When you Google centripetal acceleration + density, nothing really shows up.
1. Does centripetal acceleration depend on the density of the ball at the end of the string.
2. If so, what density equation then equals v squared / r
Centripetal force involves the mass of the object, but not volume or density. But centripetal acceleration does not involve mass, or volume, or density.
An object moving in a circle requires, at each moment, an acceleration perpendicular to the tangential velocity. That is because of Newton's 1st Law. Paraphrased: "An object in motion tends to continue in motion in a straight line." To bend that straight line, an acceleration toward the center of the circle is needed. Toward the center of the circle means that it is perpendicular to the tangential velocity at any point of the object's path.
An orbiting satellite requires an acceleration perpendicular to the tangential velocity. Acceleration means rate of change of velocity. You're familiar with acceleration that is in the same direction as the velocity. It's harder to see how the phrase "rate of change of velocity" applies when the acceleration is perpendicular to the velocity. But think about satellites. Two satellites of different mass can occupy the same orbit. They have the same velocity and same altitude, so they need the same acceleration. The results of calculating v^2/r for a large one and a small one, as long as they're in the same orbit, are identical. It could be the Space Shuttle and an astronaut, nearby but unattached, wearing a suit and jet pack. The shuttle and the astronaut both need the same centripetal acceleration to follow the circular path.
Let me repeat what I said in the first paragraph. Centripetal force does involve mass. Think of Newton's 2nd: F = m*a. Mass times acceleration is a force. So the shuttle requires more centripetal force than does the nearby astronaut to follow the circular path. And gravity provides the proper force to yield the required acceleration.
I hope this helps,