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##.