What Is the Moment About the Origin Due to the Applied Force F?

Concept in physics

Non to be confused with Momentum.

For the mathematical concept, see Moment (mathematics). For the moment of a force, sometimes shortened to "moment", see Torque.

In physics, a moment is a mathematical expression involving the production of a distance and physical quantity. Moments are ordinarily defined with respect to a fixed reference point and refer to physical quantities located some altitude from the reference point. For example, the moment of force, often called torque, is the production of a force on an object and the distance from the reference indicate to the object. In principle, any physical quantity tin exist multiplied past a altitude to produce a moment. Commonly used quantities include forces, masses, and electric accuse distributions. The moment can be used to describe a concrete quantity's location or arrangement.

Elaboration [edit]

In its most basic class, a moment is the product of the distance to a point, raised to a power, and a concrete quantity (such every bit force or electrical charge) at that point:

${\displaystyle \mu _{n}=r^{n}\,Q,}$

where ${\displaystyle Q}$ is the physical quantity such every bit a force practical at a point, or a point accuse, or a betoken mass, etc. If the quantity is not full-bodied solely at a single bespeak, the moment is the integral of that quantity'south density over infinite:

${\displaystyle \mu _{n}=\int r^{n}\rho (r)\,dr}$

where ${\displaystyle \rho }$ is the distribution of the density of charge, mass, or whatever quantity is existence considered.

More complex forms take into business relationship the angular relationships between the altitude and the physical quantity, but the above equations capture the essential feature of a moment, namely the existence of an underlying ${\displaystyle r^{n}\rho (r)}$ or equivalent term. This implies that there are multiple moments (1 for each value of n) and that the moment generally depends on the reference point from which the distance ${\displaystyle r}$ is measured, although for certain moments (technically, the lowest not-zero moment) this dependence vanishes and the moment becomes independent of the reference point.

Each value of n corresponds to a dissimilar moment: the 1st moment corresponds to n = 1; the second moment to northward = two, etc. The 0th moment (north = 0) is sometimes chosen the monopole moment; the 1st moment (n = ane) is sometimes chosen the dipole moment, and the 2nd moment (n = 2) is sometimes called the quadrupole moment, especially in the context of electric charge distributions.

Examples [edit]

Moments of mass:

Multipole moments [edit]

Assuming a density function that is finite and localized to a detail region, outside that region a 1/r potential may be expressed as a series of spherical harmonics:

${\displaystyle \Phi (\mathbf {r} )=\int {\frac {\rho (\mathbf {r'} )}{|\mathbf {r} -\mathbf {r'} |}}\,d^{3}r'=\sum _{\ell =0}^{\infty }\sum _{k=-\ell }^{\ell }\left({\frac {4\pi }{2\ell +1}}\correct)q_{\ell m}\,{\frac {Y_{\ell m}(\theta ,\varphi )}{r^{\ell +1}}}}$

The coefficients ${\displaystyle q_{\ell yard}}$ are known as multipole moments, and accept the form:

${\displaystyle q_{\ell one thousand}=\int (r')^{\ell }\,\rho (\mathbf {r'} )\,Y_{\ell m}^{*}(\theta ',\varphi ')\,d^{three}r'}$

where ${\displaystyle \mathbf {r} '}$ expressed in spherical coordinates ${\displaystyle \left(r',\varphi ',\theta '\right)}$ is a variable of integration. A more complete treatment may exist establish in pages describing multipole expansion or spherical multipole moments. (Notation: the convention in the above equations was taken from Jackson^[1] – the conventions used in the referenced pages may exist slightly unlike.)

When ${\displaystyle \rho }$ represents an electric charge density, the ${\displaystyle q_{lm}}$ are, in a sense, projections of the moments of electrical charge: ${\displaystyle q_{00}}$ is the monopole moment; the ${\displaystyle q_{1m}}$ are projections of the dipole moment, the ${\displaystyle q_{2m}}$ are projections of the quadrupole moment, etc.

Applications of multipole moments [edit]

The multipole expansion applies to 1/r scalar potentials, examples of which include the electric potential and the gravitational potential. For these potentials, the expression can be used to gauge the strength of a field produced by a localized distribution of charges (or mass) by calculating the first few moments. For sufficiently large r, a reasonable approximation tin can be obtained from merely the monopole and dipole moments. Higher allegiance can be achieved by calculating higher gild moments. Extensions of the technique can be used to calculate interaction energies and intermolecular forces.

The technique tin also be used to make up one's mind the properties of an unknown distribution ${\displaystyle \rho }$ . Measurements pertaining to multipole moments may be taken and used to infer properties of the underlying distribution. This technique applies to pocket-size objects such every bit molecules,^{[2] [3]} but has also been applied to the universe itself,^[4] being for example the technique employed past the WMAP and Planck experiments to analyze the cosmic microwave background radiation.

History [edit]

The concept of moment in physics is derived from the mathematical concept of moments.^[v] The principle of moments is derived from Archimedes' discovery of the operating principle of the lever. In the lever i applies a force, in his mean solar day almost often human muscle, to an arm, a axle of some sort. Archimedes noted that the amount of force applied to the object, the moment of force, is defined as One thousand = rF, where F is the applied force, and r is the distance from the practical forcefulness to object. However, historical evolution of the term 'moment' and its use in different branches of science, such as mathematics, physics and technology, is unclear.

Federico Commandino, in 1565, translated into Latin from Archimedes:

The heart of gravity of each solid effigy is that signal within it, about which on all sides parts of equal moment stand up. ^[vi]

This was evidently the offset employ of the give-and-take moment (Latin, momentorum) in the sense which nosotros now know it: a moment most a centre of rotation.^[7]

The word moment was first used in Mechanics in its at present rather one-time-fashioned sense of 'importance' or 'consequence,' and the moment of a force near an centrality meant the importance of the force with respect to its power to generate in affair rotation about the axis... But the word 'moment' has too come to exist used by analogy in a purely technical sense, in such expressions as the 'moment of a mass about an centrality,' or 'the moment of an area with respect to a plane,' which require definition in each instance. In those instances there is not always any corresponding physical idea, and such phrases stand up, both historically and scientifically, on a different footing. – A. Grand. Worthington, 1920^[8]

See as well [edit]

• Torque (or moment of force), see besides the commodity couple (mechanics)
• Moment (mathematics)
• Mechanical equilibrium, applies when an object is balanced and so that the sum of the clockwise moments near a pivot is equal to the sum of the anticlockwise moments about the same pivot
• Moment of inertia ${\displaystyle \left(I=\Sigma mr^{ii}\right)}$ , coordinating to mass in discussions of rotational motion. It is a measure of an object's resistance to changes in its rotation charge per unit
• Moment of momentum ${\displaystyle (\mathbf {L} =\mathbf {r} \times m\mathbf {v} )}$ , the rotational analog of linear momentum.
• Magnetic moment ${\displaystyle \left(\mathbf {\mu } =I\mathbf {A} \correct)}$ , a dipole moment measuring the strength and direction of a magnetic source.
• Electrical dipole moment, a dipole moment measuring the accuse difference and direction between 2 or more charges. For example, the electrical dipole moment betwixt a charge of –q and q separated by a distance of d is ${\displaystyle (\mathbf {p} =q\mathbf {d} )}$
• Bending moment, a moment that results in the bending of a structural chemical element
• First moment of expanse, a property of an object related to its resistance to shear stress
• Second moment of area, a belongings of an object related to its resistance to bending and deflection
• Polar moment of inertia, a property of an object related to its resistance to torsion
• Image moments, statistical properties of an image
• Seismic moment, quantity used to mensurate the size of an earthquake
• Plasma moments, fluid clarification of plasma in terms of density, velocity and pressure
• List of surface area moments of inertia
• List of moments of inertia
• Multipole expansion
• Spherical multipole moments

References [edit]

1. ^ J. D. Jackson, Classical Electrodynamics, 2d edition, Wiley, New York, (1975). p. 137
2. ^ Spackman, M. A. (1992). "Molecular electrical moments from ten-ray diffraction data". Chemical Reviews. 92 (8): 1769–1797. doi:10.1021/cr00016a005.
3. ^ Dittrich and Jayatilaka, Reliable Measurements of Dipole Moments from Single-Crystal Diffraction Data and Cess of an In-Crystal Enhancement , Electron Density and Chemical Bonding II, Theoretical Charge Density Studies, Stalke, D. (Ed), 2012, https://world wide web.springer.com/978-three-642-30807-9
4. ^ Baumann, Daniel (2009). "TASI Lectures on Inflation". arXiv:0907.5424 [hep-th].
5. ^ Robertson, D.G.E.; Caldwell, G.Due east.; Hamill, J.; Kamen, 1000.; and Whittlesey, S.N. (2004) Research Methods in Biomechanics. Champaign, IL: Human being Kinetics Publ., p. 285.
6. ^ Commandini, Federici (1565). Liber de Centro Gravitatis Solidorum. , (at Google books)
7. ^ Crew, Henry; Smith, Keith Kuenzi (1930). Mechanics for Students of Physics and Engineering. The Macmillan Company, New York. p. 25.
8. ^ Worthington, Arthur M. (1920). Dynamics of Rotation. Longmans, Green and Co., London. p. 7. , (at Google books)

External links [edit]

• [1] A dictionary definition of moment.

Look up moment in Wiktionary, the free lexicon.

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Source: https://en.wikipedia.org/wiki/Moment_(physics)

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