Related: Circular Roller Coaster

This problem is related to modeling a circular roller coaster with friction. Starting with the substitution


we can eventually reduce the EOM to a first-order differential equation of the form $\dot{\theta}=f(\theta)$. See the exact solution here:

Prior, T. and Mele, E. J. (2007). A block slipping on a sphere with friction: Exact and perturbative solutions. American Journal of Physics, 75, 423–426.

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