Centripetal acceleration vector

[jhae2.718]

$\renewcommand{\vec}[1]{\boldsymbol #1}$

From this last result, we have the centripetal acceleration given by $-r\dot{\theta}^2\hat{\vec{b}}_1$ (this result may look more familiar if we define $\omega \equiv \dot{\theta}$ and write $-r\dot{\theta}^2\hat{\vec{b}}_1 = - r\omega^2\hat{\vec{b}}_1 = -v^2/r\hat{\vec{b}}_1$). But we know that $\hat{\vec{b}}_1$ points outwards, so the centripetal acceleration must point toward the center.

Upon rereading my earlier post, I feel as if I should clarify that the $v$ in $-v^2/r\hat{\vec{b}}_1$ is not the magnitude of the inertial velocity vector, but the magnitude of the transverse velocity $r\dot{\theta} = r\omega$.