Is there an equation that relates angular acceleration to centripetal acceleration? Tangential to centripetal?

Question

Is there an equation that relates tangential and centripetal acceleration? I ask this question because it's been on my mind ever since I solved a problem involving the giant swing ride commonly seen at amusement parks. The question involved finding the speed $v$ with given symbols instead of actual values, which can be done using a force diagram. I was just thinking though, it may be useful to find an equation that relates both the radial and tangential acceleration of a particle in circular motion.

Answer 1

In general, the acceleration vector for planar motion in polar coordinates is given by $$\mathbf a=(\ddot r-r\dot\theta^2)\hat r+(r\ddot\theta+2\dot r\dot\theta)\hat\theta$$

For circular motion we have $\dot r=\ddot r=0$, so that the centripetal acceleration becomes $$a_r=r\dot\theta^2$$ and the tangential acceleration becomes$^*$ $$a_\theta=r\ddot\theta$$

Therefore, one could argue that these values are related through the $r$ variable such that $$\frac{a_r}{\dot\theta^2}=\frac{a_\theta}{\ddot\theta}$$

I should say that this relation is more "descriptive" than "constraining" in the sense that it doesn't necessarily mean that the centripetal acceleration determines the tangential acceleration or vice versa. It really only says that these two ratios have to be equal, as $a_c$ depends on $\dot\theta^2$ in the same way $a_\theta$ depends on $\ddot\theta$. I know this probably isn't as insightful as you were hoping for, but it's what we have.

Therefore, in terms of how useful this relation is for doing physics problems, I'm not so sure. Both accelerations are easy to determine if you already know $\dot\theta$ and $\ddot\theta$, but I digress.

Note that the angular acceleration is just $\ddot\theta$, so if you want to bring that in too then that is a pretty simple thing to do.

Also note that if there is no tangential acceleration for our circular motion we get an undefined value mathematically, but this makes sense physically. You can have any centripetal acceleration for uniform circular motion.

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You might be more familiar with notation $a_c=r\omega^2=v^2/r$ and $a_\theta=r\alpha$. I could have started here, but I wanted to give a more general approach starting from "the beginning" with general planar motion in polar coordinates. This is usually my starting point for investigating questions like these.