# 3.35

exercise-3-35

exercise-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.