When is $a_c = v^2/r$ valid? All the derivations I've seen of the formula in the title, namely $a_c = \frac{v^2}{r}$ rely on the tangential speed $v$ being constant. However, I watched this video:
https://www.khanacademy.org/science/physics/centripetal-force-and-gravitation/centripetal-forces/v/yo-yo-in-vertical-circle-example
and in it he happily uses the formula even though the tangential speed is not constant. For example, it will be much lower at the top of the circle than at the bottom. Is it true that the formula holds even for non-constant tangential speeds?
 A: I treat 2D first and then comment on 3D.
The complete formula for acceleration in a plane,
treated in plane polar coordinates, is
$$
{\bf a} = (\ddot{r} - r \dot{\theta}^2) \hat{\bf r}
+ (r \ddot{\theta} + 2 \dot{r} \dot{\theta} ) \hat{\boldsymbol \theta}.
$$
Using now that the velocity is
$$
{\bf v} = v_r \hat{\bf r} + v_\theta \hat{\boldsymbol \theta}
=
v_r \hat{\bf r} + r \dot{\theta}  \hat{\boldsymbol \theta}
$$
we can write the acceleration as
$$
{\bf a} = \left(\ddot{r} - \frac{v_\theta^2}{r} \right)\hat{\bf r}
+ \left(r \ddot{\theta} + 2 \dot{r} \dot{\theta} \right) \hat{\boldsymbol \theta}
$$
The four terms in the expression for acceleration may be understood or named as follows

*

*$\ddot{r} \hat{\bf r}$ radial acceleration owing to motion in the radial direction


*$- \frac{v_\theta^2}{r} \hat{\bf r}$ the centripetal acceleration which causes any transverse velocity to change direction (and thus follow a circle if the other terms were absent)


*$r \ddot{\theta} \hat{\boldsymbol \theta}$ acceleration around a circle associated with second derivative of the angle.


*$2 \dot{r} \dot{\theta} \hat{\boldsymbol \theta}$
the Coriolis acceleration, which is present only when both $r$ and $\theta$ are changing.
Three dimensions
At any point on a trajectory you can define a plane singled out by the velocity and acceleration vectors. In that plane you can set up rectangular coordinates and polar coordinates in two dimensions, and then the above formulae apply.
A: It holds for any infinitesimal piece of the circle. On each position $a_c$ and $v$ will both be different but such that the formula holds everywhere.
