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A rotating frame of reference is a coordinate system that describes how physics appears when measured against a hypothetical network of rigid rulers extending from a rotating body. If a body is described as travelling in a straight line with respect to an inertial reference frame, then when the same physics is described from a rotating reference frame, the same body's motion will usually seem to be following a curved path. A rotating reference frame is often classed as being a special case of an accelerated reference frame. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...
In theoretical physics, an accelerated reference frame is usually a coordinate system or frame of reference, that undergoes a constant and continual change in velocity over time as judged from an inertial frame. ...
In a rotating frame, someone attempting to describe the Newtonian mechanics would see the Second Law (force = mass*acceleration) modified by the addition of two forces which arise as artefacts of their choice of frame of reference. These two extra forces are described below.
Position transformation formulae If we have two frames of reference, one rotating and the other not; the place where an event occurs in one frame are obtained by applying a rotation to the original co-ordinates to obtain the coordinates in the other frame. The required angle of rotation varies linearly with time. A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...
Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...
This article is about angles in geometry. ...
To obtain cartesian co-ordinates from polar co-ordinates, replace Cartesian means of or relating to the French philosopher and mathematician René Descartes. ...
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 where r is the radius, θ is the angle in the non rotating frame, and Θ is the angle in the rotating frame. Clearly the radius stays the same in the rotating frame whereas the angle increases with time. The angle in the rotating frame is given by:  where ω is the angular velocity of the rotation.
Transforming back to cartesian coordinates gives:
 and these two equations tell us how positions are related to one another in the two frames.
Relation between velocities in the two frames If (x,y,z) are the co-ordinates of a body, and are hence functions of time, the time-derivatives of positions (X,Y) in the rotating frame depend on those of in the stationary frame (x and y). So,   Note that (X,Y) in the equations are velocities and not positions and have units s − 1.
Relation between accelerations in the two frames Applying differentiation to the equations for velocities, d2X/dt2 = (d2x/dt2).Cos(w.t) +(d2y/dt2).Sin(w.t) -(dx/dt).Sin(w.t) +(dy/dt).Cos(w.t)+(dY/dt) = (d2x/dt2).Cos(w.t) +(d2y/dt2).Sin(w.t) +2.(dY/dt) +X. d2Y/dt2 = (d2y/dt2).Cos(w.t) -(d2x/dt2).Sin(w.t) -((dy/dt).Sin(w.t) +(dx/dt).Cos(w.t)) -(dX/dt) = (d2y/dt2).Cos(w.t) -(d2x/dt2).Sin(w.t) -2.(dX/dt) +Y Note in these equations, the last X and Y are now accelerations having the dimensions s^-2
These equations enable us to describe the acceleration of the body, as seen by the rotating frame, as the sum of three terms:
1. Rotated actual acceleration
This is given by the components of the acceleration seen by the non-rotating frame, by the same transformation aas for position. This exactly equals the force acting on the body, as described by the rotating frame, divided by the body's mass. The z-component is unaffected by the rotation but can be included as a natural component of acceleration.
2. Coriolis term which is 2.w/radian times the result of rotating the velocity of the body (as seen by the rotating frame) through a right angle in the sense which maps a purely Y-wards vector into a purely X-wards one. The name comes from someone who famously documented this effect. The axial component of the velocity is ignored in this term. Gaspard-Gustave de Coriolis (May 21, 1792- September 19, 1843), French engineer and scientist. ...
3. The centrifugal term: which is the body's position, ignoring axial component, scaled by the square of w/radian. This term means flying away from the centre. The expression centrifugal force is used to express that if an object is being swung around on a string the object seems to be pulling on the string. ...
Explanation of effects To describe the physics in the rotating frame , an observer thus sees force = mass.acceleration modified by, in effect, the addition of two forces which arise as artefacts of their choice of frame of reference. General relativity is quite happy to let us use the rotating frame, and effectively regards the two artificial forces as part of gravitation.
Exploiting the vector outer product
In three dimensions (and, only in three dimensions) there is a product operator (which actually depends on your metric) that combines two 3-vectors, antisymmetrically, to produce a third.
Specifically, if the [x,y,z] components of vectors u and v are [a,b,c] and [e,f,g], respectively, then those of their outer product, u^v, are [b.g-c.f, c.e-a.g, a.f-b.e]. This relies on x, y, z being orthonormal co-ordinates for our metric; along with a presumption that the rotation which takes the y axis to the x axis appears clockwise when viewed from a position with positive z co-ordinate.
Crucially, u^v is perpendicular to both u and v, applying any scaling to either u or v scales u^v to the same degree and v^u = -u^v.
Now, both the Coriolis and centrifugal terms in our transformed acceleration have a zero component in the z-direction, i.e. parallel to the axis. The Coriolis term is perpendicular to the transformed velocity's projection into the plane perpendicular to the axis and is, consequently, perpendicular to the transformed velocity itself (which is the sum of its given projection and its component parallel to the axis; adding two vectors perpendicular to the Coriolis term yields another). So Coriolis term should be compared with the outer product of the transformed velocity and some vector in the direction of the axis. Since the Coriolis term is also proportional to w/radian, we can use this as the scale of the axial vector.
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