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Tag Archives: Physics

Reduced mass and vass

Here we take the reduced mass and show the parallel reduced vass.

In physics, the reduced mass is the “effective” inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem.

Given two object bodies, one with mass m1 and the other with mass m2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass:

where the force on this mass is given by the force between the two bodies.

Given two subject bodies, one with vass n1 and the other with vass n2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of vass:

where the force on this vass is given by the surge between the two bodies.

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Causes for subjects and objects

This continues posts such as the one here related to Aristotle’s four kinds of cause:

final cause formal cause
efficient cause material cause

A subject is a form with purposes. An object is a material with mechanisms. Objects exist in space-time. Subjects exist in time-space.

The upper causes apply to subjects, who have purposes and plans, destinations and routes. The lower causes apply to objects, which have mechanisms and materials, forces and masses. Though subjects can be considered as objects and objects as subjects.

why what
subjects: final cause formal cause
objects: efficient cause material cause

Dynamics is the study of why motion happens, whereas kinematics studies only what motion happens. Kinematics is the material for dynamics. The combination of kinematics and dynamics is called mechanics, but this implies that only objects are considered. If subjects are included, then an alternative term is needed, such as kinedynamics.

Mass and vass

In Isaac Newton’s Principia, Definition 1 states:

Quantity of matter is a measure of matter that arises from its density and volume jointly. (The Principia: The Authoritative Translation and Guide, Bernard Cohen, Anne Whitman, and Julia Budenz. University of California Press, 2016, p.403)

Today density is defined as mass per unit volume, which would make this definition circular. However, when Newton wrote, density was expressed as a relative quantity. (p.90) If we look at mass as the product of density and volume, a complementary measure arises: vass.

Density is a ratio, and ratios may be expressed as fractions in two ways: the ratio of nonzero quantities A:B is equivalent to either A/B or B/A. So instead of density as mass per unit volume we could just as well define its inverse, rarity, as volume per unit of mass. (See Max Jammer’s Concepts of Mass in Classical and Modern Physics, p.27.)

Then the rarity per unit of volume equals the vass, which is the inverse of mass. In SI units, that equates to (m³/kg) / m³ which equals 1/kg.

Mass is also defined as the ratio of force to acceleration, reflecting Newton’s second law. Force is the time rate of change of momentum. A complementary definition would be the space rate of change of fulmentum, which equals the vass.

Inertial mass is the resistance of an object to a change in its state of motion when a net force is applied. A complementary concept is the nonresistance of a subject to a change in its condition of movement when a net surge is applied.

If mass is the “quantity of matter,” what is vass the quantity of? Quantity of matter means how much of a material object there is. Vass answers how much of a material subject there is, which is measured inversely to the mass as subject and object are inverses.

6D invariant interval

Since one may associate either the arclength (travel length) or the arctime (travel time) with direction, one might think that the full coordinates for every event are of the form (s, t, ê), with arclength s, arctime t, and unit vector ê. Since the direction is a function of either the arclength or the arctime, the coordinates would be either (s, t, ê(s)) or (s, t, ê(t)).

However, since s = ∫ || r′(τ) || , where the integral is from 0 to t, and t = ∫ || w′(σ) || , where the integral is from 0 to s (see here), this reduces to either (t, r) or (s, w).

But science seeks unification and so must combine these forms into one. In that case, both s and t are redundant, and the full coordinates for every event are of the form [r, w]. That is, there are three dimensions of space and three dimensions of time. The arclength and arctime are implicit, and may be made explicit through integration.

The standard exposition of special relativity looks at one dimension of space and one dimension of time. This is convenient and makes Δs = Δx and Δt = Δw1. But in general Δs and Δt will either be measured directly or found through integration.

What is the distance-like invariant interval then between two events? The interval in length units (proper length) is (dσ)² = (cdw)² – (dr)²,  where c is the speed of light. The interval in time units (proper time) is (dτ)² = (dw)² – (dr/c)².

This appears different from special relativity because it substitutes the vector dw for the scalar dt. However, the scalar (dt)² = (dw1)² + (dw2)² + (dw3)² so there is no discrepancy.

In order to demonstrate that this interval is invariant for two observers traveling at different rates, one must either convert dw to dt or convert dr to ds, which reduces the six dimensions to four.

The intervals above may be generalized for general relativity with the relation L = cP √(–gμν dxμ dxν), where P is the path, gμν is the metric tensor, and there are six coordinates xμ and xν.

Physics of subjects

If a stone rolls down a hill, we would say it is simply following the law of gravitation. It is not “going somewhere” as if it had a destination – that would require nature to have a soul, a view that died out in the early modern period. But if a person or an animal or even a seed pod moves down a hill, we expect it to be going somewhere, to have a destination or purpose.

That is the difference between a subject in motion and an object in motion. At a minimum, an object must have some starting point, at least from our observation, but need not have a destination or purpose for all we know. On the other hand, a subject need not have a known starting point but at a minimum there must be some movement toward a destination or end, else they would not be a subject.

This simple difference leads to a different formulation of space, time, and matter for subjects and objects. Modern physics has been entirely focused on bodies as objects, particles, or waves. In contrast, the physics of subjects will focus on bodies as subjects (somebodies), transicles, and networks.

Since there is a destination, something about its location must be known. At a minimum there must exist a route or path for the subject to traverse to reach their destination. Even if the length of the path is not known, one can at least measure the progress made toward reaching the destination by measuring the space rate of movement, called the pace.

The difference between speed, the time rate of motion, and the pace is the difference between taking space or time as the independent variable. For objects their motion from a point in time is what is given and so time is the independent variable. For subjects space is the independent variable since their movement toward a destination in space is given.

That means for subjects the dependent variable is time, which is measured along with the direction of movement, which results in three dimensions of time. Space is confined to the path of movement, which may be rectified as a line for linear referencing. Examples of a linear reference are the milepoint (MP) and kilometric point (PK) on a map or sign.

Objects have chronologies. Subjects have a destinations. But subjects are like objects in some ways, and objects are like subjects in some ways. For example, a projectile is an object that has been launched by a subject toward a destination.

Mechanistic sciences such as physics study objects. Teleological sciences such as economics study subjects. The physics of subjects is physics for the social sciences.

For more, see the other posts on this website about time-space, with 3D time and 1D space.

Terms for motion again

Previous posts deal with terms for motion, such as here. Further thoughts are below.

When someone asks about the length of a trip, they are not asking for the distance between the origin and destination of the trip – that is the magnitude of the displacement. They are asking about the length of the route taken. Mathematically, travel length is the arc length of the curve of the route.

The length of a trip in time, or travel time, is the duration of a trip. Time is a kind of length, not a distance; an arc length, not a straight-line distance.

The magnitude of a displacement is the distance between two points. Call the magnitude of a distimement between two points in time the distime. This is the shortest-length travel time between them, which depends on the mode of travel.

We have the expression “as the crow flies” to distinguish the straight-line distance between two points from the travel length. Physicists would say “as light travels” to indicate the straight-line (geodesic) distance or time between two events.

While physicists may convert time and space dimensions (by multiplying time by the speed of light, or dividing length by the speed of light), this does not change the character of the dimensions. Only if the time and space dimensions are switched does their character change.

Measurement of space and time

The various ways of measuring space and time are parallel.

Measuring space:

  1. A ruler measures length, that is, the distance between two points in space (A to B).
  2. An ruler turned upside-down measures length backwards (B to A).
  3. A tripmeter measures the travel distance of a vehicle trip.
  4. An odometer measures the cumulative travel distance.Odometer 12,000
  5. A measuring wheel measures the travel distance of a wheel being pushed.
  6. A road map measures travel distance of a standard vehicle. See Geodistance.

Measuring time:

  1. A stopwatch measures time, that is, the duration between two points in time (A to B).
  2. A timer measures the time counting down from a set time, i.e., backwards (B to A).
  3. A GPS watch or time clock measures the duration of an activity, such as running or working. GPS watch
  4. A GPS watch (or smartphone app) measures cumulative travel time (or flight time).
  5. A measuring wheel with a stopwatch measures the travel time of a wheel being pushed.
  6. A clock measures travel time synchronized with a standard motion.

Note that #2 shows time can be measured backwards. Space and time can both be counted up or counted down. There’s nothing magical about it.

Motion and its interpretation

Say you’re standing near the bottom of a hill and see a small rock rolling down. How should the motion of the rock be interpreted? It could be that the rock happened to brake loose and roll down the hill. Or it could be that someone took the rock and rolled it down the hill. The motion observed could be exactly the same in either case. The only difference is the interpretation.

One interpretation would be called “natural” or “mechanistic”. In it motion occurs because of happenstance and the laws of motion. So the rock just happened to roll for reasons which are intractable and therefore considered chance. But once the rock started to roll, its trajectory followed the laws of motion.

Another interpretation would be called “artificial” or “teleological”. In it motion occurs because it fulfills a purpose in a way that accords with the laws of motion. The rock purposely moved toward an intended target or along an intended trajectory. The rock itself need not have any conscious intention; either the intention is that of an external agent or of an internal predisposition.

In the mechanistic interpretation time is the independent variable. When and where the rock starts to roll is a matter of happenstance or whatever – one doesn’t know or doesn’t care. In the teleological interpretation space is the independent variable. The placement of the rock, its initial motion, and its intended target or trajectory are a matter of independent purpose – of some internal predisposition or some external agent.

The statement of the laws of motion are mathematically equivalent in either case but the interpretation of the variables differs symmetrically. To translate from one to the other interpretation interchange the following: 3D space ↔ 3D time, scalar time ↔ scalar space, object ↔ subject, and mass ↔ vass. The laws of motion are formally the same for both interpretations. Only the meaning of the symbols changes.

From this exercise we learn that science determines the form of physical laws but not their interpretation. It would introduce metaphysics to specify that only one interpretation is valid. Science is not metaphysics but it allows metaphysics. Instead of excluding metaphysics, science affirms all metaphysical interpretations consistent with its laws. Science is pluralistic.

Actual and possible motion

This post continues the topic posted here.

The action motion of a particle or rigid body may be measured by the scalar (or 1D) rate of motion, expressed as a speed or a pace. The numerator of a speed is the measured length or travel distance, and the denominator is the unit of time or the measured travel time. The pace is the inverse.

The possible motion of a particle or rigid body is limited to three dimensions of motion. This may be represented as a rate of motion in a Euclidean space of three-dimensions. The rate of motion is the speed or pace. With the direction of motion this is the velocity or legerity.

If time (travel time) is held constant, this becomes a spatial 3D Euclidean geometry, commonly called “space”. If length (travel distance) is held constant, this becomes a temporal 3D Euclidean geometry, which may be called “time-world”.

The combination of space and time-world is a 6D Euclidean geometry of possible representations. This is not the extent of motion but the extent of the representation of motion, the next level of abstraction.

Motion vs. movement

The English words motion and movement are similar. They both have to do with “changing position or going from one place to another.” (Collins English Dictionary)

Then what’s the difference? Here are a few ways of putting it:

motion is used to describe physical properties, while movement is used to describe the qualities of motion. Ref.

motion doesn’t always imply a purpose, and movement usually does. Ref.

The difference is very fine. I would say that movement is déplacement d’un lieu à un autre [displacement from one place to another] whereas motion is le fait de ne pas rester immobile [not to stand still]. But usage and context are crucial. Ref.

People may not be consistent about it but for the purposes here they can be distinguished. Motion is the general term in kinetics, the study of motion. It says nothing about the purpose of a motion, or its origin and destination. Something just happens to change place.

However, movement includes some purpose, some origin and destination. A movement is a complete motion, from beginning to end. So movement would be preferred in the arts and social sciences and motion in the natural sciences.

Physics studies motion. Transportation studies movement. They may both speak about something changing position but there is a different perspective.

A movement is an entity, a thing, not just a change as a motion is. A motion can be studied abstractly but a movement is not fully abstract because it is an entity.

A body has its motion and a movement has its figure. A body is flesh-and-blood 3D, with motion only adding a thin 1D time perspective. A movement has 3D animation and life, with a figure only adding a thin 1D space perspective.