Mass distribution
Encyclopedia
Mass distribution is a term used in physics
and mechanics
and describes the spatial distribution of mass
within a solid
body. In principle, it is relevant also for gas
es or liquid
s, but on earth their mass distribution is almost homogeneous.
mass distribution has decisive influence on the development e.g. of nebula
e, star
s and planet
s.
The mass distribution of a solid defines its center of gravity
and influences its dynamical behaviour - e.g. the oscillation
s and eventual rotation
.
. This allows point masses, line masses, surface masses, as well as masses given by a volume density function. Alternatively the latter can be generalized to a distribution
. For example, a point mass is represented by a delta function defined in 3-dimensional space. A surface mass on a surface given by the equation f(x,y,z) = 0 may be represented by a density distribution g(x,y,z) δ (f(x,y,z)), where
is the mass per unit area.
The mathematical model
ling can be done by potential theory
, by numerical methods (e.g. a great number of mass points), or by theoretical equilibrium
figures.
solids are affected considerably by the mass distribution, either if they are homogeneous or inhomogeneous - see Torque
, moment of inertia
, wobble
, imbalance and stability
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
and describes the spatial distribution of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
within a solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
body. In principle, it is relevant also for gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
es or liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
s, but on earth their mass distribution is almost homogeneous.
Astronomy
In astronomyAstronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
mass distribution has decisive influence on the development e.g. of nebula
Nebula
A nebula is an interstellar cloud of dust, hydrogen gas, helium gas and other ionized gases...
e, star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s and planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s.
The mass distribution of a solid defines its center of gravity
Center of gravity
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...
and influences its dynamical behaviour - e.g. the oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
s and eventual rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
.
Mathematical modelling
A mass distribution can be modeled as a measureMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
. This allows point masses, line masses, surface masses, as well as masses given by a volume density function. Alternatively the latter can be generalized to a distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
. For example, a point mass is represented by a delta function defined in 3-dimensional space. A surface mass on a surface given by the equation f(x,y,z) = 0 may be represented by a density distribution g(x,y,z) δ (f(x,y,z)), where
is the mass per unit area.The mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
ling can be done by potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, by numerical methods (e.g. a great number of mass points), or by theoretical equilibrium
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
figures.
Rotating solids
RotatingRotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
solids are affected considerably by the mass distribution, either if they are homogeneous or inhomogeneous - see Torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
, moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
, wobble
Wobble
Wobble or wobbles can mean:* Wobble, to shake, or wiggle.* Wobble or wobbling effect, an artifact in video known as screen tearing* Wobble base pair, a type of basepair in genetics translation...
, imbalance and stability
Stability
-Mathematics:*Stability theory, the study of the stability of solutions to differential equations and dynamical systems**Lyapunov stability**Structural stability*Stability of a point in geometric invariant theory....
.
Related topics
- gravity, mass function, Bouguer plate, critical massCritical massA critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The critical mass of a fissionable material depends upon its nuclear properties A critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The...
- angular momentumAngular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
, asymmetryAsymmetryAsymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....
, geoidGeoidThe geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...
, isostasyIsostasyIsostasy is a term used in geology to refer to the state of gravitational equilibrium between the earth's lithosphere and asthenosphere such that the tectonic plates "float" at an elevation which depends on their thickness and density. This concept is invoked to explain how different topographic...
, polar motionPolar motionPolar motion of the earth is the movement of Earth's rotational axis across its surface. This is measured with respect to a reference frame in which the solid Earth is fixed... - mass spectrumMass spectrumA mass spectrum is an intensity vs. m/z plot representing a chemical analysis. Hence, the mass spectrum of a sample is a pattern representing the distribution of ions by mass in a sample. It is a histogram usually acquired using an instrument called a mass spectrometer...
, molar mass distributionMolar mass distributionIn linear polymers the individual polymer chains rarely have exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar mass distribution in a polymer describes the relationship between the number of moles of each polymer species...
, atmosphereAtmosphereAn atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...