Rotation

Categories: Celestial mechanics | Euclidean geometry

This article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation).
Image:Rotation illustration.png
Rotation of a planar figure around a point
Image:Flight dynamics.jpg
The principal axes of rotation in space

Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. For a three-dimensional body, one can have rotation around an axis or around a point.

An example of rotation around an axis is the movement of the propeller of an aircraft. A door attached to the wall by two or more hinges rotates around the axis going through the hinges. A body attached to a fixed point by a nonbending wire can move in space only by rotations around that point.

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More details

Rotation is the change of orientation of an object. The term may either refer to the process, or to the resulting change in orientation relative to the starting or reference orientation. If it refers to the process, the simplest case is rotation of a rigid body about a fixed axis of rotation: a line such that each point of the body moves in a plane perpendicular to that line, in a circle centered at the intersection of the plane and the line. The angle of rotation is a function of time which is the same for every point of the body. In 2D a point is center of rotation.

The fixed point can be within the body, in which case the body is said to rotate upon itself, or spin.

An object may allow rotation with respect to an attached other object by means of one or more hinges (e.g. a door, scissors, a hinge joint). Gear couplings and universal joints connect two driveshafts at an angle (rotating about their own axes) to transmit torque, by two pairs of hinges allowing rotation about two other axes. A ball-and-socket joint, e.g. in the shoulder, allows rotations about all three axes.

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Mathematics

In the three-dimensional space, the possible moves of a rigid body are rotations and translations. They form E+(3), the subgroup of direct isometries of the Euclidean group E(3).

A result of Euler's rotation theorem is that every combination of these is the combination of just one rotation about some axis, and a translation along the axis (with as special cases pure rotation and pure translation). An arbitrary rotation with a given point fixed is given by the formula for a rotation about an axis through the origin; just add an arbitrary translation to get an arbitrary move of a rigid object. It can be decomposed into rotations about three fixed axes through that point, in terms of flight dynamics pitch, roll and yaw. See also degrees of freedom (engineering).

In mathematics also more abstract rotations are considered: rotations are any norm-preserving (isometric), orientation-preserving linear transformations on a vector space with an inner product (or, more generally, corresponding affine transformations). (Note that the word "orientation" is used here in a meaning different from that above.)

In the case of only linear transformations, these rotations form the rotation group: a group of so called special orthogonal matrices (orthogonal means the transpose is the inverse; this implies that the determinant equals 1 or − 1; special means that the determinant equals 1, not −1 ) designated by the name SO(n) in case of an n-dimensional vector space.

In n-dimensional space, E+(n ) is the subgroup of direct isometries of the Euclidean group E(n ).

A principal rotation is a linear transformation with an n−2-dimensional affine subspace of fixed points, keeping distances to that space fixed. Thus in 4D we have rotation about a plane; in this case about a plane through the origin, more generally about any plane.

Clifford-style rotations are the other members of SO(n). They are combinations of orthogonal principal rotations.

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Astronomy

In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all rotate around their axes, while planets also rotate about a star such as the Sun, and moons also rotate about a planet. The motion of the components of galaxies is complex, but it usually includes a rotation component.

One consequence of the rotation of a planet is the phenomenon of precession. Precession has the overall effect of introducing a long-term "wobble" in the movement of the axis of a planet. For example, the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is currently 66.5 degrees, but this angle has slowly changed over time due to the action of precession.

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Physics

The speed of rotation is given by the angular frequency (rad/s) or frequency (turns/s, turns/min), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by torque. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia.

The angular velocity vector also describes the direction of the axis of rotation. Similarly the torque is a vector.

According to the right-hand rule, moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like a screw.

See also rotational energy.

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Amusement rides

Many amusement rides provide rotation. A Ferris wheel and observation wheel have a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centrifugal force. In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where the centrifugal force keeps people in their seats.

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See also

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External links

de:Rotation (Physik) fr:Rotation nl:Rotatie (voorwerp) ja:自転 sl:Vrtenje