3.35
Related: Circular Roller Coaster
This problem is related to modeling a circular roller coaster with friction. Starting with the substitution
$$\omega(\theta)=\dot{\theta}$$$$\ddot{\theta}=\frac{d\omega}{dt}=\frac{d\omega}{d\theta}\frac{d\theta}{dt}=\frac{d\omega}{d\theta}\omega$$
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.