# Question about acceleration equations that can be used for a single object

Can the centripetal acceleration equation $$a=\frac{v^2}r$$ and the acceleration due to gravity equation $$a=\frac{Gm}{r^2}$$ be used interchangeably to find the acceleration of a single object?

If the object is undergoing uniform circular motion in a gravitational field of $$-Gm/r^2\,\hat r$$, then its acceleration will have a magnitude of $$v^2/r$$ towards the center of the circle.
If the object is not undergoing uniform circular motion, then $$a_c=v^2/r$$ is only one component of the total acceleration. However, if the acceleration is due to gravity, then the magnitude of the acceleration is always going to be $$Gm/r^2$$.
Uniform circular motion has centripetal acceleration $$v^2/r$$ as simple kinematics, regardless of what dynamics are causing that acceleration. For example, it might be due the tension in a string rather than gravity.
$$GM/r^2$$ is dynamics: the acceleration caused by the gravityational force of a mass $$M$$. This acceleration does not necessarily have anything to do with uniform circular motion. For example, during gravitational radial infall the acceleration is not $$v^2/r$$.