Rigid body

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Rigid body

In physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, a rigid body is an idealization of a solid body

Physical body

In physics, a physical body is a collection of masses, taken to be one. For example, a cricket ball can be considered an object but the ball also consists of many particles ....
of finite size in which deformation

Deformation

In materials science, deformation is a change in the shape or size of an object due to an applied force . This can be a result of tensile strength forces, compressive strength forces, Simple shear, bending or torsion ....
is neglected. In other words, the distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
between any two given point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s of a rigid body remains constant in time regardless of external force

Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
s exerted on it. Even though such an object cannot physically exist due to relativity

Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
, objects can normally be assumed to be perfectly rigid if they are not moving near the speed of light

Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
.

In classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics

Quantum mechanics

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Encyclopedia

In physics

Physics

Physical body

Deformation

Distance

Point (geometry)

Force

Special relativity

Speed of light

Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
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 rigid bodies (see classification of molecules as rigid rotors

Rotational spectroscopy

Rotational spectroscopy or microwave spectroscopy studies the Absorption and Emission electromagnetic radiation by molecules associated with a corresponding change in the Rigid rotor#Quantum mechanical rigid rotor quantum number of the molecule....
).

Kinematics

Linear and angular position

The position of a rigid body is the position

Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different and mathematically more convenient approach is used. The position of the whole body is represented by:

the linear position or position of the body, namely the position of one of the particles of the body, specifically chosen as a reference point (for instance its center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
or its centroid
Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment about the line....
, or the origin of a coordinate system
Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
fixed to the body), together with
the angular position (or orientation) of the body.

Thus, the position of a rigid body has two components: linear and angular, respectively. The same is true for other kinematic

Kinematics

Kinematics is a branch of classical mechanics which describes the motion of objects without consideration of the causes leading to the motion....
and kinetic

Dynamics (mechanics)

In physics the term dynamics customarily refers to the time evolution of physical processes. These processes may be microscopic as in particle physics, kinetic theory, and chemical reactions, or macroscopic as in the predictions of statistical mechanics and nonequilibrium thermodynamics....
quantities describing the motion of a rigid body, such as velocity

Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
, acceleration

Acceleration

File:Acceleration.JPGFile:Acceleration components.JPGIn physics, and more specifically kinematics, acceleration is the change in velocity over time....
, momentum

Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
, impulse

Impulse

In classical mechanics, an impulse is defined as the integral of a force with respect to time. When a force is applied to a rigid body it changes the momentum of that body....
, and kinetic energy

Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
.

The linear position

Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
can be represented by a vector

Position vector

clude>A position, location or radius vector is a vector which represents the position of an object in Space#Classical_mechanics in relation to an arbitrary reference Point_....
with its tail at an arbitrary reference point in space

Space

Space is the boundless, three-dimensional extent in which Physical body and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physics usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime....
(often the origin of a chosen coordinate system

Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
) and its tip at a point of interest on the rigid body (often its center of mass

Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
or centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment about the line....
).

There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles

Euler angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles....
, a quaternion

Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
, or a direction cosine matrix (also referred to as a rotation matrix

Rotation matrix

In matrix theory, a rotation matrix is a real number square matrix whose transpose is its invertible matrix and whose determinant is 1 The matrix is so-called because it geometrically corresponds to a linear map that sends vectors to a corresponding vector rotated about the origin by a fixed angle....
).

In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation
Translation (physics)

In physics, translation is movement that changes the displacement of an object, as opposed to rotation. For example, according to Whittaker:...
and rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
, respectively. Indeed, the position of a rigid body can be viewed as an hypothetic translation and rotation (roto-translation) of the body starting from an hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).

Linear and angular velocity

Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
(also called linear velocity) and angular velocity

Angular velocity

Frame of reference

A frame of reference in physics, may refer to a coordinate system or Cartesian coordinate system within which to measure the position, orientation , and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an Observer ....
.

The linear velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
of a rigid body is a vector quantity, equal to the time rate of change

Time derivative

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as ....
of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity

Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
. However, when motion

Motion (physics)

In physics, motion means a constant change in the location of a body. Change in motion is the result of applied force. Motion is typically described in terms of velocity, acceleration, Displacement , and time....
involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if and when they happen to lay on an axis parallel to the instantaneous axis of rotation.

Angular velocity
Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed, and axis about which an object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, degrees per hour, etc....
is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating. All points on a rigid body experience the same angular velocity

Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed, and axis about which an object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, degrees per hour, etc....
at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation. The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change

Time derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
to obtain the angular velocity.

Kinematical equations

Addition theorem for angular velocity

The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect D:

.

In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.

Addition theorem for position

For any set of three points P, Q, and R, the position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R:

.

Mathematical definition of velocity

The velocity of point P in reference frame N is defined using the time derivative

Time derivative

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as ....
in N of the position vector from O to P:

where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/dt operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N.

Mathematical definition of acceleration

The acceleration of point P in reference frame N is defined using the time derivative

Time derivative

Velocity of two points fixed on a rigid body

For two points P and Q that are fixed on a rigid body B, where B has an angular velocity in the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N:

.

Acceleration of two points fixed on a rigid body

By differentiating the equation for the Velocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as

where is the angular acceleration of B in the reference frame N.

Velocity of one point moving on a rigid body

If the point R is moving in rigid body B while B moves in reference frame N, then the velocity of R in N is

.

where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest. This relation is often combined with the relation for the Velocity of two points fixed on a rigid body.

Acceleration of one point moving on a rigid body

The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by

where Q is the point fixed in B that instantaneously coincident with R at the instant of interest. This equation is often combined with Acceleration of two points fixed on a rigid body.

Other quantities

If C is the origin of a local coordinate system

Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
L, attached to the body,

the spatial or twist
Screw theory

Screw theory was developed by Sir Robert Stawell Ball in 1876, for application in kinematics and statics of mechanisms . It is a way to express displacements, velocities, forces and torques in three dimensional space, combining both rotational and translational parts....
acceleration of a rigid body is defined as the spatial acceleration
Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity....
of C (as opposed to material acceleration above);

where

represents the position of the point/particle with respect to the reference point of the body in terms of the local coordinate system L (the rigidity of the body means that this does not depend on time)
is the orientation
Orientation (rigid body)

The orientation in space of an axis , segment of axis, directed axis, or segment of directed axis is defined by the angles it forms with the axes of a Frame of reference, or other equivalent methods, such as direction cosines....
matrix, an orthogonal matrix
Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
with determinant 1, representing the orientation
Orientation (rigid body)

The orientation in space of an axis , segment of axis, directed axis, or segment of directed axis is defined by the angles it forms with the axes of a Frame of reference, or other equivalent methods, such as direction cosines....
(angular position) of the local coordinate system L, with respect to the arbitrary reference orientation of another coordinate system G. Think of this matrix as three orthogonal unit vectors, one in each column, which define the orientation of the axes of L with respect to G.
represents the angular velocity
Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed, and axis about which an object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, degrees per hour, etc....
of the rigid body
represents the total velocity of the point/particle
represents the total acceleration of the point/particle
represents the angular acceleration
Angular acceleration

Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha ....
of the rigid body
represents the spatial acceleration
Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity....
of the point/particle
represents the spatial acceleration
Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity....
of the rigid body (i.e. the spatial acceleration of the origin of L)

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.

Vehicle

Vehicle

Vehicles, derived from the Latin word, vehiculum, are non-living means of transport. Most often they are manufactured , although some other means of transport which are not made by humans also may be called vehicles; examples include icebergs and floating tree trunks....
s, 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

Winding number

In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point....
with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
.

Kinetics

Any point that is rigidly connected to the body can be used as reference point (origin of coordinate system 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:

the center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
of the whole system, which generally has the simplest motion for a body moving freely in space;
a point such that the translational motion is zero or simplified, e.g. on an axle
Axle

An axle is a central shaft for a rotation wheel or gear. In some cases the axle may be fixed in position with a bearing or bushing sitting inside the hole in the wheel or gear to allow the wheel or gear to rotate around the axle....
or hinge
Hinge

A hinge is a type of Bearing that connects two solid objects, typically allowing only a limited angle of rotation between them. Two objects connected by an ideal hinge rotate relative to each other about a fixed axis of rotation ....
, at the center of a ball and socket joint
Ball and socket joint

A ball and socket joint is a joint in which the distal bone is capable of motion around an indefinite number of axes, which have one common center....
, etc.

When the center of mass is used as reference point:

The (linear) momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
is independent of the rotational motion. At any time it is equal to the total mass of the rigid body times the translational velocity.
The angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
with respect to the center of mass is the same as without translation: at any time it is equal to the inertia tensor
Moment of inertia

Moment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate....
times the angular velocity. When the angular velocity is expressed with respect to a coordinate system coinciding with the principal axes
Moment of inertia

Moment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate....
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
Torque

Torque is the tendency of a force to rotate an object about an axis . Just as a force is a push or a pull, a torque can be thought of as a twist....
is the inertia tensor times the angular acceleration
Angular acceleration

Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha ....
.
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
Precession

Precession refers to a change in the direction of the axis of a rotation object. In physics, there are two types of precession, torque-free and torque-induced, the latter being discussed here in more detail....
.
The net external force on the rigid body is always equal to the total mass times the translational acceleration (i.e., Newton's second law holds for the translational motion, even when the net external torque is nonzero, and/or the body rotates).
The total kinetic energy
Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
is simply the sum of translational and rotational energy
Rotational energy

The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its Kinetic energy#Rotation in systems....
.

Geometry

Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the other. A rigid body is called chiral

Chirality (mathematics)

In geometry, a figure is chiral 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....
if its mirror image

Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
is different in that sense, i.e., if it has either no symmetry

Symmetry

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 perfection....
or its symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
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 S_2n

Point groups in three dimensions

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 of a sphere....
, of which the case n = 1 is inversion symmetry.

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

Through and through

Through and through describes a situation where an object, real or imaginary, passes completely through another object, also real or imaginary. The phrase has several common uses:...
image is achiral. We can distinguish again two cases:

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

Configuration space

The configuration space

Configuration space

Configuration space in physics In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints....
of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying manifold

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
of the rotation group

Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
SO(3). The configuration space of a nonfixed (with non-zero translational motion) rigid body is E⁺(3)

Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
, the subgroup of direct isometries

Euclidean group

Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
and rotation

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
s).

Rigid body

Kinematics

Linear and angular position

Linear and angular velocity

Kinematical equations

Addition theorem for angular velocity

Addition theorem for position

Mathematical definition of velocity

Mathematical definition of acceleration

Velocity of two points fixed on a rigid body

Acceleration of two points fixed on a rigid body

Velocity of one point moving on a rigid body

Acceleration of one point moving on a rigid body

Other quantities

Kinetics

Geometry

Configuration space

See also