Relative angular acceleration

Question

Let us consider a particle which is rotating in a circle of radius $R$ with a uniform angular velocity of $ω$. We are observing this particle from a frame rotating about the same axis with uniform angular speed $ω'$. Then what'll be the acceleration of particle with respect to us.

If if consider individual angular accelerations of particles, that are $ω^2R$ and $ω'^2R$, then relative acceleration comes out to be $ω^2R$ - $ω'^2R$. Which is wrong. But I don't know how?

But, if we consider relative angular velocity, which comes out to be $(ω-ω')$, then relative angular acceleration equals $(ω-ω')^2R$, which is apparently correct. My question is why taking relative angular velocity is correct, but taking relative angular acceleration is wrong?

Answer 1

One can indeed calculate relative angular acceleration, however, the method you have used is not correct.

For example, consider two objects each with its own acceleration along some given coordinate axis. Then we can conclude that the relative acceleration is $$\vec a_\text{rel}=\vec a-\vec a^\prime,$$ where $\vec a$ and $\vec a^\prime$ are the accelerations of the respect objects with respect to a stationary observer. This will work when the non-inertial frames are simply aligned like this, however, for non-inertial frames with more complex configurations, the relationship does not hold true.

Consider the two objects rotating as per your example, we may write their angular velocities such that $\vec \omega=\omega\mathbf{\hat k}$ and $\vec\omega^\prime=\omega^\prime\mathbf{\hat k}.$ The relative angular velocity is: $\vec\omega_{\text{rel}}=(\omega-\omega^\prime)\mathbf{\hat k}$. Similarly one can consider the relative angular acceleration: $\vec\alpha_\text{rel}=(\alpha-\alpha^\prime)\mathbf{\hat k}$, where $\vec\alpha$ is defined as: $$\vec\alpha={d\vec\omega\over dt}.$$ However, the relative acceleration that you sought was not the relative angular acceleration but the relative linear acceleration, which is not so easy to compute, however, this should explain why you calculation of relative velocity worked while the relative acceleration calculation failed.