Comparing Galileo’s Continuum With the Continuums for Mechanics and Spacetime
Galileo’s continuum for one, two, or three dimensional objects has an infinite numbe
r of indivisible parts, which are not countable. However, in the continuums for mechanics and spacetime, as described below, have a finite number of divisible parts, which are countable. Who is defining the continuum correctly, Galileo or the physical scientists?
The Continuum for Mechanics: Described in Wikipedia, the Free Encyclopedia (click)
Materials, such as solids, liquids and gases, are composed by molecules separated by empty space. Additionally, in a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming materials as a continuum, i.e. the matter in the body is uniformly and continuously distributed filling all the region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal small elements with properties being those of the bulk material.
The concept of continuum is a macroscopic physical model, and its validity depends on the type of problem and the scale of the physical phenomena under consideration. A material may be assumed as a continuum when the distance between the real physical particles is very small compared to the dimension of the problem. Such is the case when analyzing the deformation behavior of soil deposits, i.e. settlement under a foundation, in soil mechanics. A given volume of soil is generally formed by discrete solid particles (grains) of minerals which are packed in a certain manner leaving voids between them, i.e. granular media. In this sense, soils defeat the definition of a continuum. However, in order to simplify the deformation analysis of the soil, the volume of soil can be assumed as a continuum knowing that the dimensions of particular grain particles are very small compared with the scale of the problem, i.e. the size of the foundation and the volume of the soil mass that is influenced by the foundation load (meters) is greater than the particular soil particles (millimeters).
The validity of the continuum assumption needs to be verified with experimental testing and measurements on the real material under consideration and under similar loading conditions
The Continuum for Spacetime: Described in Wikipedia, the Free Encyclopedia (click)
In physics, spacetime is any mathematical model that combines space and time into a single construct called the spacetime continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of the fourth dimension. According to Euclidean space perception, the universe has three dimensions of space, and one dimension of time. By combining space and time into a single manifold, physicists have significantly simplified a large amount of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.
In classical mechanics, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer. In relativistic contexts, however, time cannot be separated from the three dimensions of space because the rate at which time passes depends on an object's velocity relative to the speed of light, and also the strength of intense gravitational fields which can slow the passage of time, and as such is dependant on the state of motion of the observer and is therefore not universal.
r of indivisible parts, which are not countable. However, in the continuums for mechanics and spacetime, as described below, have a finite number of divisible parts, which are countable. Who is defining the continuum correctly, Galileo or the physical scientists?The Continuum for Mechanics: Described in Wikipedia, the Free Encyclopedia (click)
Materials, such as solids, liquids and gases, are composed by molecules separated by empty space. Additionally, in a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming materials as a continuum, i.e. the matter in the body is uniformly and continuously distributed filling all the region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal small elements with properties being those of the bulk material.
The concept of continuum is a macroscopic physical model, and its validity depends on the type of problem and the scale of the physical phenomena under consideration. A material may be assumed as a continuum when the distance between the real physical particles is very small compared to the dimension of the problem. Such is the case when analyzing the deformation behavior of soil deposits, i.e. settlement under a foundation, in soil mechanics. A given volume of soil is generally formed by discrete solid particles (grains) of minerals which are packed in a certain manner leaving voids between them, i.e. granular media. In this sense, soils defeat the definition of a continuum. However, in order to simplify the deformation analysis of the soil, the volume of soil can be assumed as a continuum knowing that the dimensions of particular grain particles are very small compared with the scale of the problem, i.e. the size of the foundation and the volume of the soil mass that is influenced by the foundation load (meters) is greater than the particular soil particles (millimeters).
The validity of the continuum assumption needs to be verified with experimental testing and measurements on the real material under consideration and under similar loading conditions
The Continuum for Spacetime: Described in Wikipedia, the Free Encyclopedia (click)
In physics, spacetime is any mathematical model that combines space and time into a single construct called the spacetime continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of the fourth dimension. According to Euclidean space perception, the universe has three dimensions of space, and one dimension of time. By combining space and time into a single manifold, physicists have significantly simplified a large amount of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.
In classical mechanics, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer. In relativistic contexts, however, time cannot be separated from the three dimensions of space because the rate at which time passes depends on an object's velocity relative to the speed of light, and also the strength of intense gravitational fields which can slow the passage of time, and as such is dependant on the state of motion of the observer and is therefore not universal.
posted by George Shollenberger @ 2:20 PM
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