moment


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moment,

in physics and engineering, term designating the product of a quantity and a distance (or some power of the distance) to some point associated with that quantity. The most theoretically useful moments are moments of masses, areas, lines, and forces, including magnetic force. The concept of torque (propensity to turn about a point) is the moment of force. If a force tends to rotate a body about some point, then the moment, or turning effect, is the product of the force and the distance from the point to the direction of the force. The application of this concept is illustrated by pushing open a door: the farther from the hinge the push is applied, the less force is required. The principle of the moment of a force is perhaps best seen in the use of a leverlever,
simple machine consisting of a bar supported at some stationary point along its length and used to overcome resistance at a second point by application of force at a third point. The stationary point of a lever is known as its fulcrum.
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. Extensions of this concept are important in mechanics, in topics such as inertia, center of gravity, equilibrium, and stability of structures, and in architectural problems. The moment of inertia of a body about a point is the sum, for each particle in the body, of the mass of the particle and the square of its distance from the point. The angular momentum of a body about a fixed axis is equal to the product of the momentum and the length of the moment arm (distance from the body to the axis). A torque acting on a rigid body acts to change its angular momentum by producing an angular acceleration.

Moment

 

a mathematical concept that plays an important role in mechanics and probability theory. If we have a system of point masses m1, m2, … (mi > 0) lying on a line and if their abscissas with respect to some origin O are equal to x1, x2, …, respectively, then the sum

x1km1 + x2km2 + … = Σ ixikmi

is called the kth order moment of the system about the point O. In mechanics the first-order moment is called the static moment and the second-order moment is called the moment of inertia. If in the expression for the moment all the abscissas are replaced by their absolute values, a quantity called the absolute moment is obtained. The point with abscissa (Σiximi)/(Σimi) is called the center of mass of a given system of masses. Moments computed about the center of mass are called central moments. The central, first-order moment for any system is equal to zero. Of all the moments of inertia, the central moment of inertia is the smallest. Chebyshev’s inequality states that the sum of the masses located at a distance greater than a from the point O does not exceed the system’s moment of inertia with respect to O divided by a2. If a mass distribution has a density f(x) ≧ 0, then the integral

is called the moment of order k, provided that it is absolutely convergent. In the case of an arbitrary mass distribution, the sums in the expressions for the moments are replaced by Stieltjes integrals; the Stieltjes integral first arose in this way. All the definitions and theorems mentioned above remain valid in this case.

In probability theory, the different possible values of a random quantity play the role of abscissas, and the corresponding probabilities take the place of the masses. The first-order moment is called the mathematical expectation value of the given random quantity, and the central, second-order moment is called its dispersion. The first-order moment is always the abscissa of the center, since the total mass is unity. Chebyshev’s inequality, referred to above, plays an extremely important role in probability theory. In mathematical statistics, moments usually function as basic, statistical quantities summarizing the characteristics of distributions.

The problem in mathematical analysis of the characterization of the properties of a function f(x) by the properties of the sequence of its moments

is called the problem of moments. This problem was first considered by P. L. Chebyshev in 1874 in connection with his studies in probability theory (the attempt to prove the central limit theorem). Later, powerful new methods of mathematical analysis were developed in the investigation of this problem.

REFERENCES

Chebyshev, P. L. Izbr. trudy. Moscow, 1955.
Markov, A. A. Izbr. trudy. Moscow, 1951.
Gnedenko, B. V. Kurs teorii veroiatnostei, 5th ed. Moscow, 1969.
Loève, M. Teoriia veroiatnostei. Moscow, 1962. (Translated from English.)

moment

[′mō·mənt]
(mechanics)
Static moment of some quantity, except in the term “moment of inertia.”
(statistics)
The n th moment of a distribution ƒ(x) about a point x0 is the expected value of (x - x0) n , that is, the integral of (x - x0) n d ƒ(x), where d ƒ(x) is the probability of some quantity's occurrence; the first moment is the mean of the distribution, while the variance may be found in terms of the first and second moments.

moment

The property by which a force tends to cause a body, to which it is applied, to rotate about a point or line; equal in magnitude to the product of the force and the perpendicular distance of the point from the line of action of the force.

moment

Physics
a. a tendency to produce motion, esp rotation about a point or axis
b. the product of a physical quantity, such as force or mass, and its distance from a fixed reference point
References in classic literature ?
It will meet the moon four days after its discharge, precisely at midnight on the 4th of December, at the moment of its transit across the zenith.
The members of the Gun Club ought, therefore, without delay, to commence the works necessary for such an experiment, and to be prepared to set to work at the moment determined upon; for, if they should suffer this 4th of December to go by, they will not find the moon again under the same conditions of perigee and of zenith until eighteen years and eleven days afterward.
The doorway was dark and gloomy at any time; but just at this moment it was rendered doubly so by the fact that the thunder- storm had just broken, and the rain was coming down in torrents.
The man moved on up the stairs; a moment later the prince passed up them, too.
I eyed it wistfully for a moment, and then, unable any longer to stand on ceremony, plunged my hand into the yielding mass, and to the boisterous mirth of the natives drew it forth laden with the poee-poee, which adhered in lengthy strings to every finger.
At last the wrath of the chief evaporated, and in a few moments he was as placid as ever.
After he had left I stood and talked with Dejah Thoris, and she stretched her dear hand through those cruel bars that I might hold it until the last moment.
In a moment a dozen men broke through the now blinding smoke to my side.
For a moment he looked away from his own work and took the opportunity to light a fresh cigarette.
Richardson, in a few moments, staggered to his feet.
Those details Vronsky learned later; at the moment all he saw was that just under him, where Frou-Frou must alight, Diana's legs or head might be in the way.
At the very moment when Vronsky thought that now was the time to overtake Mahotin, Frou-Frou herself, understanding his thoughts, without any incitement on his part, gained ground considerably, and began getting alongside of Mahotin on the most favorable side, close to the inner cord.