Consider a light rod of length l with its upper end clamped such...

For a block attached to a light rod, the rod can provide both tension and compression. Thus, the rod will not slacken like a string.

To complete the vertical circle, the minimum velocity at the highest point can be zero, i.e., $v_{top} = 0$.

Using conservation of mechanical energy between the lowest and highest points:

$\dfrac{1}{2}mv^2 = \dfrac{1}{2}mv_{top}^2 + mg(2l)$

Substituting $v_{top} = 0$:

$\dfrac{1}{2}mv^2 = mg(2l)$

$v^2 = 4gl$

$v = 2\sqrt{gl}$