Most problems in dynamics can be solved by using energy methods. This method is particularly helpful with complex systems. Energy will be used for the problem to demonstrate how it can be used.
First, the total energy, both potential and kinetic, needs to be determined for the system. The gravitational potential energy V of the marble is
V = mgh
The kinetic energy of the marble, including both linear and rotational, is
One problem is that there is three variables, h, v, and dφ/dt that describe the marble motion and position. Energy methods can only have one unknown variable, so the three must be related.
The marble's vertical location h is directly related to angle θ by
h = (R - r) (1 - cosθ)
The marble linear velocity is related to the angular velocity dθ/dt through the arc as
Finally, the angle φ can be expressed in terms of the angle θ through the relationship
Adding potential and kinetic energy (total energy), and using h, v, and dφ/dt relationships give