vol 4: Glossary
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Physical dimension
Numbers and measurement seem to be as old as written history. A large proportion of the documents we have recovered from the era when writing began are accounts and inventories. We can see the measurements of trade as the origin of measurement in science. Scientists, like accountants, like to know what is really there.
Physics is based on measurement and uses numbers and arithmetic to model the phenomena it measures. At the more precise end of physics, it is not unusual for quantities to be accurate to ten or more decimal places. Such results suggest that the universe is an incarnation of mathematics.
Physical quantities share the properties of numbers and obey all the rules of arithmetic. From the point of view of arithmetic, it makes no difference whether we are counting eggs or sheep, but it makes no sense to a farmer to add eggs to sheep. Incommensurable sets of items must be accounted for separately.
In the design of an accounting system, it is important to choose a set of accounts that is useful. Various accounts may be consolidated in various ways, and by assigning a money value to every item in the accounting system, we can eventualy arrive at a single number 'bottom line'.
Physics deals with the simplest levels of differentiation in the universe. It has been found that there are three fundamental classes of measurable quantity in the universe, mass M, length L and time T. These are called dimensions. In classical physics, we cannot compare masses with lengths, or lengths with times, but must account for them separately.
Other physical quantities can be described by combinations of these three dimensions.
Area is calculated as length x length, giving it a dimension of L2.. Volume has dimension L3. Density, being mass per unit volume, has dimension M / L3 usually written ML-3.
Velocity is defined as distance travelled per unit time, so its dimension is LT-1. Acceleration is the rate of change of velocity per unit time, so it has the dimension of velocity / time, ie LT-2.
Momentum, the product of mass by velocity, has dimension MLT-1.
A unit of force produces a unit of momentum in a unit of time. From this we deduce that the dimension of force is MLT-2.
Energy or work is measured by the product of force and distance, ie E = Fs, so the dimension of energy becomes ML2T-2. Kinetic energy is the product of mass by velocity squared, so its dimension is also ML2T-2, as we would expect. Mass energy is measured by Einstein's equation E = Mc 2, again leading to ML2T-2.
The angular momentum of a particle in orbit is the product of its mass by the radius of its orbit, which is a length. The dimension of angular momentum is consequently ML2T-1.
Action is measured by the product of energy and time, giving it the dimension ML2T-1. This happens to be the same as angular momentum, pointing to a connection between these classically disparate quantities that are united in the quantum mechanical notion of spin.
The physics of heat requires us to introduce two new quantities, temperature and entropy.
The kinetic theory of heat tells us that temperature is a measure of the average kinetic energy of the atoms or molecules of a substance. Temperature and energy are related by Boltzmann's constant, k the energy per molecule per degree. Assuming that k is a pure number, we assign to temperature the dimension of energy, ML2T-2.
Thermodynamic entropy is defined as the ratio of the energy transferred between two bodies to the temperature at which the transfer is accomplished. It is has therefore the dimension of energy divided by energy, that is it is a pure number. Boltzmann found that thermodynamic entropy S, of a substance was related to a count of the different internal arrangements or complexions W of the substance by the formula S = k log W. This expression which equates entropy to a count also suggests that entropy is dimensionless.
The theory of information, like accounting in money values, seems capable of reducing the whole of physics to pure numbers. The thermodynamic concept of entropy carries over from thermodynamics to information theory because it is dimensionless. Khinchin
The classical concepts of dimension may be combined with the 'natural units' of relativity and quantum mechanics to give us the Planck scale.
Books
| Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' Amazon back |
Click on an "Amazon" link in the booklist at the foot of the page to buy the book, see more details or search for similar items
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