \right)

The requirement that p \ll mc is called a slow motion condition. An ethereal particle like a photon cannot move slowly because m = 0 so the condition cannot be satisfied by any value of the momentum.

The Lorentz Factor

The mass and momentum may also be be combined to specify yet another quantity

\gamma \equiv \dfrac{1}{\sqrt{ \; 1 - \left( p/mc  \right)^{2} \; \rule{0px}{10px} }}

Hendrik Antoon Lorentz, 1853β€”1928.

where  c is a constant. This number  \gamma is called the Lorentz factor after the Dutch physicist Hendrik Lorentz. His original work1H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat, page 225. Published by B. G. Teubner at Leipzig, 1909. expressed  \gamma differently. But later, after discussing velocity, we will see that the forgoing definition is equivalent. The Lorentz factor is used to classify particles. For example, when the slow motion condition applies, then  \gamma \simeq 1. But if  \gamma \gg 1, then we say that a particle is relativistic. To make a useful approximation for the Lorentz factor we expand the square root into a binomial series

\gamma = 1 + \dfrac{p^{2}}{2m^{2}c^{2}}  +  \dfrac{3p^{4}}{8m^{4}c^{4}} + \dfrac{15p^{6}}{48m^{6}c^{6}} + \ldots

Note that this is a little different from the previous series. But for non-relativistic motion, terms become progressively smaller, and the Lorentz factor is roughly given by just the first two summands. Then we can substitute these terms back into the mechanical energy approximation to obtain

E \simeq \gamma mc^{2}

Energy Measurements

Consider some laboratory experiments to measure  E and let us review some terms often used to compare these 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.

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^{k} \ \ldots \ E^{N}. These observed values are related to  E, the theoretical concept of mechanical energy, by

E = \widetilde{E}         \pm \delta \! E

where  \widetilde{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  \widetilde{E} are interchangeable thus reconciling theory and observation. Usually the experimental average is determined from the arithmetic mean of the set of observations

\displaystyle \widetilde{E} = \dfrac{1}{N} \sum_{k=1}^{N} \; E^{k}

and the experimental uncertainty is represented by their standard deviation

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

Another important number is the coefficient of variation in the data which is defined by the ratio \delta \! E / \widetilde{E}. The inverse of this quantity is known as the signal to noise ratio

\varsigma = 10 \log{ \left(\widetilde{E} / \delta \! E \right) } (dB)

expressed on a logarithmic scale in units of decibels .

The mass depends on a difference between two quantities in tension with each other, somewhat like the warp and weft of this ikat weaving from Borneo.
Bidang, Ketunggau people. Kalimantan Indonesia, 20th century 58 x 124 cm. Photograph by D Dunlop.
References
1H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat, page 225. Published by B. G. Teubner at Leipzig, 1909.