# Dimensionality Dilemma: Dimensional Analysis Invalidates my Mathematical Model

I am trying to derive an equation that describes the rotational motion of an "auto-unravelling system": systems comprised of a material (string, chain, cloth etc.) wound around a cylinder and left to unwind under the weight of the hanging segment of material.

Everything up until the red text is correct dimensionally speaking. The issue arises when I attempt to get rid of the $\frac{dl}{dt}$ term: the rate of change in the length of chain left on the roll. I want to put in terms of angular displacement (theta).

Note that my unravelling system is composed of small chained beads wrapped around a toilet roll in such a way that we can assume each full turn to be a loop of beads. Thus the entire system is just a series of concentric loops.

The issue is that the dimensions of $\frac{dl}{dt}$ are $\frac{metres}{second}$ but the dimensions of kwR (w-omega being angular velocity) is $\frac{radian*metres}{second}$ (k is simply an error coefficient to account for error between calculated length under the aforementioned assumption and empirical length)

QUESTION: How do I augment my equation so that the dimensions match up or do I need to find another way to describes $\frac{dl}{dt}$ in terms of omega? If a new method is required could you point me in the right direction?