This paper presents a mathematical model for a mechanical computer called the Turing Tumble. We show that our model called Turing Tumble Model (TTM) is computationally universal under the assumptions that a configuration of TTM is sufficiently large and that local interactions between elements can be transferred without limitations. The Turing Tumble has a strict constraint, based on gravity, since signals can only move from top to bottom. We introduce a uniform scheme that takes into account this restriction in directionality to construct universal machines in the TTM based on directed acyclic graphs. This model may be useful for implementing computers that exploit mechanical interactions in nature, especially those on micrometer-scales.