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The instantaneous power of the Coriolis force, evaluated with the velocity in the same rotating reference frame, vanishes:This follows from the perpendicularity of the cross product and the dot product definition of power. The Coriolis force alone does no work and cannot change the rotating-frame kinetic energy $m|{\mathbf{v}}_{\mathrm{rot}}{|}^2/2$. It can change the direction of the relative velocity. Actual forces and the centrifugal force can still change that kinetic energy, as the free particle in rotating coordinates illustrates. Conservation of an appropriate total energy has the additional conditions stated under energy balance in a uniformly rotating frame.

Apply the rotating-frame derivative formula first to $r$, then to $v^{\prime }=[dr/dt{]}_{S^{\prime }}$. Since the angular velocity is constant in either frame,The two apparent forces in the rotating equation are the Coriolis force and the centrifugal acceleration multiplied by mass. ThusSet ${\varphi }_{\mathrm{eff}}=\varphi -\frac{m}{2}|\omega \times r{|}^2$. The supplied gradient identity gives $\nabla {\varphi }_{\mathrm{eff}}=\nabla \varphi +m\omega \times (\omega \times r)$. Taking the scalar product with $v^{\prime }$ removes $G$, since it is orthogonal to $v^{\prime }$, and removes the Coriolis force. Therefore the energy balance in a uniformly rotating frame isThe displayed quantity is conserved when the potential is stationary in the rotating coordinates. This includes a static inertial potential invariant under rotations about the chosen axis, such as vertical gravity in part (b). If the word conservative only means a time-independent inertial potential, that extra rotational invariance is needed; a conservative force alone does not imply the claimed conservation.

For an explicit counterexample to that unrestricted interpretation, take unit mass, an inertial force $F=e_x$, $\varphi =-x$, no $G$, rotation $\omega =\omega e_z$, and inertial trajectory $r(t)=(t^2/2,1,0)$ with initial velocity zero. It satisfies Newton's second law. Substituting $v^{\prime }=\dot{r}-\omega \times r$ into the proposed energy gives $E=\omega t$, which is not constant for nonzero rotation. The qualified conservation law above is the one used for the hoop.