Question

Derive the analogues effective force relation in rotating frame and explain all the
terms and discuss circumstances where some of these terms vanishes for different
settings.

Answer 1

In a rotating frame of reference Newton's second law looks like

$$m\textbf{a}=\textbf{F}-2m\omega\times \textbf{v}-m\dot{\omega}\times \textbf{r}-m\omega\times(\omega\times \textbf{r}).$$

In this equation $\textbf{a}$ and $\textbf{v}$ are the velocity and the acceleration relative to the rotating frame. The other terms are fictitious forces. For instance, $-2m\omega\times \textbf{v}$ is the famous Coriolis force, and $-m\omega\times(\omega\times \textbf{r})$ as you may guess is the centrifugal force. We can see that if the body does not move, the Coriolis force vanishes. If the system does not rotate, we get Newton's second law in its common everyday form. At the centre of the rotating frame, i.e. at $\textbf{r}=0$ the centrifugal force vanishes.