Rotation

Related Topics:
Rotational_Molding

Rotation is the movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself; spin implies a quick rotation, perhaps freely-moving with angular momentum. A circular motion about an external point, for example the Earth around the Sun, is more properly called orbital revolution.

Mathematics

Mathematically, a rotation is a rigid body movement which keeps a point fixed; unlike a translation. This definition is applicable both for rotations in a plane (two dimensions) and in space (three dimensions). It turns out that a rotation in the three-dimensional space keeps fixed not just a single point, but rather an entire line; that is to say, any rotation in the three dimensional space is a rotation around an axis. This is a consequence of Euler's rotation theorem.

Any rigid body movement is in fact either a rotation, or a translation, or a combination of the two.

If one does a rotation around a point (axis), followed by another rotation around the same point (axis), the total result is yet another rotation. The reverse (inverse) of a rotation is also a rotation. It follows that the rotations around a point or axis form a group. If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may not be a rotation anymore; it can for example be a translation.

Principal axes of rotation

Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as pitch, roll and yaw.

Astronomy

In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all spin around on their axes (the plural of axis).

This rotation induces a centrifugal acceleration which slightly counteracts the effect of gravity the closer one is to the equator. One effect is that an object weighs slightly less at the equator. Another is that the Earth is slightly deformed into a oblate spheroid.

Another consequence of the rotation of a planet is the phenomenon of precession. Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is 66.5 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole star.)

Revolution

Moons also revolve about their planet, planets revolve about their star (such as the Earth around the Sun); and stars slowly revolve about their galaxial center. The motion of the components of galaxies is complex, but it usually includes a rotation component.

The Moon makes one complete rotation during one complete revolution around the Earth (a effect called tidal locking) so that the same side of the Moon always faces the Earth (the other side is called the far side of the Moon).

Retrograde rotation

Most planets in our solar system, including Earth, spin in the same direction as they orbit the Sun. The exceptions are Venus, Uranus, and Pluto. Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. Venus may be thought of as rotating slowly backwards (or being "upside down").

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, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw.

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.

Sports

Rotation, usually called spin, plays a role in many sports. Topspin and backspin in tennis. English, follow and draw in billiards and pool. Curve balls in baseball and spin bowling in cricket.

External links

Product of Rotations at cut-the-knot
When a Triangle is Equilateral at cut-the-knot
Rotation (Interactive Animation) at Math is Fun

Euclidean geometry | Celestial mechanics

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