“moment of inertia”的概念、定义、翻译、参考文献-科学参考

moment of inertia

Physics

Symbol I. The moment of inertia of a massive body about an axis is the sum of all the products formed by multiplying the magnitude of each element of mass (δ‎m) by the square of its distance (r) from the line, i.e. I_m=Σ‎r²δ‎m. It is the analogue in rotational dynamics of mass in linear dynamics. The basic equation is T=Iα‎, where T is the torque causing angular acceleration α‎ about the specified axis.

Mathematics

A quantity relating to a rigid body and a given axis, derived from the way in which the mass of the rigid body is distributed relative to the axis. It arises in the calculation of the kinetic energy and the angular momentum of the rigid body in general motion. It replaces the role of mass in formulae for linear motion.

For a planar rigid body, rotating with angular speed ω‎ about an axis perpendicular to the plane, the moment of inertia is given by

$I = \sum_{i} m_{i} r_{i}^{2} or I = \int_{R} ρ (r) {| r |}^{2} d A,$

where the first expression is for a discrete collection of particles of mass m_i at distance r_i from the axis, and the second expression is for a continuous distribution of matter within a region R of density ρ‎(r), where r is the position vector of a point from the axis. The angular momentum of the body equals Iω‎, and its (rotational) kinetic energy equals ½Iω‎².

Note that the same rigid body will have different moments of inertia for different axes. For example, the moment of inertia of a uniform disc will be greater for an axis through a point of the circumference than through the centre, as the mass is distributed farther from the axis. Moments of inertia about other axes may be calculated by using the parallel axis theorem and the perpendicular axis theorem. For a three-dimensional rigid body, moments of inertia I_xx, I_yy, and I_zz can be defined for the x‐axis, the y‐axis, and the z‐axis. Here r_i and r denote the distance and displacement from the axis. See inertia matrix, product of inertia.

For a list of various moments of inertia, see appendix 3.

Astronomy

For a system of several bodies, the mass of each body multiplied by the square of its distance from the centre of the system, the products all being added together. For a single solid body, the moment of inertia is the sum of the mass of each particle in that body multiplied by the square of its distance from the body’s axis of rotation. The shape of the Moon, for example, is an ellipsoid with three different axes, the longest of them pointing towards the Earth. The moments of inertia of the Moon about these three axes are slightly different, which can produce measurable effects in the orbit of an artificial lunar satellite. Moment of inertia is a measure of a body’s ability to resist a change in its state of rest or angular velocity.

Chemistry

A quantity associated with a body that is rotating about an axis. If a body consists of i particles of mass m_i a perpendicular distance r_i from an axis of rotation, the moment of inertia I of the body about that axis of rotation is given by I = ∑ m_i r_i². The moment of inertia of a body is an important quantity because it is the analogue for rotational motion of mass for linear motion.
In a molecule all rotational motion can be analysed using three perpendicular axes of rotation. Each of these has a moment of inertia associated with it. To a first approximation, a molecule can be regarded as a rigid rotor, i.e. a body that is not distorted by its rotation. There are four types of rigid rotor:
In a spherical top all three moments of inertia are equal (e.g. SF₆).
In a symmetric top two of the moments of inertia are equal (e.g. NH₃).
In an asymmetric top all three moments of inertia are different (e.g. H₂O).
In a linear rotor the moment of inertia about the axis of the molecule is zero (e.g. HCl, CO₂).
The type of rotor depends on the symmetry of the molecule. If a molecule has cubic or icosahedral symmetry it is a spherical top. If it has a threefold or higher axis of symmetry it is a symmetric top. If it does not have a threefold or higher axis of symmetry it is an asymmetric top. All linear molecules (and hence all diatomic molecules) are linear rotors.
In reality, molecules are not rigid rotors because of the centrifugal forces associated with the rotation. The effect of such forces can be taken into account by adding a correction term to the rigid rotor model.

Geology and Earth Sciences

The kinematic properties of a rotating body; a measure of the rotational inertia of an object around a specific axis of rotation, in units of kg/m². In the Earth, the principal moment of inertia lies close to the axis of rotation and passes through the centre of mass of the Earth. Changes in the mass distribution, e.g. ice sheets, seasonal atmospheric changes, etc., cause changes in the location of the moments of inertia.

单词	moment of inertia
释义	moment of inertia Physics Symbol I. The moment of inertia of a massive body about an axis is the sum of all the products formed by multiplying the magnitude of each element of mass (δ‎m) by the square of its distance (r) from the line, i.e. I_m=Σ‎r²δ‎m. It is the analogue in rotational dynamics of mass in linear dynamics. The basic equation is T=Iα‎, where T is the torque causing angular acceleration α‎ about the specified axis. Mathematics A quantity relating to a rigid body and a given axis, derived from the way in which the mass of the rigid body is distributed relative to the axis. It arises in the calculation of the kinetic energy and the angular momentum of the rigid body in general motion. It replaces the role of mass in formulae for linear motion. For a planar rigid body, rotating with angular speed ω‎ about an axis perpendicular to the plane, the moment of inertia is given by $I = \sum_{i} m_{i} r_{i}^{2} or I = \int_{R} ρ (r) {\| r \|}^{2} d A,$ where the first expression is for a discrete collection of particles of mass m_i at distance r_i from the axis, and the second expression is for a continuous distribution of matter within a region R of density ρ‎(r), where r is the position vector of a point from the axis. The angular momentum of the body equals Iω‎, and its (rotational) kinetic energy equals ½Iω‎². Note that the same rigid body will have different moments of inertia for different axes. For example, the moment of inertia of a uniform disc will be greater for an axis through a point of the circumference than through the centre, as the mass is distributed farther from the axis. Moments of inertia about other axes may be calculated by using the parallel axis theorem and the perpendicular axis theorem. For a three-dimensional rigid body, moments of inertia I_xx, I_yy, and I_zz can be defined for the x‐axis, the y‐axis, and the z‐axis. Here r_i and r denote the distance and displacement from the axis. See inertia matrix, product of inertia. For a list of various moments of inertia, see appendix 3. Astronomy For a system of several bodies, the mass of each body multiplied by the square of its distance from the centre of the system, the products all being added together. For a single solid body, the moment of inertia is the sum of the mass of each particle in that body multiplied by the square of its distance from the body’s axis of rotation. The shape of the Moon, for example, is an ellipsoid with three different axes, the longest of them pointing towards the Earth. The moments of inertia of the Moon about these three axes are slightly different, which can produce measurable effects in the orbit of an artificial lunar satellite. Moment of inertia is a measure of a body’s ability to resist a change in its state of rest or angular velocity. Chemistry A quantity associated with a body that is rotating about an axis. If a body consists of i particles of mass m_i a perpendicular distance r_i from an axis of rotation, the moment of inertia I of the body about that axis of rotation is given by I = ∑ m_i r_i². The moment of inertia of a body is an important quantity because it is the analogue for rotational motion of mass for linear motion. In a molecule all rotational motion can be analysed using three perpendicular axes of rotation. Each of these has a moment of inertia associated with it. To a first approximation, a molecule can be regarded as a rigid rotor, i.e. a body that is not distorted by its rotation. There are four types of rigid rotor: In a spherical top all three moments of inertia are equal (e.g. SF₆). In a symmetric top two of the moments of inertia are equal (e.g. NH₃). In an asymmetric top all three moments of inertia are different (e.g. H₂O). In a linear rotor the moment of inertia about the axis of the molecule is zero (e.g. HCl, CO₂). The type of rotor depends on the symmetry of the molecule. If a molecule has cubic or icosahedral symmetry it is a spherical top. If it has a threefold or higher axis of symmetry it is a symmetric top. If it does not have a threefold or higher axis of symmetry it is an asymmetric top. All linear molecules (and hence all diatomic molecules) are linear rotors. In reality, molecules are not rigid rotors because of the centrifugal forces associated with the rotation. The effect of such forces can be taken into account by adding a correction term to the rigid rotor model. Geology and Earth Sciences The kinematic properties of a rotating body; a measure of the rotational inertia of an object around a specific axis of rotation, in units of kg/m². In the Earth, the principal moment of inertia lies close to the axis of rotation and passes through the centre of mass of the Earth. Changes in the mass distribution, e.g. ice sheets, seasonal atmospheric changes, etc., cause changes in the location of the moments of inertia.
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