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rigid body | A Wisdom Archive on rigid body | | rigid body A selection of articles related to rigid body | |
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rigid body
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| ARTICLES RELATED TO rigid body | | | |  rigid body: Encyclopedia II - Euclidean group - SubgroupsTypes of subgroups of E(n):
Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category.
Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically ...
See also:Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Subgroups |
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| | | | | rigid body: Encyclopedia II - Euclidean group - Overview of isometries in up to three dimensions E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:
E(1) - 1:
E+(1):
identity - 0
translation - 1
those not preserving orientation:
reflection in a point - 1
E(2) - 3:
E+(2):
identity - 0
translation - 2
rotation about a point - 3
those not preserving orientation:
reflection in a line - 2
reflection in a line ...
See also:Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Overview of isometries in up to three dimensions |
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| | | | | rigid body: Encyclopedia II - Euclidean group - Relation to the affine group The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. Instead of by a pair (A, b), Euclidean group elements can also be represented as square matrices of size n + 1, as explained for the affine group.
In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometry. All affine theorems apply; the extra fa ...
See also:Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Relation to the affine group |
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| | | | | | | | rigid body: Encyclopedia II - Virtual work - Virtual work principle for a deformable body Consider now the free body diagram of a deformable body, which is composed of an infinite number of differential cubes as shown in the figure. Let's define two unrelated states for the body:
The -State (Fig.a): This shows external surface forces T, body forces f, and internal stresses in equilibrium.
The -State (Fig.b): This shows continuous displacements and consistent strains .
The superscript * emphasizes that the two states are unrelated. Other than the above stated con ...
See also:Virtual work, Virtual work - Virtual work principle for a particle, Virtual work - Virtual work principle for a rigid body, Virtual work - Virtual work principle for a deformable body, Virtual work - Principle of virtual displacements, Virtual work - Principle of virtual forces, Virtual work - Alternative forms, Virtual work - Bibliography Read more here: » Virtual work: Encyclopedia II - Virtual work - Virtual work principle for a deformable body |
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| | | | | | | | rigid body: Encyclopedia II - Rotation group - Orthogonal matrices Like any linear transformation, a rotation can always be represented by a matrix. Let R be a given rotation. With respect to the standard basis (e1,e2,e3) of R3 the columns of R are given by (Re1,Re2,Re3). Since the standard basis is orthonormal, the columns of R form another orthonormal basis. This orthonormality condi ...
See also:Rotation group, Rotation group - Properties, Rotation group - Orthogonal matrices, Rotation group - Axis of rotation, Rotation group - Topology, Rotation group - Representations of rotations, Rotation group - Generalizations Read more here: » Rotation group: Encyclopedia II - Rotation group - Orthogonal matrices |
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| | | | | | | rigid body: Encyclopedia II - Trajectory - Examples
Trajectory - Uniform gravity no drag or wind.
The case of uniform gravity, disregarding drag and wind, yields a trajectory which is a parabola. To model this, one chooses V = mgz, where g (gee) is the so-called acceleration of gravity. This gives the equations of motion
Simplifications are made for the sake of studying the basics. The actual situation, at least on the surface of Earth ...
See also:Trajectory, Trajectory - Physics of trajectories, Trajectory - Examples, Trajectory - Uniform gravity no drag or wind, Trajectory - Uphill/downhill in uniform gravity in a vacuum, Trajectory - Orbitting objects Read more here: » Trajectory: Encyclopedia II - Trajectory - Examples |
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| | | | | rigid body: Encyclopedia II - Rotation group - Generalizations The rotation group generalizes quite naturally to n-dimensional Euclidean space, Rn. The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).
In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite sig ...
See also:Rotation group, Rotation group - Properties, Rotation group - Orthogonal matrices, Rotation group - Axis of rotation, Rotation group - Topology, Rotation group - Representations of rotations, Rotation group - Generalizations Read more here: » Rotation group: Encyclopedia II - Rotation group - Generalizations |
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| | | | | rigid body: Encyclopedia II - Trajectory - Physics of trajectories One of the remarkable achievements of Newtonian mechanics was the derivation of the laws of Kepler, in the case of the gravitational field of a single point mass (representing the Sun). The trajectory is a conic section, like an ellipse or a parabola. This agrees with the observed orbits of planets and comets, to a reasonably good approximation. Although if a comet passes close to the Sun, then it is also influenced by other forces, such as the solar wind and radiation pressure, which modify the or ...
See also:Trajectory, Trajectory - Physics of trajectories, Trajectory - Examples, Trajectory - Uniform gravity no drag or wind, Trajectory - Uphill/downhill in uniform gravity in a vacuum, Trajectory - Orbiting objects Read more here: » Trajectory: Encyclopedia II - Trajectory - Physics of trajectories |
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| | | | | rigid body: Encyclopedia II - Trajectory - Examples
Trajectory - Uniform gravity no drag or wind.
The case of uniform gravity, disregarding drag and wind, yields a trajectory which is a parabola. To model this, one chooses V = mgz, where g (gee) is the acceleration of gravity. This gives the equations of motion
Simplifications are made for the sake of studying the basics. The actual situation, at least on the surface of Earth, is consi ...
See also:Trajectory, Trajectory - Physics of trajectories, Trajectory - Examples, Trajectory - Uniform gravity no drag or wind, Trajectory - Uphill/downhill in uniform gravity in a vacuum, Trajectory - Orbiting objects Read more here: » Trajectory: Encyclopedia II - Trajectory - Examples |
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| | | | | rigid body: Encyclopedia II - Rotation around a fixed axis - Speed of rotation angular acceleration and torque The speed of rotation is given by the angular frequency (rad/s) or frequency / rotational speed / revolutions per minute (turns/s, turns/min), or period (seconds, days, etc.).
With one direction of rotation considered positive, the sign of the angular frequency indicates the direction of rotation.
The time-rate of change of angular frequency is the scalar version of angular acceleration (rad/s²). This change is caused by the scalar version of the torque, which can have a positive or negative value in accordance with the conven ...
See also:Rotation around a fixed axis, Rotation around a fixed axis - Speed of rotation angular acceleration and torque, Rotation around a fixed axis - Vectors, Rotation around a fixed axis - Centripetal force, Rotation around a fixed axis - Constant angular speed Read more here: » Rotation around a fixed axis: Encyclopedia II - Rotation around a fixed axis - Speed of rotation angular acceleration and torque |
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| | | | | rigid body: Encyclopedia II - Torque - Units Torque has dimensions of force times distance and the SI units of torque are stated as "newton-metres". Even though the order of "newton" and "metre" are mathematically interchangeable, the BIPM (Bureau International des Poids et Mesures) specifies that the order should be N·m not m·N[1].
The joule, the SI unit for energy or work, is also defined as 1 N·m, but this unit is not used for torque. Since energy can be thought of as the result of "force dot distance", energy is always a scalar whereas torque is "force cross distance" and ...
See also:Torque, Torque - Units, Torque - Special cases and other facts, Torque - Moment arm formula, Torque - Force at an angle, Torque - Static equilibrium, Torque - Torque as a function of time, Torque - Machine torque, Torque - Relationship between torque and power, Torque - Conversion to other units, Torque - Derivation Read more here: » Torque: Encyclopedia II - Torque - Units |
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| | | | | rigid body: Encyclopedia II - Torque - Special cases and other facts
Torque - Moment arm formula.
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
The construction of the "moment arm" is shown in the figure below, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three-dimensional cases. If the force is perpendicular to the displacemen ...
See also:Torque, Torque - Units, Torque - Special cases and other facts, Torque - Moment arm formula, Torque - Force at an angle, Torque - Static equilibrium, Torque - Torque as a function of time, Torque - Machine torque, Torque - Relationship between torque and power, Torque - Conversion to other units, Torque - Derivation Read more here: » Torque: Encyclopedia II - Torque - Special cases and other facts |
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| | | | | rigid body: Encyclopedia II - Torque - Machine torque Torque is part of the basic specification of an engine: the power output of an engine is expressed as its torque multiplied by its rotational speed. Internal-combustion engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000–6,000 rpm for a small car). The varying torque output over that range can be measured with a dynamometer, and shown as a torque curve. The peak of that torque curve usually occurs somewhat below the overall power peak. The torque peak cannot, by definition, ...
See also:Torque, Torque - Units, Torque - Special cases and other facts, Torque - Moment arm formula, Torque - Force at an angle, Torque - Static equilibrium, Torque - Torque as a function of time, Torque - Machine torque, Torque - Relationship between torque and power, Torque - Conversion to other units, Torque - Derivation Read more here: » Torque: Encyclopedia II - Torque - Machine torque |
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