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.

My derivation so far

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?


I don't quite see any problems with dimensions: radians are dimensionless.

  • $\begingroup$ so the dimensions of w is just 1/s ? $\endgroup$ – Aniekan Umoren Feb 9 '18 at 2:19
  • $\begingroup$ @AniekanUmoren : Yes, the dimension of angular veloсity is 1/s. $\endgroup$ – akhmeteli Feb 9 '18 at 2:23

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