Centripetal force disappearing when switching between inertial frames

Question

Suppose we have a particle moving in a circular motion governed by the equation $\vec{r} = \cos(t) \hat{i} + \sin(t) \hat{j}$, then we notice that velocity at $t=0$ is given as: $\vec{\dot{r}} = \hat{j}$, from this we can write centripetal acceleration as:

$$ \ddot{r_c} = \frac{| \hat{j}|^2}{ |\vec{r}|}$$

by the formula:

$$ \vec{\ddot{r_c} } = \frac{|\vec{\dot{r} }|^2}{|\vec{r}|^2}$$

But, suppose we a particle moving with velocity $\hat{j}$ located somewhere, then in this particles from the velocity of the particle in circular motion would be zero and hence the centripetal acceleration of the particle in circular motion is zero.

So, switches between frames seems to have cancelled out the force, this leads me to the question: In which frame do we put the velocity in for the centripetal acceleration formula?

Answer 1

from this we can write centripetal acceleration as: $$ \ddot{r_c} = \frac{| \hat{j}|^2}{ |\vec{r}|}$$ by the formula: $$ \vec{\ddot{r_c} } = \frac{|\vec{\dot{r} }|^2}{|\vec{r}|^2}$$

These formulas are not general formulas but only apply for the specific case of uniform circular motion. While $\vec{r} = \cos(t) \hat{i} + \sin(t) \hat{j}$ is uniform circular motion $\vec{R} = \cos(t) \hat{i} + (\sin(t)+t) \hat{j}$ is not. Therefore, the equations for uniform circular motion do not apply in the transformed frame since the motion is not uniform circular motion in that frame.

Note, although the motion is not uniform circular motion and therefore the acceleration is not centripetal, both frames do agree on the magnitude and the direction of the acceleration itself. So it is not that the force disappears, just that calling the force "centripetal" no longer makes sense.

Answer 2

Your initial equation assumes a particular coordinate system. If you move to a different system, all of your relations will change.