Rotation

The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or autorotation).

Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps at least one point fixed.

This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity.

Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude.

The ratio of torque τ to the angular acceleration α is given by the moment of inertia:

Since translation involves displacement of rigid bodies while preserving the orientation of the body, in the case of curvilinear translation, all the points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation.

By Euler's theorem, any change in orientation can be described by rotation about an axis through a chosen reference point.

[6] Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through instantaneous center of circle and perpendicular to the plane of motion.

In contrast, a rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion.

In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.

According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant.

In astronomy, rotation is a commonly observed phenomenon; it includes both spin (auto-rotation) and orbital revolution.

The rotation rate of planets in the solar system was first measured by tracking visual features.

Stellar rotation is measured through Doppler shift or by tracking active surface features.

This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equator.

Earth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles.

Another is that over time the Earth is slightly deformed into an oblate spheroid; a similar equatorial bulge develops for other planets.

Currently the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is 23.44 degrees, but this angle changes slowly (over thousands of years).

Most planets in the Solar System, including Earth, spin in the same direction as they orbit the Sun.

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.

In flight dynamics, the principal rotations described with Euler angles above are known as pitch, roll and yaw.

The term rotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff.

A Ferris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically.

In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.

Rotation of a ball or other object, usually called spin, plays a role in many sports, including topspin and backspin in tennis, English, follow and draw in billiards and pool, curve balls in baseball, spin bowling in cricket, flying disc sports, etc.

Table tennis paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball.

Rotation of a player or performer one or more times around a horizontal axis may be called a flip, roll, somersault, heli, etc.

in gymnastics, waterskiing, or many other sports, or a one-and-a-half, two-and-a-half, gainer (starting facing away from the water), etc.

A combination of vertical and horizontal rotation (back flip with 360°) is called a möbius in waterskiing freestyle jumping.