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The position of a rigid body is determined by the position of its center of mass and by its orientation (at least six parameters in total).
The position of a rigid body is determined by the position of its center of mass and by its orientation (at least six parameters in total). This article deals with orientation of reference axes or frames

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physics, a physical body (sometimes called simply a body or even an object) is a collection of Masses taken to be one In Materials science, deformation is a change in the shape or size of an object due to an applied force. 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. Distance is a numerical description of how far apart objects are In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In Physics, a force is whatever can cause an object with Mass to Accelerate. 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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons For instance, in quantum mechanics molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors). In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by Rotational spectroscopy or microwave spectroscopy studies the absorption and emission Electromagnetic radiation (typically in the Microwave

Contents

Kinematics

Position

The position of a rigid body can be described by a combination of a translation and a rotation from a given reference position. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation For this purpose a reference frame is chosen that is rigidly connected to the body (see also below). This is typically referred to as a "local" reference frame (L). The position of its origin and the orientation of its axes with respect to a given "global" or "world" reference frame (G) represent the position of the body. This article deals with orientation of reference axes or frames The position of G not necessarily coincides with the initial position of L.

Thus, the position of a rigid body has two components: linear and angular, respectively. Each can be represented by a vector. The angular position is also called orientation. This article deals with orientation of reference axes or frames There are several methods to describe numerically the orientation of a rigid body (see orientation). This article deals with orientation of reference axes or frames In general, if the rigid body moves, both its linear and angular position vary with time. In the kinematic sense, these changes are referred to as translation and rotation, respectively. Translation is the interpreting of the meaning of a text and the subsequent production of an equivalent text likewise called a " translation A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

All the points of the body change their position during a rotation about a fixed axis, except for those lying on the rotation axis. If the rigid body has any rotational symmetry, not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation.

In two dimensions the situation is similar. In one dimension a "rigid body" can not move (continuously change) from one orientation to the other.

Other quantities

If C is the origin of the local reference frame L,

For any point/particle of a moving rigid 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) + \boldsymbol\omega(t) \times (\mathbf{r}(t,\mathbf{r}_0) - \mathbf{r}_c(t)) = \mathbf{v}_c(t) + \boldsymbol\omega(t) \times A(t) \mathbf{r}_0
\mathbf{a}(t,\mathbf{r}_0) = \mathbf{a}_c(t) + \boldsymbol\alpha(t) \times A(t) \mathbf{r}_0 + \boldsymbol\omega(t) \times \boldsymbol\omega(t) \times A(t) \mathbf{r}_0
\boldsymbol\psi(t,\mathbf{r}_0) = \mathbf{a}(t,\mathbf{r}_0) - \boldsymbol\omega(t) \times \mathbf{v}(t,\mathbf{r}_0) = \boldsymbol\psi_c(t) + \boldsymbol\alpha(t) \times A(t) \mathbf{r}_0

where

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

Vehicles, walking people, etc. Vehicles, derived from the Latin word vehiculum, are non-living Means of transport. 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. The term winding number may also refer to the Rotation number of an Iterated map. Compare the amount of rotation associated with the vertices of a polygon. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit

Kinetics

Main article: Rigid body dynamics

Any point that is rigidly connected to the body can be used as reference point (origin of frame L) to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on the choice).

However, depending on the application, a convenient choice may be:

When the center of mass is used as reference point:

Geometry

Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the other. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or A rigid body is called chiral if its mirror image is different in that sense, i. 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 "Mirror Image" is an episode of the Television series The Twilight Zone. e. , if it has either no symmetry or its symmetry group contains only proper rotations. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is 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. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group

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:

A sheet with a through and through image is achiral. Through and through describes a situation where an object real or imaginary passes completely through another object also real or imaginary We can distinguish again two cases:

Configuration space

The configuration space of a rigid body with one point fixed (i. "Configuration space" may also refer to PCI Configuration Space. e. , a body with zero translational motion) is given by the underlying manifold of the rotation group SO(3). A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be This article is about rotations in three-dimensional Euclidean space This article is about rotations in three-dimensional Euclidean space The configuration space of a nonfixed (with non-zero translational motion) rigid body is E+(3), the subgroup of direct isometries of the Euclidean group in three dimensions (combinations of translations and rotations). In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional In Euclidean geometry, a translation is moving every point a constant distance in a specified direction A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

See also

References

Dictionary

rigid body

-noun

  1. (physics) an idealized solid whose size and shape are fixed and remain unaltered when forces are applied; used in Newtonian mechanics to model real objects
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