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Frame of reference

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. Two observers may choose to use different frames of reference to investigate a common system. This definition applies to "classical" physics, i.e. before the general theory of relativity. In relativity, it can happen that a spacetime cannot be covered by (described in terms of) a rigid reference frame, but gravity can cause distortions that vary from place to place. (see, for example [1] which demonstrates experimentally that the rotation of the Earth pulls inertial reference frames near it in a circular motion whose rotational speed must fall off at large distances. Simply put, a set of locally inertial reference frames at varying distances from the Earth's axis are twisting up kind of like molasses stirred by a central rotator.)

The measurements that an observer makes about a system generally depend on the observer's frame of reference (examples are given below). However, the principle of relativity states that, even though a set of measurements may depend on an observer's particular frame of reference, the observed physical events must still follow the same physical laws in all inertial frames of reference.

For example, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).

Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving - for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Now assume Candace is driving her car in the opposite direction. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same - in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a' = a - A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a' = a + A in the negative y-direction - a larger value than Alfred's measurement.

Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another reference frame. The speed of light is considered to be the only true constant between moving frames of reference.

Nomenclature and notation

When working a problem involving one or more frames of reference it is common to designate an inertial frame of reference.

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x' , y' , a' .

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r'. From the geometry of the situation, we get

$\vec r = \vec R + \vec r'$

Taking the first and second derivatives of this, we obtain

$\vec v = \vec V + \vec v'$
$\vec a = \vec A + \vec a'$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as

$\vec F = m\vec a = m\vec A + m\vec a'$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decellerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or decelleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in tangential direction, the coriolis effect). In actual fact the force exerted on the object that keeps the object's motion in sync with the rotating frame elicits manifestation of inertia. If there is insufficient force to keep the object's motion in sync with the rotating frame, then seen from the perspective of the rotating frame there is an apparent accelaration. Whenever manifestation of inertia appears to act as a force it is labeled as a fictitious force. Inertia is very much real, of course, but unlike force it never accelerates an object.

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation

$\vec a = \vec a' + \dot\vec\omega \times \vec r' + 2\vec\omega \times \vec v' + \vec\omega \times (\vec\omega \times \vec r') + \vec A_0$

Multiplying through by the mass m gives

$\vec F' = \vec F_\mathrm{physical} + \vec F'_\mathrm{transverse} + \vec F'_\mathrm{coriolis} + \vec F'_\mathrm{centripetal} - m\vec A_0$

where

$\vec F'_\mathrm{transverse} = -m\dot\vec\omega \times \vec r'$
$\vec F'_\mathrm{coriolis} = -2m\vec\omega \times \vec v'$ (Coriolis force)
$\vec F'_\mathrm{centrifugal} = -m\vec\omega \times (\vec\omega \times \vec r')=m(\omega^2 \vec r'- (\vec\omega \cdot \vec r')\vec\omega)$ (centrifugal force)