Rotation

From Example Problems
Jump to navigation Jump to search
This article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation).
File:800px-Nightly Rotation above San Jose International Airport small.jpg
Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1.3 sec). The center of rotation is at the left-center of the photo
File:Rotation illustration.png
Rotation of a planar figure around a point

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, the rotation is around an axis — it amounts to rotation in each plane perpendicular to the axis around the intersection of the plane and the axis.

An example of rotation of a planar figure around a point 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.

File:Axes changes.png
Rotation as change in orientation.

If the axis of rotation is within the body, the body is said to rotate upon itself, or spin. Among such rotations, the simplest case is that of constant angular frequency (see also below).

Allowed rotations

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.

See also: rolling, axle, wheel, rolling, gyration, gyroscope, rotations in anatomy

Mathematics

Template:See details

Mathematically, a rotation is a rigid body movement which keeps a point fixed, unlike a translation. 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 the Euler's rotation theorem, and is no longer true in higher dimensions.

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 followed by rotation around another point, the overall movement may not be a rotation anymore.

File:Flight dynamics.jpg
The principal axes of rotation in space

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.

See also: curl, cyclic permutation, Euler angles, rigid body, rotation around a fixed axis, rotation group, rotation matrix

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.

See also: orbital period, oblate

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, angular momentum, angular velocity, centrifugal force, centripetal force, circular motion, circular orbit, Coriolis effect, spin, rigid body angular momentum

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.

External links

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