= Solution
Let the <rotating frame> have origin <acceleration> $\mathbf A_0$ and <angular velocity> $\boldsymbol\Omega(t)$ relative to an <inertial frame>. For a <vector> referred to the moving axes, the transport rule is
$$
\left(\frac{d\mathbf a}{dt}\right)_{\mathrm I}
=\left(\frac{d\mathbf a}{dt}\right)_{\mathrm R}+\boldsymbol\Omega\times\mathbf a.
$$
Apply it twice to position. With rotating-frame <velocity> $\mathbf v$ and <acceleration> $\mathbf a_R$,
$$
\mathbf a_I=\mathbf A_0+\mathbf a_R+2\boldsymbol\Omega\times\mathbf v
+\dot{\boldsymbol\Omega}\times\mathbf r
+\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r).
$$
Thus Newton's equation expressed in the <rotating frame> is
$$
\boxed{m\mathbf a_R=\mathbf F_{\mathrm{real}}-m\mathbf A_0
-2m\boldsymbol\Omega\times\mathbf v-m\dot{\boldsymbol\Omega}\times\mathbf r
-m\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r).}
$$
The added terms are translational, Coriolis, Euler and centrifugal <fictitious forces>. They reflect the frame's <acceleration>, rather than additional interactions with nearby objects.
The <centrifugal force> points outward from the <rotation> axis and, for constant $\boldsymbol\Omega$, has potential $-m|\boldsymbol\Omega\times\mathbf r|^2/2$. The <Coriolis force> is perpendicular to <velocity>, so does no <work> on the particle measured in that <rotating frame>; it changes direction rather than speed directly. The <Euler force> is present when the frame's <rotation> rate or direction changes.
On Earth, gravity and <centrifugal acceleration> combine into effective gravity. For slow horizontal motion at latitude $\lambda$, the vertical component of Earth's <rotation> gives Coriolis parameter $f=2\Omega\sin\lambda$: an initially northward motion deflects eastward in the northern hemisphere. This supplies a useful local explanation of rotating flows and Foucault <precession>. A bucket of rotating <liquid> illustrates the <centrifugal potential>: its steady free surface is an equipotential of $gz-\Omega^2r^2/2$, giving $z=\Omega^2r^2/(2g)+\mathrm{constant}$. These equations require a declared frame; adding a <centrifugal force> while retaining inertial-frame <acceleration> would count the same kinematic effect twice.
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