The pendulum-peg problem (part I)

Consider the following experiment

A ball of mass m is attached to a string of length L, and is pulled so that the ball is an initial height D from the lowest point in its swing. A rod is clamped at a height h above the lowest point so that when the pendulum swings down, the string wraps around the rod, and the ball swings around in a circle of smaller radius.

As the ball swings over the top of the peg, it is pulled downwards by both the string and by gravity. This downwards pull provides the force needed to keep the ball moving in a circle of radius h around the peg.

Your challenge:

Determine the smallest height D from which the ball can be released in order that it complete the small circle without the string going slack. That means that as the ball swings over the top of the peg, the string provides almost no tension, so that all the inwards force is due to gravity:

Which means that the minimum possible speed at the top of the loop must be