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Encyclopedia > Rigid body

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. In classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules (consisting of the point masses: electrons and nuclei) are often seen as a rigid bodies, see classification of molecules as rigid rotors. Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... A physical body is an object which can be described by the theories of classical mechanics, or quantum mechanics, and experimented upon by physical instruments. ... In engineering mechanics, deformation is a change in shape due to an applied force. ... Distance is a numerical description of how far apart things lie. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... In physics, force is an influence that may cause a body to accelerate. ... Fig. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ... Rotational spectroscopy or microwave spectroscopy studies the absorption and emission of electromagnetic radiation (typically in the microwave region of the electromagnetic spectrum) by molecules associated with a corresponding change in the rotational quantum number of the molecule. ...


The configuration space of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying manifold of the rotation group SO(3). The configuration space of a nonfixed (with non-zero translational motion) rigid body is E+(n), the subgroup of direct isometries of the Euclidean group (combinations of translations and rotations). In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... A sphere rotating around its axis. ...


For any point/particle of a moving (translating and rotating) body we have

mathbf{r}(t,mathbf{r}_0) = mathbf{r}_c(t) + A(t) mathbf{r}_0
mathbf{v}(t,mathbf{r}_0) = mathbf{v}_c(t) + boldsymbolomega(t) times (mathbf{r}(t,mathbf{r}_0) - mathbf{r}_c(t)) = mathbf{v}_c(t) + boldsymbolomega(t) times A(t) mathbf{r}_0
dot{A}(t)mathbf{r}_0 = boldsymbolomega(t) times A(t)mathbf{r}_0

where

  • mathbf{r}(t) is the position of the point/particle at time t,
  • mathbf{r}_c(t) is the position of a reference point of the body at time t,
  • A(t), is the orientation, an orthogonal matrix with determinant 1
  • dot{A}(t), is the time derivative of A(t),, see Newton's notation for differentiation and derivative of a matrix.
  • mathbf{r}_0 is the position of the point/particle with respect to the reference point of the body in a reference orientation, for instance mathbf{r}_0 =mathbf{r}(0), (the reference orientation is the one at initial time)
  • boldsymbolomega is the angular velocity
  • mathbf{v} is the total velocity of the point/particle
  • mathbf{v}_c is the translational velocity

To describe the motion, the reference point mathbf{r}_0 can be any point that is rigidly connected to the body (the translation vector depends on the choice). Depending on the application a convenient choice may be: Changing orientation is the same as moving the coordinate axes. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... Newtons notation for differentiation involved placing a dash/dot over the function name, which he termed the fluxion. ... Angular velocity describes the speed of rotation. ...

  • the center of mass of the whole system; properties:
    • the (linear) momentum: the total mass of the rigid body times the translational velocity. The net external force on the rigid body is the total mass times the translational acceleration (i.e., Newton's second law holds for the translational motion). The linear momentum is independent of the rotational motion.
    • the angular momentum with respect to the center of mass is the same as without translation: at any time it is equal to the inertia tensor times the angular velocity. When the angular velocity is expressed with respect to the principal axes frame of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; the torque is the inertia tensor times the angular acceleration.
    • possible motions in the absence of external forces are translation with constant velocity, steady rotation about a fixed principal axis, and also torque-free precession.
    • the total kinetic energy is simply the sum of translation and rotational energy
  • a point such that the translational motion is zero or simplified, e.g on an axle or hinge, at the center of a ball-and-socket joint, etc.

When the cross product In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... In classical mechanics, momentum (pl. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... Gyroscope. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², English units lbs ft2) quantifies the rotational inertia of a rigid body, i. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², English units lbs ft2) quantifies the rotational inertia of a rigid body, i. ... It has been suggested that this article or section be merged with Moment (physics). ... лInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non... Precession refers to a change in the direction of the axis of a rotating object. ... Kinetic energy is the energy by virtue of the motion of an object. ... The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy. ... An axle is a central shaft for a rotating wheel or gear. ... A hinge is a mechanical device that connects two solid objects, allowing rotation between them. ... A joint (from French joint) (articulation) is the location at which two bones make contact (articulate). ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ...

boldsymbolomega(t) times A(t)mathbf{r}_0

is written as a matrix multiplication, this matrix is a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements, In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ...

boldsymbolomega(t) times A(t)mathbf{r}_0 = begin{pmatrix} 0 & -omega_z(t) & omega_y(t)  omega_z(t) & 0 & -omega_x(t)  -omega_y(t) & omega_x(t) & 0  end{pmatrix} A(t)mathbf{r}_0.

In 2D the matrix A(t) simply represents a rotation in the xy-plane by an angle which is the integral of the scalar angular velocity over time.


Vehicles, walking people, etc. usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is the winding number with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon. The Trikke is a Human Powered Vehicle (HPV) This article is about the means of transport. ... A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ... Look up polygon in Wiktionary, the free dictionary. ...


The orientation can also be described in a different way, e.g. as a unit-quaternion-valued function of time. Although the latter is specific up to a factor -1, it would be reasonable to choose it continuously. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...


Two rigid bodies are said to be different (not copies) is that there is no proper rotation from one to the other. A rigid body is called chiral if its mirror image is different in that sense, i.e., if it has either no symmetry or its symmetry group contains only proper rotations. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version. The latter applies for S2n, of which the case n = 1 is inversion symmetry. Two rigid bodies are said to be equal (copies of each other) if there is a mathematical proper rotation, that applied to one, gives the same physical state as the other, or correspondingly, if removal of one object and a combination of rotations of the other to the position of... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... This article is about the Twilight Zone episode. ... Sphere symmetry group o. ... The symmetry group of an object (e. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...


For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases:

  • the sheet surface with the image is not symmetric - in this case the two sides are different, but the mirror image of the object is the same, after a rotation by 180° about the axis perpendicular to the mirror plane.
  • the sheet surface with the image has a symmetry axis - in this case the two sides are the same, and the mirror image of the object is also the same, again after a rotation by 180° about the axis perpendicular to the mirror plane.

A sheet with a through and through image is achiral. We can distinguish again two cases: An image may be through and through in the following cases: ink or paint penetrating to the other side inlaying with another material, stained glass, patchwork, woodwork, linoleum, marble, etc. ...

  • the sheet surface with the image has no symmetry axis - the two sides are different
  • the sheet surface with the image has a symmetry axis - the two sides are the same

See also


  Results from FactBites:
 
Rigid body - Wikipedia, the free encyclopedia (761 words)
In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected.
The configuration space of a rigid body with one point fixed is given by the underlying manifold of the rotation group SO(3).
Two rigid bodies are said to be different (not copies) is that there is no proper rotation from one to the other.
  More results at FactBites »

 
 

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