Understanding Research Papers

Equilibrium curve setup. Treat chain as a bit of a rope, so

• Horizontal change of a "bead": $dx$
• Length of a "bead": $ds$
• Angle by which the bead is above the ground: $\theta$
• Mass density: $\lambda$

And hence, $ds = \frac{dx}{sin(\theta)}$, and mass is $\lambda ds$ as it is indeed the mass density times length.

Apparently the acceleration due to gravity could be ignored… Not sure why/how.

It seems, it stedy-state does not have an tangent acceleration (circular motion, only acceleration is perpendicular), we know?? that the tangent force has to balance gravity. And hence, Biggins claims:

$$\begin{equation} \frac{d}{dx} T(x) = \frac{\lambda g}{sin \theta} cos(\theta) \end{equation}$$