# 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 $$Gm/r^2$$.

You are confusing kinematics and dynamics.

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 $$v^2/r$$.

The two accelerations are never conceptually interchangeable. They happen to be numerically equal when you have an object in uniform circular motion due to gravity. This is of course no coincidence: the force of gravitational attraction is causing the centripetal acceleration.