# Apparent contradiction regarding centrifugal force

Consider an inertial reference frame $$I$$, and a point object $$K$$ which is at rest with respect to $$I$$. $$K$$ is not at the origin of $$I$$.

Consider another reference frame $$I’$$, with the same origin as $$I$$, but rotating along a specified axis with angular velocity $$\omega$$ with respect to $$I$$.

From the reference frame $$I’$$, the object $$K$$ appears to be revolving around the origin of $$I’$$ with angular velocity $$\omega$$.

In the inertial frame $$I$$, no external force was needed on the body $$K$$ to describe its motion.

On the other hand, in the non-inertial frame $$I’$$, we need a pseudo-force to describe the body using Newtonian laws. Since in this frame, the body is revolving, so the required pseudo-force is centripetal.

However, in a uniformly rotating non-inertial frame, the effective pseudo-force should be centrifugal, instead of centripetal. So we have a contradiction.

I am sure I am missing something, but what?

If $$K$$ would be at rest in $$I'$$ then one would indeed observe a centrifugal force from $$I'$$. From $$I$$ one would see a real force that makes $$K$$ co-rotate (such as friction).
If $$K$$ is now at rest in $$I$$, then from $$I'$$ you would see that $$K$$ has a velocity $$\mathbf{v}$$ in the azimuthal direction. The fictitious force acting on $$K$$ is then the Coriolis force $$\mathbf{F}_{\text{Coriolis}} = -2m\boldsymbol{\omega}\times\mathbf{v}$$. You will find that it points in the negative radial direction.
• So the Coriolis force provides $2m\omega^2r$ radially inwards and the centrifugal force provides $m\omega^2r$ radially outwards, right? So, effectively we get a centripetal supply of $m\omega^2r$? Commented Jun 11 at 16:21
• Yes in this case, if we use cylindrical coordinates, $\boldsymbol{\omega} = \omega\hat{\mathbf{z}}$ for example and $\mathbf{v}$ is purely azimuthal (in the negative $\phi$-direction). So with the RHR you would find that $\mathbf{F}_{\text{Coriolis}}$ is purely in the negative $r$-direction with a magnitude $2m\omega v$. Commented Jun 11 at 17:14