Rotating reference frame: Difference between revisions

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The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) frame. Then, by taking time derivatives, formulas are derived that relate the velocity of the particle as seen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames.

=== Relation between coordinatespositions in the two frames ===

To derive these fictitious forces, it's helpful to be able to convert between the coordinates <math>\left(x', y', z'\right)</math> of the rotating reference frame and the coordinates <math>(x, y, z)</math> of an [[inertial reference frame]] with the same origin.<ref group=note>So <math>x', y', z'</math> are functions of <math>x, y, z,</math> and time <math>t.</math> Similarly <math>x, y, z</math> are functions of <math>x', y', z',</math> and <math>t.</math> That these reference frames have the same origin means that for all <math>t,</math> <math>\left(x', y', z'\right) = (0, 0, 0)</math> if and only if <math>(x, y, z) = (0, 0, 0).</math></ref>

If the rotation is about the <math>z</math> axis with a constant [[angular velocity]] <math>\Omega</math> (so <math>z' = z</math> and <math>\frac{\mathrm{d} \theta}{\mathrm{d} t} \equiv \Omega,</math> which implies <math>\theta(t) = \Omega t + \theta_0</math> for some constant <math>\theta_0</math> where <math>\theta(t)</math> denotes the angle in the <math>xyx-y</math> -plane formed at time <math>t</math> by <math>\left(x', y'\right)</math> and the <math>(x,y</math>-axis),

and if the two reference frames coincide at time <math>t = 0</math> (meaning <math>\left(x', y', z'\right) = (x, y, z)</math> when <math>t = 0,</math> so take <math>\theta_0 = 0</math> or some other integer multiple of <math>2\pi</math>), the transformation from rotating coordinates to inertial coordinates can be written

<math display=block>x = x'\cos(\theta(t)) - y'\sin(\theta(t))</math>