Editing Inertial frame of reference (section)

===Primed frames===

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

: <math>\mathbf r = \mathbf R + \mathbf r'.</math>

Taking the first and second derivatives of this with respect to time

: <math>\mathbf v = \mathbf V + \mathbf v',</math>
: <math>\mathbf a = \mathbf A + \mathbf a'.</math>

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, [[Newton's laws of motion#Newton.27s second law|Newton's second law]] can be written as

: <math>\mathbf F = m\mathbf a = m\mathbf A + m\mathbf a'.</math>

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 decelerating 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 deceleration), 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 force (rotating reference frame)|centrifugal]] direction, or in a direction orthogonal to an object's motion, the [[Coriolis effect]]).

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 (see [[Fictitious force]] for a derivation):

: <math>\mathbf a = \mathbf a' + \dot{\boldsymbol\omega} \times \mathbf r' + 2\boldsymbol\omega \times \mathbf v' + \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') + \mathbf A_0,</math>

or, to solve for the acceleration in the accelerated frame,

: <math>\mathbf a' = \mathbf a - \dot{\boldsymbol\omega} \times \mathbf r' - 2\boldsymbol\omega \times \mathbf v' - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') - \mathbf A_0.</math>

Multiplying through by the mass ''m'' gives

: <math>\mathbf F' = \mathbf F_\mathrm{physical} + \mathbf F'_\mathrm{Euler} + \mathbf F'_\mathrm{Coriolis} + \mathbf F'_\mathrm{centripetal} - m\mathbf A_0,</math>

where

: <math>\mathbf F'_\mathrm{Euler} = -m\dot{\boldsymbol\omega} \times \mathbf r'</math> ([[Euler force]]),
: <math>\mathbf F'_\mathrm{Coriolis} = -2m\boldsymbol\omega \times \mathbf v'</math> ([[Coriolis force]]),
: <math>\mathbf F'_\mathrm{centrifugal} = -m\boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') = m(\omega^2 \mathbf r' - (\boldsymbol\omega \cdot \mathbf r')\boldsymbol\omega)</math> ([[Centrifugal force (rotating reference frame)|centrifugal force]]).